AI Class 12 Questions and Answers, CBSE Class 12 AI Important Questions with Answers, Artificial Intelligence Class 12 Important Questions and Answers, Class 12 AI Viva Questions, CBSE Class 12 Artificial Intelligence Questions and Answers.

Artificial Intelligence Class 12 Questions and Answers

AI Important Questions Class 12 | Artificial Intelligence Class 12 Important Questions and Answers

AI Class 12 Important Questions PART A Employability Skills

1. Communication Skills Class 12 Questions and Answers
2. Self Management Skills Class 12 Questions and Answers
3. Basic ICT Skills Class 12 Questions and Answers (Information and Communication Technology Skills)
4. Entrepreneurial Skills Class 12 Questions and Answers
5. Green Skills Class 12 Questions and Answers

AI Class 12 Questions and Answers PART B Subject Specific Skills

1. Capstone Project Class 12 Questions and Answers
2. Model Lifecycle Class 12 Questions and Answers
3. Storyrelling Through Data Class 12 Questions and Answers

CBSE Class 12 AI Sample Papers

• CBSE Sample Papers for Class 12 AI Set 1
• CBSE Sample Papers for Class 12 AI Set 2
• CBSE Sample Papers for Class 12 AI Set 3
• CBSE Class 12 AI Question Paper 2023
• CBSE Class 12 AI Question Paper 2024

Also Read

• Class 12 AI MCQ

Important Questions for Class 12 AI CBSE, AI Important Questions Class 12 CBSE, Class 12th Artificial Intelligence Important Questions, Class 12 AI Practical Questions.

The post Class 12 AI Important Questions | CBSE Class 12 Artificial Intelligence Questions and Answers appeared first on Learn CBSE.

Students must start practicing the questions from Class 12 AI Important Questions and CBSE Sample Papers for Class 12 AI Set 1 are designed as per the revised syllabus.

CBSE Sample Papers for Class 12 AI Set 1 with Solutions

Max. Time: 2 Hours
Max.Marks:50

General Instructions:

1. Please read the instructions carefully.
2. This Question Paper consists of 21 questions in two sections: Section A & Section B.
3. Section A has Objective type questions whereas Section B contains Subjective type questions.
4. Out of the given (5+16=) 21 questions, a candidate has to answer (5+10=) 15 questions in the allotted (maximum) time of 2 hours.
5. All questions of a particular section must be attempted in the correct order.
6. SECTIONA-OBJECTIVE TYPE QUESTIONS (24MARKS):
i. This section has o5 questions.
ii. Mavks allotted are mentioned against each question/part.
iii. There is no negative marking.
iv. Do as per the instructions given.
7. SECTIONB -SUBJECTIVE TYPE QUESTIONS (26 MARKS):
i. This section has 16 questions.
ii. A candidate has to do 10 questions.
iii. Do as per the instructions given.
iv. Marks allotted are mentioned against each question/part.

SECTION-A
Objective type Questions:

Question 1.
Answer any 4 out of the given 6 questions on Employability Skills (1 × 4 = 4 marks)

i. An indirect object answers the questions, such as
a. What?
b. Why?
c. To whom?
d. Which?
Answer:
a. What?

ii. “I enjoy singing and practice it as it gives me a lot of pleasure. It also works as a stress buster.” The sentence given above is an example of
a. Intrinsic motivation
b. Extrinsic motivation
c. External motivation
d. Lack of motivation
Answer:
a. Intrinsic motivation

iii. Statement 1: Personality traits are defined as relatively lasting patterns of thoughts, feelings and behaviors. Statement 2: They distinguish individuals from one another.
a. Both Statement 1 and Statement 2 are correct
b. Both Statement 1 and Statement 2 are incorrect
c. Statement 1 is correct but Statement 2 is incorrect
d. Statement 2 is correct but Statement 1 is incorrect
Answer:
a. Both Statement 1 and Statement 2 are correct

iv. Match the following:

a. A-2 ; B-3 ; C-1
b. A-3 ; B-1 ; C-2
c. A-1 ; B-3 ; C-2
d. A-1 ; B-2 ; C-3
Answer:
b.

v. After 20 years of being a successful entrepreneur in Lucknow, Ravi decided to move back to his village. He opened a clothing store but people in his village said that the style did not match what they wanted. They wanted more variety and brighter colours. So, he sold off the old clothes to his friend in a city and bought good quality clothes from a local seller.
Which attitude can you witness in Ravi?
a. Interpersonal skills
b. Perseverancequad
c. Competitiveness
Barrier
Answer:
b. Perseverancequad

vi. Electronic waste, as known as e-waste, is generated when any electronic or electrical equipment becomes unfit for the intended use or if it has crossed its expiry date. Due to rapid technological advancements and the production of newer electronic equipment, the old ones get easily replaced with new models. It has particularly led to an exponential increase in e-waste in India.
Which of the following is the correct way to handle e-waste?
a. Sell the e-waste to a local scrap dealer
b. Throw the e-waste in the dustbin
c. Dispose-off the e-waste with the help of a certified partner
d. Dump the e-waste in the local landfill
Answer:
c. Dispose-off the e-waste with the help of a certified partner

Question 2.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. In Design Thinking, _______ phase involves gathering user feedback on the prototypes you’ve created as well as obtaining a better understanding of your users.
a) Prototype
b) Test
c) Ideate
d) Empathize
Answer:
b) Test

ii. _______ is the first step involved in telling an effective data story.
(a) Creating visuals
(b) Adding narrative
(c) Understanding the Audience
(d) Gathering data
Answer:
(c) Understanding the Audience

iii. Match the following

a) 1= D, 2=B, 3=E, 4=A, 5=C
b) 1=C, 2=D, 3=B, 4=E, 5=A
c) 1=D, 2=B, 3=C, 4=E, 5=A
d) 1=E, 2=A, 3=D, 4=C, 5=B
Answer:

iv. Identify two AI development tools from the following:
1) DataRobot
2) Python
3) Scikit Learn
4) Watson Studio
(a) 1 & 2
(b) 2 & 3
(c) 1 & 3
(d) 1 & 4
Answer:
(d) 1 & 4

v. You want to predict future house prices. The price is a continuous value, and therefore we want to do regression. Which loss function should be used here?
(a) RMSE
(b) MSE
(c) Exponential error
(d) MAE
Answer:
(b) MSE

vi. The design phase of the AI Model Life Cycle is an _______ process.
(a) compact
(b) permanent
(c) periodic
(d) iterative
Answer:
(d) iterative

Question 3.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. Techniques like descriptive statistics and visualisations can be applied to datasets after the original data gathering to analyse the content. To close the gap, additional data collecting may be required. Identify the stage of this analytic approach.
(a) Data Requirements
(b) Data Gathering
(c) Data Understanding
(d) Data Preparation
Answer:
(c) Data Understanding

ii. Stories create _____ experiences that transport the audience to another space and time.
(a) unpleasant
(b) tedious
(c) repetitive
(d) engaging
Answer:
(d) engaging

iii. In this phase, we define the project’s strategic business objectives and desired outcomes, align all stakeholders’ expectations as well as establish success metrics. Identify this phase of the AI Model Life Cycle.
(a) Design
(b) Scoping
(c) Evaluation
(d) Data
Collection
Answer:
(b) Scoping

iv. Assertion (A): Stories that combine statistics and analytics are more persuasive.
Reason (R): When we talk about data storytelling, we’re talking about stories in which data plays a central role.
Select the appropriate option for the statements given above:
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not the correct explanation of A
(c) A is true but R is false
(d) A is False but R is true
Answer:
(b) Both A and R are true and R is not the correct explanation of A

v. Which of the following is not a feature of RMSE?
(a) It tells about the accuracy of the model.
(b) Higher value means hyper parameters need to be tweaked
(c) Lower RMSE values are not good for the AI model.
(d) RMSE is a measure of how evenly distributed residual errors are.
Answer:
(c) Lower RMSE values are not good for the AI model.

vi. Once you have got an AI model that’s ready for production, AI engineers then _____ a trained model, making it available for external inference requests.
(a) Evaluate
(b) Test
(c) Deploy
(d) Redesign
Answer:
(c) Deploy

Question 4.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. Data Validation for human biases is conducted in _________ phase of AI Model Life Cycle.
(a) Scoping
(b) Data Collection
(c) Design
(d) Testing
Answer:
(d) Testing

ii. Identify the following icons:
(a)

(b)

Answer:
(a) Narrative
(b) Data

iii. Which of the following is a disadvantage of Cross Validation Technique?
(a) Cross-validation provides insight into how the model will generalize to a new dataset.
(b) Cross-validation aids in determining a more accurate model prediction performance estimate.
(c) As twe need to train on many training sets, cross-validation is computationally expensive.
(d) Cross-validation could result in more precise models.
Answer:
(c) As twe need to train on many training sets, cross-validation is computationally expensive.

iv. Hyper parameters are parameters whose values govern the learning process.
(a) True
(b) False
Answer:
(a) True

v. The steps that assist in finding compelling stories in the data sets are as follows. Arrange them in proper order:
1) Visualize the data.
2) Examine data relationships.
3) Get the data and organise it.
4) Create a simple narrative embedded with conflict.
(a) 1-2-3-4
(b) 2-3-1-4
(c) 4-1-3-2
(d) 3-1-2-4
Answer:
(d) 3-1-2-4

vi. Choose the difference between Regression and Classification Loss functions from the following:
(a) Regression functions predict a quantity, and classification functions predict a label.
(b) Regression functions predict a label, and classification functions predict a quantity.
(c) Regression functions predict a qualitative value, and classification functions predict a label.
(d) Regression functions predict a label, and classification functions predict a qualitative value.
Answer:
(a) Regression functions predict a quantity, and classification functions predict a label.

Question 5.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. Stories change the way that we interact with data, transforming it from a dry collection of _____ to something that can be entertaining, thought provoking, and inspiring change.
(a) visuals
(b) points
(c) images
(d) facts
Answer:
(d) facts

ii. A good model should have an _____ value less than 180.
(a) RMSE
(b) MSE
(c) Focal Loss
(d) MAE
Answer:
(a) RMSE

iii. Which of the following is incorrect?
1) Testing data is the one on which we train and fit our model basically to fit the parameters
2) Training data is used only to assess performance of model
3) Testing data is the unseen data for which predictions have to be made
a) 1) and 3) only
b) 1) and 2) only
c) 2) and 3) only
d) 1), 2) and 3)
Answer:
b) 1) and 2) only

iv. Choose an example of an AI predictive model.
(a) YouTube
(b) Spam detection
(c) weather forecast
(d) Sentiment Analysis
Answer:
(c) weather forecast

v. Which of the following are the objectives of the testing team in AI modelling?
1) Model Validation
2) Security compliance
3) Understanding data
4) Minimizing bias
a. (1), (2) and (3)
b. (2), (3) and (4)
c. (1), (3) and (4)
d. (1), (2) and (4)
Answer:
d. (1), (2) and (4)

vi. If AI techniques are to be applied to a dataset, the data must have a
a. association
b. relationship
c. pattern
d. Either a and b
Answer:
c. pattern

SECTION B :
Subjective type Questions

Answer any 3 out of the given 5 questions on Employability Skills ( 2× 3=6 marks) Answer each question in 20-30 words.

Question 6.
What does ‘P’ stand for in the acronym ‘RESPECT’ used to ensure active
listening? Explain.
Answer:
P in Respect stands for – Pay attention and focus on what the speaker is saying.
(1 mark for the word ‘Pay’; and 1 mark for explanation)

Question 7.
Shikha is an elderly woman. She stays with her family. She has a habit of washing her hands at least 20 × a day. Even after washing her hands, she feels they are not clean, and continues rubbing or washing them. She neither talks to her grandchildren, nor does she participate in any family activity. Suggest her one way in which she can overcome her personality disorder.
Answer:
Steps to overcome personality disorders

• Talk to someone. Most often, it helps to share your feelings.
• Look after your physical health. A healthy body can help you maintain a healthy mind.
• Build confidence in your ability to handle difficult situations.
• Engage in hobbies, such as music, dance and painting. These have a therapeutic effect.
• Stay positive by choosing words like ‘challenges’ instead of ‘problems’.
  (Any 2 of the above -1 mark each)

Question 8.
Differentiate between the terms worksheet and workbook.
Answer:
A worksheet is a collection of cells in the form of a grid (a network of lines that intersect each other, making rectangles). When you open a spreadsheet for the first time, you see a blank worksheet with the name ‘Sheet1’. A workbook is a spreadsheet that has one or more worksheets. (1 mark each)

Question 9.
Name two government schemes that help small-scale entrepreneurs gain capital.
Answer:

• Government Mudra Yojana
• Credit Guarantee Scheme
• Stand-Up India Scheme (Any 2;1 mark each)

Question 10.
The greening of economy presents a major opportunity to start new businesses, develop new markets and lower energy costs. Write any two benefits of green jobs.
Answer:

increase the efficiency of energy and raw material.
reduce greenhouse gas emissions.
control waste and pollution.
protect and restore ecosystems.
support adaptation to the effects of climate change.
(Any 2;1 mark each)

Answer any 4 out of the given 6 questions in 20-30 words each ( 2× 4=8 marks)

Question 11.
Define capstone project.
Answer:
A capstone project is a project where students must research a topic independently to find a deep understanding of the subject matter. It gives an opportunity for the student to integrate all their knowledge and demonstrate it through a comprehensive project.

Question 12.
Can MSE be a negative value? Give reasons.
Answer:
The MSE value cannot be negative. The difference between projected and actual values are always squared. As a result, all outcomes are either positive or negative.

Question 13.
What is the importance of a narrative in a story?
Answer:
Stories are more likely to drive action than are statistics and numbers. Therefore, when told in the form of a narrative, it reduces ambiguity, connects data with context, and describes a specific interpretation communicating the important messages in most effective ways.

Question 14.
Imagine that you want to create your first app. Create a list of questions you would develop to decompose this task.
Answer:
To decompose this task, you would need to know the answer to a series of smaller problems:
what kind of app you want to create?
what will your app will look like?
who is the target audience for your app?
what will the graphics will look like?
what audio will you include?

Question 15.
Differentiate between training set and test set.
Answer:
A training set is a set of historical data in which the outcomes are already known. Train Dataset: Used to fit the machine learning model. Test Dataset: Used to evaluate the fit machine learning model.

Question 16.
“Once the relevant projects have been selected and properly scoped, the next step of the machine learning lifecycle is the Design or Build phase.” Briefly explain this phase
Answer:
The Design phase is essentially an iterative process comprising all the steps relevant to building the AI or machine learning model: data acquisition, exploration, preparation, cleaning, feature engineering, testing and running a set of models to try to predict behaviors or discover insights in the data.

Answer any 3 out of the given 5 questions in 50-80 words each (4 × 3 = 12 marks)

Question 17.
Explain how data storytelling can bring about change using a diagram.
Answer:
Data storytelling is a structured approach for communicating insights drawn from data, and invariably involves a combination of three key elements: data, visuals, and narrative. When the narrative is accompanied with data, it helps to explain the audience what’s happening in the data and why a particular insight has been generated.

When visuals are applied to data, they can enlighten the audience to the insights that they wouldn’t perceive without the charts or graphs. Finally, when narrative and visuals are merged together, they can engage or even entertain an audience. When you combine the right visuals and narrative with the right data, you have a data story that can influence and drive change.

Question 18.
Consider the following data

Regression Equation: 0.7x + 17.8 Calculate the RMSE (Root means Square Error) for the above data.
Answer:

Question 19.
List the considerations which data scientists have to keep in mind during the testing
stage.
Answer:
Considerations:
The volume of test data can be large, which presents complexities.
Human biases in selecting test data can adversely impact the testing phase, therefore, data validation is important.
Your testing team should test the AI and ML algorithms keeping model validation, successful learnability, and algorithm effectiveness in mind. Regulatory compliance testing and security testing are important since the system might dealwith sensitive data, moreover, the large volume of data makes performance testing crucial.

Question 20.
Explain the Cross Validation Procedure? In which situation is it better than a Train
Test Split?
Answer:
In cross-validation, we run our modeling process on different subsets of the data to get multiple measures of model quality. In k-fold cross-validation, the original dataset is equally divided into k subparts or folds. Out of the k-folds, for each iteration, one group is selected as test data, and the remaining (k-1) groups are selected as training data. This process is repeated for k times. The final accuracy of the model is calculated by taking the mean accuracy. When the dataset is smaller, cross-validation procedure should be selected for higher accuracy.

Question 21.
(a) List the steps of creating an effective data story.
(b) Which of the following is a better data story? Give reasons.
Option A:

(Source: https://www.crazvegg.com/blog/data-storytelling-5-steps-charts/)
Option B:

Answer:
(a) The steps involved in telling an effective datastory are given below:

• Understanding the audience
• Choosing the right data and visualizations
• Drawing attention to key information
• Developing a narrative
• Engaging your audience

(b) Option A is a better data story. Reasons
1. The insight given at the bottom of the visual is giving a clear idea about context which is not therein

Option B.
Option A visual provides a compelling narrative.

The post CBSE Sample Papers for Class 12 AI Set 1 with Solutions appeared first on Learn CBSE.

Students must start practicing the questions from Class 12 AI Important Questions and CBSE Sample Papers for Class 12 AI Set 2 are designed as per the revised syllabus.

CBSE Sample Papers for Class 12 AI Set 2 with Solutions

Max. Time: 2 Hours
Max.Marks:50

General Instructions:

1. Please read the instructions carefully.
2. This Question Paper consists of 21 questions in two sections: Section A & Section B.
3. Section A has Objective type questions whereas Section B contains Subjective type questions.
4. Out of the given (5+16=) 21 questions, a candidate has to answer (5+10=) 15 questions in the allotted (maximum) time of 2 hours.
5. All questions of a particular section must be attempted in the correct order.
6. SECTIONA-OBJECTIVE TYPE QUESTIONS (24MARKS):
i. This section has o5 questions.
ii. Mavks allotted are mentioned against each question/part.
iii. There is no negative marking.
iv. Do as per the instructions given.
7. SECTIONB -SUBJECTIVE TYPE QUESTIONS (26 MARKS):
i. This section has 16 questions.
ii. A candidate has to do 10 questions.
iii. Do as per the instructions given.
iv. Marks allotted are mentioned against each question/part.

SECTION-A

Objective type Questions:

Question 1.
Answer any 4 out of the given 6 questions on Employability Skills (1 x 4 = 4 marks)

i. Self-motivation is important because
1. It increases individual’s energy and activity.
2. It directs an individual towards specific goals.
3. It results in initiation and persistence of specific activities
4. It affects cognitive processes and learning strategics used for completing similar tasks.
a). Only 1
b). Both 1 and 2
c). 1, 3 and iv
d). All of the above
Answer:
d). All of the above

ii. For an entrepreneur, aversion to risk is:
a) A failure
b) An opportunity
c) A psychological barrier
d) An initiative
Answer:
b) An opportunity

iii. ________ Is the command to select the entire worksheet:
a) CTRL + O
b) CTRL + A
c) CTRL + C
d) CTRL + Z
Answer:
b) CTRL + A

iv. Which of the following factors influence personality?
e) Genes
f) Environmental setting
g) Culture
h) All of the above
Answer:
h) All of the above

v. A competent person is
i) Self-confident
j) Emotional
k) Dependent
I) Lazy
Answer:
i) Self-confident

vi. Which software helps in performing calculations using formulae and in analyzing the data?
a) Microsoft Outlook
b) CommCentral
c) Range
d) Open office
Answer:
All of the above

Question 2.
Answer any 5 0ut of the given 6 questions (1 × 5 = 5 marks)

i. In Design Thinking, _______ phaseinvolvesgatheringuserfeedbackontheprototypesyou’v ecreatedaswell as obtaining a better understanding of you r users.
a) Prototype
b) Test
c) Ideate
d) Empathize
Answer:
b) Test

ii. ________ Is the first step involved in telling an effective data story.
(a) Creating visuals
(b) Adding narrative
(c) Understanding the Audience
(d) Gathering data
Answer:
(c) Understanding the Audience

iii. Match the following

a) 1=D, 2=B, 3=E, 4=A, 5=C
b) 1=C, 2=D, 3=B, 4=E, 5=A
c) 1=D, 2=B, 3=C, 4=E, 5=A
d) 1=E, 2=A, 3=D, 4=C, 5=B
Answer:
a)

iv. Identify two AI development tools from the following:
1) Data Robot
2) Python
3) Scikit Learn
4) Wats on Studio
(a) 1 \ 2
(b) 2 \ 3
(c) 1 \ 3
(d) 1 \ 4
Answer:
(d) 1 \ 4

v. You want to predict future house prices. The price is a continuous value, and therefore we want to do regression. Which loss function should be used here?
(a) RMSE
(b) MSE
(c) Exponential error
(d) MAE
Answer:
(b) MSE

vi. The design phase of the AI Model Life Cycle is an ________
(a) compact
(b) permanent
(c) periodic
(d) iterative
Answer:
(d) iterative

Question 3.
Answer any 5outofthegiven6 questions (1 × 5 = 5 marks)

i. Techniqueslikedescriptivestatisticsandvisualisationscanbeappliedtodatasetsafter the original data gathering to analyse the content. To close the gap, additional data collecting may be required. Identify the stage of this analytic approach.
(a) Data Requirements
(b) Data Gathering
(c) Data Understanding
(d) Data Preparation
Answer:
(c) Data Understanding

ii. Stories create ________ experiences that transport the audience to another space and time.
(a) unpleasant
(b) tedious
(c) repetitive
(d) engaging
ins.
(d) engaging

iii. In this phase, we define the project’s strategic business objectives and desired out comes, align all stake holders’expectations aswell as establish success metrics. Identify this phase of the AI Model Life Cycle.
(a) Design
(b) Scoping
(c) Evaluation
(d) Data Collection
Answer:
(b) Scoping

iv. Assertion (A): Stories that combine statistics and analytics are more persuasive.
Reason (R): When we talk about data story telling, we’re talking about stories in which data plays a central role.
Select the appropriate option forth e statements given above:
a. Both A and R are true and R is the correct explanation of A
b. Both A and Rare true and R is not the correct explanation of A
c. A is true but R is false
d. A is False but R is true
Answer:
b. Both A and Rare true and R is not the correct explanation of A

v. Which of the following is not a feature of RMSE?
(a) It tells about the accuracy of the model.
(b) Higher value means hyper parameters need to be tweaked
(c) Lower RMSE values are not good for the AI model.
(d) RMSE is a measure of how evenly distributed residual errors are.
Answer:
-V

vi. Once you have got an AI model that’s ready for production, AI engineers then a trained model, making it available for external inference requests.
(a) Evaluate
(b) Test
(c) Deploy
(d) Redesign
Answer:
(c) Deploy

Question 4.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. Data Validation for human biases is conducted in ________ phase of AI Model Life Cycle.
(a) Scoping
(b) Data Collection
(c) Design
(d) Testing
Answer :
(d) Testing

ii. Identify the following icons:
(a)

(b)

Answer:
(a) Narrative
(b) Data

iii. Which of the following is a disadvantage of Cross Validation Technique?
(a) Cross-validation provides insight into how the model will generalize to anew dataset.
(b) Cross-validation aids in determining a more accurate model prediction performance estimate.
(c) As we need to train on many training sets, cross-validation is computationally expensive.
(d) Cross-validation could result in more precise models.
Answer:
(c) As we need to train on many training sets, cross-validation is computationally expensive.

iv. Hyper parameters are parameters whose values govern the learning process.
(a) True
(b) False
Answer:
(a) True

v. The steps that assist in finding compelling stories in the data sets are as follows. Arrange the min proper order:
1) Visualize the data.
2) Examined at a relationships.
3) Get the data and organize it.
4) Create a simple narrative embedded with conflict.
(a) 1-2-3-4
(b) 2-3-1-4
(c) 4-1-3-2
(d) 3-1-2-4
Answer:
(d) 3-1-2-4

vi. Choose the difference between Regression and Classification Loss functions from the following:
(a) Regression functions predict a quantity, and classification functions predict a label.
(b) Regression functions predict a label, and classification functions predict a quantity.
(c) Regression functions predict a qualitative value, and classification functions predict a label.
(d) Regression functions predict a label, and classification functions predict a Qualitative value.
Answer:
(a) Regression functions predict a quantity, and classification functions predict a label.

Question 5
Answer any 5 out of the given 6 questions (1 × 5 = 5marks)

i. Stories change the way that we interact with data, transforming it from a dry collection of ________ to something that can be entertaining, thought provoking, and inspiring change.
(a) visuals
(b) points
(c) images
(d) facts
Answer:
(d) facts

ii. A good model should have an ________ value less than 180 .
(a) RMSE
(b) MSE
(c) Focal Loss
(d) MAE
Answer:
(a) RMSE

iii. Which of the following is incorrect?
1) Testing data is the one on which wet rain and fit our model basically to fit the parameters
2) Training data is used only to assess performance of model
3) Testing data is the unseen data for which prediction shave to be made
a) 1) and 3) only
b) 1) and 2) only
c) 2) and 3) only
d) 1), 2) and 3)
Answer:
b) 1) and 2) only

iv. Choose an example of an AI predictive model.
(a) YouTube
(b) Spam detection
(c) weather forecast
(d) Sentiment Analysis
Answer:
(c) weather forecast

v. Which of the following are the objectives of the testing team in AI modelling?
1) Model Validation
2) Security compliance
3) Under standing data
4) Minimizing bias
a. (1),(2) and (3)
b. (2),(3) and (4)
c. (1),(3) and(4)
d. (1),(2) and (4)
Answer:
d. (1),(2) and (4)

vi.
If AI techniques are to be applied to a data set, the data must have a
a. association
b. relationship
c. pattern
d. Either a and b
Answer:
c. pattern

SECTION – B

Subjective type Questions:

Answer any 3 out of the given 5 questions on Employability Skills (2 × 3 = 6 marks)
Answer cach question in 20-30 words.

Question 6.
Why is self-management important? Give one example.
Answer:
Self-management abilities are crucial for demonstrating your dependability as an employee. Those without these abilities could be unpredictable, which can make an employer uneasy. For instance, when someone struggles to regulate their emotions, they may lash out at a customer or say nasty things to a coworker.

Question 7.
Define any two of the following personality disorders
a. Paranoid
Answer:
Paranoid personality disorder (PPD) is a mental health condition marked by a pattern of distrust and suspicion of others without adequate reason to be suspicious. People with PPD are always on guard, believing that others are constantly trying to demean, harm or threaten them.

b.Anti-social
Answer:
Antisocial personality disorder is a particularly challenging type of personality disorder characterized by impulsive, irresponsible and often criminal behavior.
Someone with antisocial personality disorder will typically be manipulative, deceitful and reckless, and will not care for other people’s feelings.

c. Obsessive Compulsive
Answer:
Obsessive-Compulsive Disorder (OCD) is a common, chronic, and long-lasting disorder in which a person has uncontrollable, reoccurring thoughts cobsessions) and/or behaviors (compulsions) that he or she feels the urge to repeat over and over.

Question 8.
Mention two steps in which we can save a workbook.
Answer:
Ctrl+s
File→Save

Question 9.
What are entrepreneurial skills?
Answer:
Entrepreneurial skills are those that are typically related to being an entrepreneur, though anyone can acquire them. Though starting and growing a successful firm on your own is typically the definition of being an entrepreneur, those with entrepreneurial qualities can also succeed in more established businesses.

Question 10.
Name two qualities of a person with entrepreneurial skills. Explain in a sentence
each.
Answer:
1. Recognition of an opportunity
Do you know when to seize an opportunity? New possibilities are continuously presented to us in our life, but the majority of us are typically too fearful or preoccupied to see them. We see them as issues instead. An effective entrepreneur can analyze a situation and see how a viable solution might be developed.

2. Take action
But merely recognizing opportunities is insufficient. A true businessperson has the courage to act, frequently without any prior experience or frame of reference.
Instead than waiting for someone else to fix a problem they care about, they can independently and pro-actively address it.

Answer any 4 out of the given 6 questions in 20-30 words each (2 × 4 = 8 marks)

Question 11.
Define capstone project.
Answer:
A capstone project is a task that requires students to conduct independent research on a subject in order to gain a thorough understanding of it. It offers the student the chance to put all of their knowledge together and show it through an extensive project.

Question 12.
Can MisE be a negative value? Give reasons.
Answer:
The MSE value cannot be negative. The difference between projected and actual values are always squared. As a result, all outcomes are either positive or negative.

Question 13.
What is the importance of a narrative in a story?
Answer:
Stories are more likely to drive action than are statistics and numbers. Therefore, when told in the form of a narrative, it reduces ambiguity, connects data with context, and describes a specific interpretation – communicating the important messages in most effective ways.

Question 14.
Imagine that you want to create your first app. Create a list of questions you would Develop to decompose this task.
Answer:
To decompose this task, you would need to know the answer to a series of smaller problems: what kind of app you want to create?

• what will your app will look like?
• who is the target audience for your app?
• what will the graphics will look like?
• what audio will you include?

Question 15.
Differentiate between training set and test set.
Answer:
A training set is a set of historical data in which the outcomes are already known. Train Dataset: Used to fit the machine learning model. Test Dataset: Used to evaluate the fit machine learning model.

Question 16.
“Oncetherelevantprojectshavebeenselectedandproperlyscoped,thenextstep Of the machine learning lifecycle is the Design or Build phase.”Briefly explain this phase.
Answer:
The Design phase is essentially an iterative process comprising all the steps relevant to building the AI or machine learning model: data acquisition, exploration, preparation, cleaning, feature engineering, testing and running a set of models to try to predict behaviors or discover insights in the data.

Answer any 3 out of the given 5 questions in 50-80 words each (4 × 3 = 12 marks)

Question 17.
Explain how data storytelling can bring about change using a diagram.
Answer:
Data storytelling is a structured approach for communicating insights drawn from data, and invariably involves a combination of three key elements: data, visuals, and narrative. When the narrative is accompanied with data, it helps to explain the audience what’s happening in the data and why a particular insight has been generated.

When visuals are applied to data, they can enlighten the audience to the insights that they wouldn’t perceive without the charts or graphs. Finally, when narrative and visuals are merged together, they can engage or even entertain an audience. When you combine the right visuals and narrative with the right data, you have a data story that can influence and drive change.

Consider the following data

Question 18.

Regression Equation: 0.7x+17.8
Calculate the RMSE (Root means Square Error)for the above data.
Answer:

Question 19.
Listtheconsiderationswhichdatascientistshavetokeepinmindduringthetesting stage.
Answer:
Considerations:

The volume of test data can be large, which presents complexities.
Human biases in selecting test data can adversely impact the testing phase, therefore, data validation is important.
Your testing team should test the AI and ML algorithms keeping model validation, successful learn ability, and algorithm effectiveness in mind
Regulatory compliance testing and security testing are important since the system might deal with sensitive data, moreover, the large volume of data makes performance testing crucial.

Question 20.
Explain the Cross Validation Procedure? In which situation is it better than a Train Test Split?
Answer:
In cross-validation, we run our modeling process on different subsets of the data to get multiple measures of model quality. In k-fold cross-validation, the original dataset is equally divided into k subparts or folds. Out of the k -folds, for each iteration, one group is selected as test data, and the remaining (k-1) groups are selected as training data. This process is repeated for k times. The final accuracy of the model is calculated by taking the mean accuracy. When the dataset is smaller, cross-validation procedure should be selected for higher accuracy.

Question 21.
(a) List the steps of creating an effective data story.
(b) Which of the following is a better data story? Give reasons.
Option A:

Option B:

Answer:
(a) The steps involved in telling an effective data story are given below:

• Understanding the audience
• Choosing the right data and visualizations
• Drawing attention to key information
• Developing a narrative
• Engaging your audience

(b) Option A is a better data story.
Reasons:

1. The insight given at the bottom of the visual is giving a clear idea about context which is not there in Option B.
2. Option A visual provides a compelling narrative.

The post CBSE Sample Papers for Class 12 AI Set 2 with Solutions appeared first on Learn CBSE.

Students must start practicing the questions from Class 12 AI Important Questions and CBSE Sample Papers for Class 12 AI Set 3 are designed as per the revised syllabus.

CBSE Sample Papers for Class 12 AI Set 3 with Solutions

Max. Time: 2 Hours
Max.Marks:50

General Instructions:

1. Please read the instructions carefully.
2. This Question Paper consists of 21 questions in two sections: Section A & Section B.
3. Section A has Objective type questions whereas Section B contains Subjective type questions.
4. Out of the given (5+16=) 21 questions, a candidate has to answer (5+10=) 15 questions in the allotted (maximum) time of 2 hours.
5. All questions of a particular section must be attempted in the correct order.
6. SECTIONA-OBJECTIVE TYPE QUESTIONS (24MARKS):
i. This section has o5 questions.
ii. Mavks allotted are mentioned against each question/part.
iii. There is no negative marking.
iv. Do as per the instructions given.
7. SECTIONB -SUBJECTIVE TYPE QUESTIONS (26 MARKS):
i. This section has 16 questions.
ii. A candidate has to do 10 questions.
iii. Do as per the instructions given.
iv. Marks allotted are mentioned against each question/part.

SECTION-A

Objective type Questions:

Question 1.
Answer any 4 out of the given 6 questions on Employability Skills (1 × 4=4 marks)

i. Self – motivation boosts individuals’ _______ and _______. [1]
a) efficiency and productivity
b) hard work and team management
c) money and workload
d)mental pressure and efficiency
Answer:
a) efficiency and productivity

ii. Entrepreneurs who have no prior experience of business is known as _______. [1]
a) first generation entrepreneurs
b) entrepreneurs in business
c) innovators
d) startup entrepreneurs
Answer:
a) first generation entrepreneurs

iii. Which software helps in performing calculations using formula and analysing the data? [1]
a) Microsoft outlook
b) Microsoft power point
c) Open office calc
d) Libre office
Answer:
c) Open office calc

iv. Which of the following is not a quality of an entrepreneur? [1]
a) Willingness to take risks
b) Self-confidence
c) Ability to make decisions
d) Working for others
Answer:
d) Working for others

v. What is the value of the following formula: [1]
= SUM (12,20,33,48)
A -100
B-114
C-112
D-113
Answer:
D-113

vi. An entrepreneur’s primary responsibility is to secure the timely availability of all the______. [1]
A-Supplies
B-Resources
C-Money
D-Infrastructure
Answer:
B-Resources

Question 2.
Answer any 5 out of the given6 questions (1 × 5 = 5 marks)

i. Every AI project must undergo ______ steps. [1]
a. 3
b. 4
c. 5
d. 6
Answer:
d. 6

ii. Stories change the way that we interact with data, transforming it from a dry collection of _______ to something that can be entertaining, thought provoking, and inspiring change. [1]
(a) visuals
(b) points
(c) images
(d) facts
Answer:
(d) facts

iii. Match the following [1]

a) 1=D, 2=B, 3=E, 4=A, 5=C
b) 1=C, 2=E, 3=D, 4=A, 5=B
c) 1=D, 2=B, 3=C, 4=E, 5=A
d) 1=E, 2=A, 3=D, 4=C, 5=B
Answer:
b)

iv. Which of the following is not a regression loss function [1]
a. Exponential loss
b. Mean absolute error
c. Mean square error
d. Quantile loss
Answer:
a. Exponential loss

v. In AI development which framework is used? [1]
a. Scikit-learn
b. TensorFlow
c. PyCharm
d. Matplotlib
Answer:
b. TensorFlow

vi. Which process does NOT come under Capstone Project? [1]
a. AI Model
b. AI Project Cycle
c. Deployment
d. Data Gathering
Answer:
b. AI Project Cycle

Question 3.
Answer any 5outofthegiven6 questions (1 × 5 = 5 marks)

i. The robotic arm will be able to paint every corner in the automotive parts while minimizing the quantity of paint wasted in the process. Which learning technique is used in this problem? [1]
a. Supervised Learning.
b. Unsupervised Learning.
c. Reinforcement Learning.
d. Both (a) and (b)
Answer:
c. Reinforcement Learning.

ii. It can be said as a means of delivering a narrative or the art of weaving a coherent pleasant scenario around the numbers that conveys the logic. It is the final part of _______. [1]
a) data analysis.
b) data collection
c) data understanding
d) data modelling
Answer:
a) data analysis.

iii. In this phase, we define the project’s strategic business objectives and desired out comes, align all stake holders’ expectations as well as establish success metrics. Identify this phase of the AI Model LifeCycle. [1]
(a) Design
(b) Scoping
(c) Evaluation
(d) Data Collection
Answer:
(b) Scoping

iv. Assertion (A): If the collected data is not relevant, the effective AI algorithm built will collapse. (R): The whole AI model is based on data. Select the appropriate option for the statements given above: [1]
a. Both A and R are true and R is the correct explanation of A
b. Both A and R are true and R is not the correct explanation of A
c. A is true but R is false
d. A is False but R is true
Answer:
a. Both A and R are true and R is the correct explanation of A

v. Which of the following statement is not correct? [1]
(a) Python and R languages are open- source language.
(b) Azure ML Studio help in the development process.
(c) Scikit learn is the most popular framework.
(d) Reinforcement learning helps in increasing the productivity
Answer:
(d) Reinforcement learning helps in increasing the productivity

vi. Primary key to collect DATA (Data gathering process). [1]
(a) Experiment
(b) Survey
(c) Interview
(d) Observation
Answer:
(b) Survey

Question 4.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. Which of the following is FALSE about Correlation and Covariance?
a. A zero correlation does not necessarily imply independence between variables.
b. Correlation and covariance values are the same.
c. The covariance and correlation are always the same sign.
d. Correlation is the standardized version of Covariance.
Answer:
b. Correlation and covariance values are the same.

ii. Identify the following the following figures: [1]
(a)

(b)

Answer:

iii. Which of the following is not a AI development platform? [1]
(a) Microsoft Azure
(b) Google Cloud
(c) BigML
(d) R
Answer:
(d) R

iv. Deployment is the final stage of AI model life cycle. [1]
(a) True
(b) False
Answer:
(a) True

v. The steps that are involved in telling an effective story. Arrange the min proper order: 1) Highlight the important data and organize it.
2) Know about the audience.
3) Find out the appropriate data and visualization.
4) Create a simple narrative keeping your audience in mind
(a) 1-2-3-4
(b) 2-3-1-4
(c) 4-1-3-2
(d) 3-1-2-4
Answer:
(b) 2-3-1-4

vi. Choose the difference between cross-validation procedure and train-test split evaluation from following: [1]
(a) Cross-validation technique for evaluating small sample of data, and train test split for large dataset.
(b) Cross-validation evaluation divides the dataset into 2 groups, train test split organizes the data into K groups.
(c) Cross-validation evaluation gives model quick overview, train test split summaries the model’s ability.
(d) Cross-validation evaluation evaluate the model on new data that was not used to train the model, train test split evaluates the model against the training set.
Answer:
(a) Cross-validation technique for evaluating small sample of data, and train test split for large dataset.

Question 5.
Answer any 5 out of the given 6 questions (1 × 5 = 5 marks)

i. Which of the following is not an example of hyperparameter? [1]
a) The method of train-test split evaluation
b) The number of iterations required to train a neural network
c) The loss function that the model will employ
d) Number of hidden layers in a neural network
Answer:
a) The method of train-test split evaluation

ii Every AI project must undergo _______ steps? [1]
a. 3
b. 4
c. 5
d. 6
Answer:
d. 6

iii Which of the following statement is False in the case of the KNN Algorithm? [1]
i. For a very large value of K, points from other classes may be included in the neighborhood.
ii. For the very small value of K, the algorithm is very sensitive to noise.
iii. KNN is used only for classification problem statements.
iv. KNN is a lazy learner.
a. i and ii
b. ii and iii
c. iii
d. iv and i
Answer:
c. iii

iv. 10_______ is not an example of data story telling? [1]
a-Amazon story boxes
b-Refinery 21
c-National geographic channel
d-Google
Answer:
d-Google

v. Which of the following statement is correct about cross validation procesudure? [1]
i-It randomly shuffle the data set
ii-It organizes the data into K group
iii-It summarize the model ability
iv-For each distinct group it does not fit a model and test it against the test set.
a-I, ii and iii
b- ii, iii and iv
c-I, iii and iv
d-I, ii and iv
Answer:
d-I, ii and iv

vi Which of the following role a Business sponsors who need a analytical solution does not perform? [1]
a-They define the problem and its objectives
b-They are involved throughout the project c-They review interim conclusions
d-They are not keen to produce intended solutions
Answer:
d-They are not keen to produce intended solutions

SECTION-B

Subjective type Questions:

Answer any 3 out of the given 5 questions on Employability Skills ( 2 × 3=6 marks) Answer each question in 20-30 words.

Question 6.
Name any two source of motivation [2]
Answer:
Motivation comes from two places:

• Intrinsic motivation: This is when motivation comes from “internal” factors to meet personal needs. We do things we do because we enjoy them, not because we have to….
• Extrinsic motivation: This is when motivation comes from “external” factors that are given or controlled by others.

Question 7.
Briefly explain the five factor model. [2]
Answer:
Abstract. The five-factor model of personality is a hierarchical organization of personality traits in terms of five basic dimensions:
Extraversion, Agreeableness, Conscientiousness, Neuroticism, and Openness to Experience.

Question 8.
Define difference between Worksheet and Workbook? [2]
Answer:
A worksheet is the working area or page on the screen. A worksheet consists of rows and columns. A collection of worksheets is known as a workbook.

Question 9.
Write down the steps to insert picture from file. [2]
Answer:
The steps to insert a picture from a file are:

1. Select Insert-From File. Picture
2. The Insert picture dialog box appears.
3. Select the picture you want to insert and click Open.

Question 10.
What do you understand by “entrepreneurship” [2]
Answer:
An entrepreneurship, there are many views but precisely it can be ” the ability and readiness to develop, organize and run a business enterprise and with a strong will to face uncertainties in order to make a profit.

Answer any 4 out of the given 6 questions in 20-30 words each (2 × 4 = 8 marks)

Question 11.
Write the names of sum regression loss function. [2]
Answer:

• Mean Square Error/Quadratic Loss
• Mean Absolute Error
• Huber Loss/ Smooth Mean Absolute Error
• Log Cash Loss
• Quantile Loss

Question 12.
Write the names of sum classification function. [2]
Answer:

• Log Loss
• Focal Loss
• KL divergence/Relative Entropy
• Exponential Loss
• Hinge Loss

Question 13.
What are the considerations to be taken during the evaluation stage? [2]
Answer:
These are the few points to be taken into considerations:

• The volume of test data should not be huge as it may provide data complexity.
• Some times the system may deal with sensitive data so regulatory compliance and security testing are essential.
• It is also necessary for system integration testing, if the model the require data from other systems.
• The testing data must be developed by the team who are involved in the validation of the M L models.

Question 14.
“Once the relevant projects have been selected and properly scoped, the next step of the machine learning lifecycle is the Design or Build phase.” Briefly explain this phase. [2]
Answer:
Data gathering, investigation, preparation, cleaning, feature engineering, testing, and running a number of models in an attempt to anticipate behaviors or find insights in the data make up the majority of the steps involved in constructing an AI or machine learning model.

Question 15.
What are the steps involved in telling an effective data story? [2]
Answer:

• Understanding the audience
• Choosing the right data and visualizations
• Drawing attention to key information
• Developing a narrative
• Engaging your audience

Question 16.
What are the two possible graphs that can be used to represent this data? [2]

Answer:
Bar Graph, Line Graph

Answer any 3 out of the given 5 questions in 50-80 words each (4 × 3 = 12 marks)

Question 17.
What do you understand by the term over fitting, under fitting and model fit in terms of model testing. [4]
Answer:
In terms of modeling over fitting correspondence to the predictions that the model predicts on testing data and it is due to the inaccurate prediction of the model. Either due to the to much detailing or due to the noise.

The reason is the model is too complex.
Under fitting : A machine algorithm is said to be under fitting when it performs well on the training data but no on the testing data. This is due to the fewer data to build the model or due to non-linear data. That is the reason the model makes wrong predictions. Another reason is model is to simple and size of training data set not enough.

Model fit: When a model is best fitted it produces more accurate outcomes and it is essential because if the model does not fit your data correctly the outcomes will not be accurate enough to be useful for practical decision making.

Question 18.
Find the MSE of the following data, if the regression equation is: 0.8 ×+18.9: [4]

Calculate the RMSE(Root means Square Error) for the above data.
Answer:
Considering the above data, first calculate its predicted value, predictedobserve value.

Regression Equation: 0.8 x+18.9
Here, the formula of RMSE is

Putting the values on the above formula, we get RMSE
= \(\sqrt{\frac{248.05}{5}}\) =7.043
Thus answer is 7.043

Question 19.
Given below four graphs pick analytical approach based on the type of questions asked [4]

A-Which of the graph is used for statistical analysis?
B-Which of the graph is showing current status?
C-Which of the graph is used for predictive forecasting?
D-Which of the graph is used for probability of the action?
Answer:
A-The second graph shows statistical analysis that is what happened, why is this
happening?
B-The first graph shows the current status or descriptive status of a question.
C-The third graph shows predictive forecasting that is what will happen next or what if
the trend will continue type of questions.
D-The four graph is used to determine the probability of action for this the predictive
model might be used.

Question 20.
Ohat are the Shortcoming of Train-Test Split [4]
Answer:
Consider a dataset that contains 50oo rows. You can choose how many rows go to the training set and how many go to the test set using the test size argument in the train test split method. The larger the test set, the more trustworthy your model quality metrics will be. In extreme cases, you could Imagine that the test set contains just one row of data. Which alternative model Imerforms the best forecasts for a single data point when you evaluate different models will mostly depend on luck.

Normally, you’ll preserve 20 % of the dataset for testing. However, even with 1000 rows in the test set, the model scores are subject to some random chance. A model could do well on.

Question 21.
a-Supposing you are asked to develop a smart student management system for your school. List down the steps that you will follow for developing the system. [4]
Answer:
Follow the given steps to develop a smart student management system.

• First set goals
• Identify the Stakeholders
• Identify the existing measures
• Identify the ethical concerns
• Identify the data needs
• Visualise the mock data

1. b- Teacher asked Vipin and Mohan to graphically present all students marks in two subjects: Artificial Intelligence and
Mathematics. Their graphs are shown below:

Students average scoring in Artificial Intelligence is 81.8 . So, they are more inclined to Artificial Intelligence as a subject.

What do you think whose data story telling is more appealing?
Answer:
Mohan’s graph depicts a better story. The insight given at the top of the visual is giving a clear idea about context which is not there in Vipin’s graph.

The post CBSE Sample Papers for Class 12 AI Set 3 with Solutions appeared first on Learn CBSE.

Students must start practicing the questions from Class 12 AI Important Questions and CBSE Class 12 AI Question Paper 2023 are designed as per the revised syllabus.

CBSE Class 12 AI Exam Question Paper 2023 with Solutions

Question 1.
Answer any 4 out of the given 6 questions on Employability Skills (4 × 1 = 4)

I. Narcissistic personality disorder is characterized by which of the following conditions(s):
i. People have an inflated sense of their own importance.
ii. A deep need for excessive attention
iii. Admiration and lack of empathy
iv. Introvert
(a) Only I
(b)Both i and ii
(C) i,ii, and iii
(d) i,ii,iii and iv
Answer:
(d) i,ii,iii and iv

II. “S” in acronym SMART in goal setting stands for:
(a) Strong
(b) Segment
(c) Specific
(d) Special
Answer:
(c) Specific

III. _____ is an economic process, where an idea is generated or an opportunity is created, refined, developed and implemented, while being exposed to uncertainity, to realize a profit by effective utilization of resources.
(a) Entrepreneurs
(b) Entrepreneurship development
(c) Entrepreneurship
(d) Cluster intervention
Answer:
(c) Entrepreneurship

IV. In a spreadsheet software, an arrangement of cells in a vertical manner is known as:
(a) Worksheets
(b) Workbooks
(C) Rows
(d) Columns
Answer:
(d) Columns

V. Which of the following is not an example of a spreadsheet?
(a) Microsoft excel
(b) LibreOffice calc
(c) OpenOffice impress
(d) Google sheets
Answer:
(c) OpenOffice impress

VI. Which entrepreneur, out of the following, is essentially a manufacturer, who identifies the needs of customers and creates products or services to serve them?
(a) Service entrepreneur
(b) Industrial entrepreneur
(c) Agricultural entrepreneur
(d) Technical entrepreneur
Answer:
(b) Industrial entrepreneur

Question 2.
Answer any 5 out of the given 6 questions (5 × 1 = 5)

I. A training set is a set of _______ -data in which the outcomes are already known.
(a) Current
(b) Instant
(c) Historical
(d) Output
Answer:
(c) Historical

II. _____ involves a combination of three key elements, that help to explain the audience what’s happening in the data in an engaging and entertaining manner.
(a) Data analysis
(b) Data visualization
(c) Data storytelling
(d) Data narrative
Answer:
(c) Data storytelling

III. Match the following step of AI project given in column ‘A’ with its exact sequence given in column ‘B’

Answer:

IV. In AI development which of the following framework is used?
(a) Python
(b) Tensor flow
(c) Visual Basic
(d) C++
Answer:
(b) Tensor flow

V. All the algorithms in machine learning rely on minimizing or maximizing a function, called as_______.
(a) Objective function
(b) Focal loss
(c) Loss function
(d) Gradient descent
Answer:
(a) Objective function

VI. Which of the following is incorrect?
(a) The testing phase is essent
(b) The first fundamental step
(C) In scoping phase, it’s crucial to precisely define the strategic business objectives and desired outcomes of the project.
(d) Test data should include all relevant subsets of training data.
Answer:
(d) Test data should include all relevant subsets of training data.

Question 3.
Answer any 5 out of the given 6 questions. (5 × 1 = 5)

I. When visuals are applied to data, they can _______ the audience to the insights that they wouldn’t perceive without the charts or graphs.
(a) Engage
(b) Explain
(c) Enlighten
(d) Change
Answer:
(a) Engage

II. It is believed that if there is a _______ in the data, then AI development techniques may be employed.
(a) Pattern
(b) Program
(c) Language
(d) Logic
Answer:
(a) Pattern

III. Assertion (A) : It’s crucial to precisely define the strategic business objectives and desired outcomes of the project, align all the different stakeholders’ expectations, anticipate the key resources and steps, and define the success metrics.
Reason (R) : Selecting the AI or machine learning use, cases and being able to evaluate the return or investment (ROI) is critical to the success of any data project.
(a) Both (A) and (R) are true and (R) is the correct explanation of (A)
(b) Both (A) and (R) are true and ® is not the correct explanation of (A)
(C) (A) is true, but (B) is false.
(d) (A) is false, but (R) is true.
Answer:
(a) Both (A) and (R) are true and (R) is the correct explanation of (A)

IV. Which of the following shows the audience where to look and what not to miss and also keeps the audience engaged?
(a) data
(b) narrative
(c)charts
(d) story
Answer:
(d) story

V. Once the relevant projects have been selected and properly scoped, which of the following phase will be the next step of the machine learning lifecycle?
(a) Problem scoping
(b) Deployment
(c) Design
(d) Testing
Answer:
(b) Deployment

VI. Which of the following is the first stage of an AI model life cycle?
(a) Build
(b) Scoping
(c) Design
(d) Testing
Answer:
(b) Scoping

Question 4.
Answer any 5 out of the given 6 questions (5 × 1 = 5)

I. Which of the following is the most preferred language for building an AI model?
(a) Python
(b) VB
(c) Java
(d) C++
Answer:
(a) Python

II. First four steps of writing Python code to find out RMSE values of the model are given here. Arrange them in proper order-
1. Splitting the data into training and test.
2. Reading the data.
3. Fitting simple linear regression to the training set.
4. Important required libraries
(a) 2-4-1-3
(b) 4-1-2-1
(c) 1-2-3-4
(d) 4-2-1-3
Answer:
(c) 1-2-3-4

III. Which of the following is NOT true for testing?
(a) The volume of test data should be very small
(b) Data validation is important
(C) Your testing team should test the AI and ML algorithms keeping model validation
(d) Your team must create test suites that help you validate your ML models.
Answer:
(a) The volume of test data should be very small

IV. Data storytelling is a _______ approach for communicating insights drawn from data.
(a) Iterative
(b) Procedural
(c) Sequential
(d) Structural
Answer:
(c) Sequential

V. Which of the following is NOT true for train-test split evaluation?
(a) The procedure involves taking a dataset and dividing it into two subsets.
(b) The train-test procedure is appropriate when there is a larger dataset.
(c) The objective is to estimate the performance of the user.
(d) It can be used for classification or regression problems.
Answer:
(c) The objective is to estimate the performance of the user.

Question 5.
Answer any 5 out of the given 6 questions.

I. Stories create ______ experiences that transport the audience to another space and time.
(a) Visualizations
(b) Engaging
(c) Testing
(d) Necessary
Answer:
(b) Engaging

II. Which of the following is incorrect?
i. A capstone project is a project where students must research a topic independently to find a deep understanding of the subject matter.
ii. The initial part of the project of academic program is known as capstone project.
iii. The final project of an academic program, integrating all of the learning from the program is called capstone project.
(a) i and iii only
(b) ii and iii only
(c) ii only
(d) i, ii and iii
Answer:
(b) ii and iii only

III. Which of the following is an example of a development tool used in building an AI model?
(a) C+++
(b) Anaconda
(c) Linux
(d) Scratch
Answer:
(b) Anaconda

IV. Identify the correct statements from the following:
i. Data modeling focuses on developing modes that are either descriptive or predictive.
ii. A predictive model tries to yield yes/no, or stop/go type outcomes.
iii. The data scientist will use a training set for descriptive modeling.
iv. Data modeling focuses on testing the project.
(a) i, ii, and iii
(b) ii, iii and iv
(c) i and ii
(d) i and iv
Answer:
(a) i, ii, and iii

V. _______ is a design methodology that provides a solution-based approach to solving problems.
(a) Classification
(b) Design thinking
(c) Recommendation
(d) Computational
Answer:
(b) Design thinking

VI. Which of the following is a key element of a data story?
(a) Audience
(b) Visuals
(c) Engage
(d) Information
Answer:
(a) Audience

Section-B

Subjective Type Questions:

Answer any 3 out of the given 5 questions on Employability Skills. Answer each question in 2030 words:

Question 6.
With reference to ‘Five Factor Model’, mention any four parameters that describe an individual’s personality.
Answer:
The Five Factor Model, also known as the Big Five personality traits, includes the following parameters:

1. Openness to Experience: This reflects the degree of intellectual curiosity, creativity, and preference for novelty and variety.
2. Conscientiousness: This measures the degree of organization, responsibility, dependability, and goal-oriented behavior.
3. Extraversion: This indicates the level of sociability, assertiveness, talkativeness, and the tendency to seek out social stimulation.
4. Agreeableness: This assesses the individual’s degree of kindness, cooperativeness, and the tendency to be compassionate and cooperative rather than suspicious and antagonistic.
5. Neuroticism: This measures emotional stability and the tendency to experience negative emotions, such as anxiety, anger, and depression.

Question 7.
Define any two of the following personality disorders: (2)
(a) Avoidant
(b) Dependent
(C) Histrionic
Answer:
(a) Avoidant Personality Disorder:

• Individuals with Avoidant Personality Disorder (APD) tend to have a pervasive pattern of social inhibition, feelings of inadequacy, and hypersensitivity to negative evaluation.
• They are often reluctant to engage in social activities and fear criticism or rejection.
• People with APD may avoid new activities or meeting new people due to a fear of embarrassment or being socially inept.

(b) Dependent Personality Disorder:

• Dependent Personality Disorder (DPD) is characterized by a pervasive and excessive need to be taken care of.
• Individuals with this disorder often have difficulty making everyday decisions without reassurance and advice from others.
• They may feel helpless when alone and are preoccupied with fears of being abandoned.
• This dependency on others can lead to submissive and clingy behavior in relationships.

(c) Histrionic Personality Disorder:

• Histrionic Personality Disorder (HPD) is characterized by a pattern of excessive attentionseeking and emotionality.
• Individuals with HPD may be uncomfortable when they are not the center of attention and may use their physical appearance to draw attention to themselves.
• They often display rapidly shifting and shallow emotions and may be easily influenced by others.
• Relationships may be characterized by a need for constant approval and may be perceived as more intimate than they actually are.

Question 8.
Differentiate between a row and a cell of a spreadsheet. (2)
Answer:
In a spreadsheet:
Row:

• A row is a horizontal line of cells in a spreadsheet.
• It is identified by numbers along the left side of the spreadsheet, usually starting from 1 and increasing sequentially ( 1,2,3, … ).
• Each row in a spreadsheet is labeled with a row number to help identify its position.

Cell:

• A cell is a single unit or intersection point within a spreadsheet where a column and a row meet.
• It is identified by a combination of its column letter and row number. For example, cell A1 is in the first column and first row.
• Each cell can hold data, such as text, numbers, formulas, or functions.

In summary, a row is a horizontal arrangement of cells, and a cell is a single unit within the spreadsheet where data is entered or calculated. The combination of rows and columns creates a grid, and each cell in the grid has a unique address based on its column and row position.

Question 9.
Write any four qualities that motivate an entrepreneur. (2)
Answer:
Motivation plays a crucial role in driving entrepreneurs to pursue their goals and overcome challenges. Here are four qualities that often motivate entrepreneurs:
Passion: Entrepreneurs are often motivated by a deep passion for their business idea or industry. Passion fuels their commitment, perseverance, and enthusiasm, helping them stay focused on their goals.

Autonomy: The desire for autonomy and independence is a strong motivator for entrepreneurs. They seek the freedom to make their own decisions, control their destiny, and build something of their own.

Challenge and Achievement: Entrepreneurs are motivated by challenges and the opportunity to overcome obstacles. The sense of accomplishment derived from successfully navigating difficulties and achieving milestones serves as a powerful motivator.

Innovation and Creativity: The opportunity to innovate and bring creative ideas to life is a motivating factor for many entrepreneurs. The prospect of developing new solutions, disrupting industries, and contributing to positive change can drive their entrepreneurial spirit.

It’s important to note that motivation can vary among entrepreneurs, and individual motivations may encompass a combination of these and other factors.

Question 10.
Briefly define the term competency. Mention any two common Entrepreneurial Competencies.
Answer:
Competency: Competency refers to the combination of knowledge, skills, abilities, and behaviors that an individual possesses and demonstrates in order to perform effectively in a particular role or field. In the context of entrepreneurship, entrepreneurial competencies are the key qualities and capabilities that enable individuals to identify opportunities, create and manage ventures, and navigate the challenges of running a business.

Two Common, Entrepreneurial Competencies:
Opportunity Recognition:
Entrepreneurs need the competency to identify and evaluate business opportunities in the market. This involves the ability to spot gaps, unmet needs, or trends that can be turned into viable business ventures. Successful entrepreneurs possess a keen sense of observation, market understanding, and the ability to assess the feasibility of potential opportunities.

Risk Management:
Entrepreneurial ventures often involve uncertainty and risk. Competency in risk management is crucial for entrepreneurs to assess, mitigate, and take calculated risks. Entrepreneurs need to be able to make informed decisions in the face of uncertainty, manage challenges, and adapt to changing circumstances. This competency involves a combination of analytical skills, strategic thinking, and the ability to weigh potential rewards against potential risks.

Answer any 4 out of the given 6 questions in 20-30 words each (4 × 2 = 8)

Question 11.
Mention the names of any four stages of Design Thinking.
Answer:
Design Thinking typically involves a series of stages that guide the problem-solving and innovation process. While the specific number of stages can vary, a commonly used framework consists of the following four stages:

Empathize: In this stage, designers seek to understand and empathize with the needs, experiences, and perspectives of the people they are designing for. This involves engaging with users, conducting interviews, and observing behaviors to gain deep insights into their needs and challenges.

Define: The Define stage involves synthesizing the information gathered during the Empathize stage to define the core problems and challenges. Designers work to articulate the problem statement clearly and frame it in a way that guides the rest of the design process.

Ideate: Ideation is the stage where designers generate a wide range of creative ideas and potential solutions to address the defined problems. It encourages brainstorming and free-thinking, allowing for the exploration of diverse possibilities without immediate judgment.

Prototype: Prototyping involves creating tangible representations or mock-ups of potential solutions. These prototypes can be low-fidelity or high-fidelity, depending on the stage of the design process. Prototyping allows designers to test and iterate on their ideas, getting feedback and refining the solutions.

These stages are often iterative and may be revisited as designers gain more insights and refine their understanding of the problem and potential solutions.

Question 12.
Explain the term time series decomposition. (2)
Answer:
Time series decomposition is a statistical technique used to break down a time series into its underlying components. The primary goal is to separate the observed data into different systematic and non-systematic components, making it easier to understand the patterns, trends, and seasonality present in the time series data. The decomposition typically involves three main components:
Trend: The trend component represents the long-term direction or movement in the time series. It identifies the underlying pattern that may indicate whether the values are increasing, decreasing, or staying relatively constant over time.

Seasonal: The seasonal component captures repeating patterns or fluctuations that occur at regular intervals within a specific time period, such as daily, monthly, or yearly. Seasonal patterns often reflect recurring events or cycles.

Residual (or Error): The residual, also known as the error or remainder, accounts for the variability in the data that cannot be attributed to the trend or seasonal components. It represents the random or irregular fluctuations in the time series that are not part of any discernible pattern.

The time series decomposition helps analysts and data scientists to better understand the underlying structure of the data, making it easier to analyze and model. Various methods, such as additive or multiplicative decomposition, can be employed based on the nature of the time series data. Decomposition is particularly useful in time series forecasting and trend analysis, allowing for more accurate predictions and insights into the factors influencing the data over time.

Question 13.
Give two points of difference between Cross-Validation and Train-Test split.
Answer:
Cross-Validation and Train-Test Split are techniques used to assess the performance of machine learning models, but they differ in their approach to splitting the dataset for training and evaluation. Here are two points of difference between Cross-Validation and Train-Test Split:
Data Splitting: Train-Test Split: In this method, the dataset is divided into two subsets: a training set and a test set. The model is trained on the training set, and its performance is evaluated on the separate test set. The split is typically done with a fixed ratio, such as 80 % for training and 20% for testing.

Cross-Validation: Cross-Validation involves dividing the dataset into multiple folds or subsets. The model is trained on some of the folds and tested on the remaining fold. This process is repeated multiple times, with each fold serving as the test set exactly once. The results are averaged to provide a more robust evaluation.

Usage of Data: Train-Test Split: The data is split into a training set and a test set, and the model is trained exclusively on the training set. The test set is reserved for assessing the model’s performance after training.

Cross-Validation: The entire dataset is used for both training and testing across multiple iterations. Each data point serves as part of the training set and part of the test set at different stages of the cross-validation process. This ensures that every data point is used for validation at least once.

In summary, Train-Test Split involves a fixed separation of the dataset into training and test sets, while Cross-Validation systematically uses different subsets of the data for training and testing in multiple iterations, providing a more comprehensive evaluation of the model’s performance.

Question 14.
Selecting the right analytical approach depends on the questions being asked.
In the light of the above given statement, which type of questions can be asked for (2)
(A) Classification approach
(B) Descriptive approach
Answer:
(A) Classification Approach:

Questions for Classification Approach:

• Is the given observation a part of a specific category or class?
• Can we predict the class or category of a new observation based on its features?
• What factors contribute the most to the classification of observations into different classes?
• How well can the model distinguish between different classes?

Example Scenario:

• Predicting whether an email is spam or not based on its content.
• Classifying customers into high, medium, or low risk for a credit application.

Descriptive Approach:
Questions for Descriptive Approach:

• What are the key characteristics and trends in the dataset?
• How are variables distributed, and what is the central tendency?
• Can we identify patterns or anomalies in the data?
• What are the relationships between different variables?

Example Scenario:

• Analyzing the average sales per month over the past year.
• Describing the distribution of customer ages in a given dataset.

In summary, the choice between a classification approach and a descriptive approach depends on whether the primary goal is to predict categories or understand and describe patterns and trends in the data.

Question 15.
Name the main stages of AI project life cycle. (2)
Answer:
The AI project life cycle typically involves several stages, each crucial for the successful development and deployment of artificial intelligence solutions. The main stages of an AI project life cycle include:

• Problem Definition: Clearly define the problem or objective that the AI project aims to address. Identify the scope, goals, and constraints of the project.
• Data Collection: Gather relevant data that will be used for training and testing the AI model. Ensure data quality, completeness, and relevance to the problem at hand.
• Data Preparation: Clean, preprocess, and transform the collected data into a format suitable for training machine learning models. This may involve handling missing values, encoding categorical variables, and scaling features.
• Modeling: Select and apply appropriate machine learning algorithms to build predictive models’ This stage involves training the model on the prepared data and fine-tuning its parameters.
• Evaluation: Assess the performance of the trained model using evaluation metrics and validation techniques. This helps ensure that the model generalizes well to new, unseen data.
• Deployment: Integrate the trained model into the target environment, making it accessible for making predictions or decisions in real-world scenarios.
• Monitoring and Maintenance: Continuously monitor the performance of the deployed model in real-world conditions. Implement updates and maintenance as needed to address changing data patterns and ensure ongoing effectiveness.
• Feedback Loop: Establish a feedback loop to gather information from the model’s performance in the real world. Use this feedback to improve the model, update training data, and enhance overall system performance.
• Documentation: Document the entire AI project life cycle, including problem statements, data sources, modeling techniques, evaluation results, and deployment details. This documentation aids in knowledge transfer and future enhancements.

These stages are iterative, and the AI project life cycle often involves revisiting and refining certain steps as the project progresses and additional insights are gained.

Question 16.
Give two reasons why storytelling is so powerful? (2)
Answer:
Storytelling is a powerful communication tool that has been used throughout history. Two key reasons why storytelling is so powerful are:
1. Emotional Connection:
Stories have the ability to evoke emotions and create a personal connection with the audience. When information is presented in a narrative form, it engages the audience on an emotional level. Emotional connections are more memorable and impactful, leading to a deeper understanding of the message. By incorporating relatable characters, challenges, and resolutions, storytelling can resonate with the audience’s own experiences, making the message more meaningful and memorable.

2. Enhanced Understanding and Retention:
Stories provide a context and structure that help people better understand complex information. Instead of presenting facts and data in isolation, storytelling weaves information into a narrative that is easier to follow and comprehend. The narrative structure of a story, with a clear beginning, middle, and end, helps organize information in a way that the human brain naturally processes and remembers.

As a result, storytelling enhances the retention of information, making it more likely that the audience will remember and internalize the key messages conveyed in the story.

In summary, the emotional connection created through storytelling and the enhanced understanding and retention of information contribute to the power of storytelling as a communication tool. Stories have the capacity to engage, inspire, and leave a lasting impact on the audience.

Answer any 3 out of the given 5 questions in 50-8o words each. (3 × 4 = 12)

Question 17.
Give steps to break down a problem into smaller units before coding.
Answer:
Breaking down a problem into smaller units, often referred to as decomposition, is a crucial step in the software development process. It helps in managing complexity, promoting modularity, and facilitating a more systematic approach to problem-solving. Here are steps to break down a problem before coding:

• Understand the Problem: Begin by thoroughly understanding the problem statement or requirements. Identify the key objectives, constraints, and expected outcomes.
• Identify Subtasks: Break down the main problem into smaller, manageable subtasks or components. Each subtask should represent a specific aspect of the problem that can be addressed independently.
• Define Inputs and Outputs: For each subtask, clearly define the inputs required and the expected outputs. Understand how data will flow between different components and what transformations or operations are needed.
• Prioritize and Order Subtasks: Prioritize the subtasks based on their importance and dependencies. Identify any prerequisites or sequential relationships between subtasks.
• Use Modular Design: Aim for a modular design by encapsulating related functionality into modules or functions. This promotes code reusability, maintainability, and a clearer organization of the codebase.
• Identify Data Structures and Algorithms: Determine the appropriate data structures and algorithms for each subtask. Consider the efficiency and scalability of the chosen solutions.
• Pseudocode or Floweharts: Before diving into actual code, consider creating pseudocode or flowcharts for each subtask. This provides a high-level overview of the logic and helps in refining the algorithm.
• Review and Refine: Review the decomposition and design with a focus on clarity, efficiency, and adherence to the problem requirements. Refine the decomposition as needed to improve the overall structure.
• Test Each Subtask Independently: Develop and test each subtask independently to ensure that it functions correctly. This allows for early detection and resolution of issues before integrating the components.
• Integrate and Test the Whole System: Once individual subtasks are tested and verified, integrate them into the complete system. Conduct comprehensive testing to ensure that the integrated solution meets the overall requirements.

By following these steps, developers can break down a complex problem into manageable units, facilitating a more systematic and organized approach to coding. This decomposition process promotes a clear understanding of the problem and improves the overall efficiency of the development process.

Question 18.
The following is the diagram depicting the foundational methodology for data science.

The diagram is marked with A, B, C, D. Identify these four steps and briefly explain the significance of steps marked as ‘A’ and ‘B’.
Answer:

• A-depicts-Business understanding
• B-depicts data collection
• C-depicts modeling
• D-depicts deployment

A. Business understanding means is a critical and foundational phase in the data science process. Its significance lies in several key aspects:

• Alignment with Business Goals:
• Problem Definition and Refinement:
• Identification of Stakeholder Needs:
• Return on Investment (ROI):

B. Data collection- is required for the following:

• Data collection helps in finding raw material for data analysis.
• High quality data helps support in decision making.
• Data collection enables the identification of patterns, trends, and correlations within the dataset

Question 19.
Explain AI project cycle and project scoping in detail.
Answer:
The AI project cycle outlines the various stages involved in developing and deploying an artificial intelligence solution. While specific approaches may vary, a general AI project cycle typically includes the following stages:

• Project Scoping: Define the project’s goals, objectives, and scope. Identify the problem to be addressed and the expected outcomes. Set clear boundaries and constraints for the project.
• Data Collection and Preparation: Gather relevant data required for training and testing the AI model. Clean, preprocess, and format the data to ensure its quality and suitability for machine learning algorithms.
• Modeling: Select appropriate machine learning algorithms based on the nature of the problem. Train the model using the prepared data and fine-tune its parameters for optimal performance.
• Evaluation: Assess the performance of the trained model using evaluation metrics and validation techniques. Evaluate how well the model generalizes to new, unseen data.
• Deplayment: Integrate the trained model into the target environment or application. Deploy the AI solution for real-world use, making predictions or decisions based on new data.
• Monitoring and Maintenance: Continuously monitor the performance of the deployed model in the real-world context. Implement updates and maintenance as needed to adapt to changing data patterns and ensure ongoing effectiveness.
• Feedback Loop: Establish a feedback loop to gather information from the model’s performance in the real world. Use this feedback to improve the model, update training data, and enhance overall system performance.

Project Scoping:

• Project scoping is a critical initial phase in the AI project cycle. It involves defining the boundaries and parameters of the project to ensure clarity and alignment with organizational objectives. Here are key steps in project scoping:
• Define Objectives: Clearly articulate the goals and objectives of the AI project. Understand what the organization aims to achieve through the implementation of AI.
• Identify Stakeholders: Identify and involve key stakeholders, including project sponsors, end-users, and other relevant parties. Understand their expectations and requirements.
• Set Boundaries and Constraints: Define the scope of the project by setting boundaries and constraints. Clearly specify what is included and excluded from the project to manage expectations.
• Understand Resources: Assess the resources available for the project, including budget, time, and personnel. Determine the feasibility of the project within the given constraints.
• Risk Assessment: Identify potential risks and challenges associated with the project. Conduct a risk assessment to understand and mitigate potential issues that may arise during the project.
• Feasibility Study: Evaluate the technical and economic feasibility of the AI project. Assess whether the proposed solution aligns with the organization’s capabilities and financial resources.
• Define Deliverables: Clearly define the deliverables and milestones expected at various stages of the project. This includes specific outcomes, reports, and any other tangible results.
• Create a Project Plan: Develop a detailed project plan that outlines tasks, timelines, dependencies, and responsibilities. This plan serves as a roadmap for project execution.
• Communication Plan: Establish a communication plan that defines how project progress will be communicated to stakeholders. Ensure transparent and regular communication channels.
• Approval and Sign-Off: Present the project scope, objectives, and plan to stakeholders for approval and sign-off. This formalizes commitment and ensures alignment before proceeding to subsequent stages.
• By effectively scoping an AI project, organizations can set the foundation for a successful implementation, mitigate risks, and align the project with overarching business objectives.

Question 20.
(a) Mention the steps that can assist in finding compelling stories in the data sets.
Answer:
Some of the simple steps that assist in compelling data stories are :

• The story context must be a relevant one.
• It must include authentic data.
• Should be presented in compelling visuals.
• There should be a relationship between the data.

(b) Which of following (Option-A or Option-B) is a better data story? Give reasons.

Option-A

Option-B

In 2021, the total forest and tree cover in India is 80.9 million hectares, which is 24.62 % of the geographical area of the country.
Answer:
Option-B is a better data story figure because

• The textual context within the figure is more appropriate.
• The numbers represents the exactness of the data and depicts a clear picture of what the text conveys.
• It also presents geographical data with certain regions and areas.

The post CBSE Class 12 AI Question Paper 2023 with Solutions appeared first on Learn CBSE.

Students must start practicing the questions from Class 12 AI Important Questions and CBSE Class 12 AI Question Paper 2024 are designed as per the revised syllabus.

CBSE Class 12 AI Exam Question Paper 2024 with Solutions

Question 1.
Answer any 4 out of the given 6 questions on Employability Skills. (4 × 1 = 4)

(i) ________ is a form of communication that allows students to put their feelings and ideas on paper, to organize their knowledge and beliefs into convincing arguments, and to convey meaning through well-constructed text.
(a) Active listening
(b) Writing
(c) Absolute phrase
(d) Speeches
Answer:
(b) Writing

(ii) The term OCPD stands for _____.
(a) Obsessive ‘compulsive personality disorder
(b) Operational compulsive personality disorder
(c) Obsessive compulsive personality defect
(d) Organised compulsive professional disorder
Answer:
(a) Obsessive ‘compulsive personality disorder

(iii) Identify the incorrect statement from the following:
(a) Motivation and positive thinking can help us overcome fears and take up new challenges.
(b) Motivation and positive thinking can help us in ignoring our duties.
(c) An individual’s motivation may come from within or be inspired by others or events.
(d) Directing behavior towards certain motive or goal is the essence of motivation.
Answer:
(b) Motivation and positive thinking can help us in ignoring our duties.

(iv) Which of the following statements is NOT true for spreadsheet?
(a) A workbook has one or more worksheets.
(b) Large volumes of data can be easily handled and manipulated.
(c) Data cannot be easily represented in pictorial form like graphs or charts.
(d) Built-in functions make calculations easier, faster and more accurate.
Answer:
(c) Data cannot be easily represented in pictorial form like graphs or charts.

(v) Which of the following is one of the barriers that an entrepreneur may face?
(a) Self-confidence
(b) Availability of monetary resources on time
(c) Unavailability of monetary resources on time
(d) Availability of skilled labour/staff
Answer:
(c) Unavailability of monetary resources on time

(vi) A _____ is defined as one that helps bring about and maintain transition to environmentally sustainable forms of production and consumption.
(a) Blue collar job
(b) White collar job
(c) Yellow job
(d) Green job
Answer:
(d) Green job

Question 2.
Answer any 5 out of the given 6 questions (5 × 1 = 5)

(i) During Train-Test split evaluation, we usually split the data around _______ between testing and training stages.
(a) 90% – 10%
(b) 20% – 80%
(c) 100% – 0%
(d) 0% – 100%
Answer:
(b) 20% – 80%

(ii) With reference to Data storytelling, complete the given statement: “Data can be persuasive, but _____ are much more.”
(a) Machines
(b) Projects
(c) Stories
(d) Humans
Answer:
(c) Stories

(iii) _____ provides a useful abstract model for thinking about time series generally and for better understanding problems during time series analysis and forecasting.
(a) Decomposition
(b) Modelling
(c) Stage
(d) Building
Answer:
(a) Decomposition

(iv) The first fundamental step when starting an AI initiative is- \qquad
(a) Evaluation
(b) Testing
(c) Development
(d) Scoping
Answer:
(d) Scoping

(v) Which of the following is not one of the steps of an AI project life cycle?
(a) Problem definition
(b) Understanding the problem
(c) Data delivery
(d) Data gathering
Answer:
(d) Data gathering

(vi) Which of the following does not come under open language category.
(a) Linux
(b) Python
(c) R
(d) Scala
Answer:
(a) Linux

Question 3.
Answer any 5 out of the given 6 questions. (5 × 1 = 5)

(i) _____ is the last step in the AI project life cycle.
(a) Problem definition
(b) Data gathering
(c) Deployment
(d) Evaluation
Answer:
(c) Deployment

(ii) Identify the given element that makes a compelling data story and choose its correct name from the following options:

(a) Graphs
(b) Numbers
(c) Story
(d) Data
Answer:
(d) Data

(iii) In _______ phase, it’s crucial to precisely define the strategic business objectives and desired outcomes of the project.
(a) Design
(b) Deployment
(c) Testing
(d) Requirement analysis
Answer:
(d) Requirement analysis

(iv) Assertion (A) : With reference to Data storytelling, narrative is the way we simplify and make sense of a complex world.
Reason (R) : Narrative explains what is going on within the dataset.
Select the appropriate option for the statements given above:
(a) Both (A) and (R) are true and (R) is the correct explanation of (A)
(b) Both (A) and (R) are true and (R) is not the correct explanation of (A)
(c) (A) is true but (R) is false
(d) (A) is false but (R) is true
Answer:
(a) Both (A) and (R) are true and (R) is the correct explanation of (A)

(v) AI is perhaps the most transformative technology available today. At a high level, every AI project follows total–steps.
(a) Six
(b) Seven
(c) Eight
(d) Infinite
Answer:
(b) Seven

(vi) During step 3 of AI model life cycle,——-should include all relevant subsets of training data.
(a) Relevant Data
(b) Deployment
(c) Test data
(d) Scoping
Answer:
(c) Test data

Question 4.
Answer any 5 out of the given 6 questions. (5 × 1 = 5)

(i) Match the following:

(a) 1-D, 2-A, 3-B, 4-C
(b) 1-D, 2-C, 3-B, 4-A
(c) 1-B, 2-A, 3-D, 4-C
(d) 1-C, 2-B, 3-A, 4-D
Answer:
(b)

(ii) Stories that incorporate ______ and analytics are more convincing than those based entirely on anecdotes or personal experience.
(a) Suspense
(b) Humour
(c) Data
(d) Energy
Answer:
(c) Data

(iii) During modeling approach of Capstone project, the data scientist will use a _________ set for predictive modeling.
(a) Training
(b) Testing
(c) Valuable
(d) Known
Answer:
(a) Training

(iv) As Data Storytelling is a structured approach for communicating insights drawn from data, and invariably involves a combination of key elements.
When the _____ is accompanied with data, it helps to explain the audience what’s happening in the data and why a particular insight has been generated.
(a) Data
(b) Visuals
(c) Narrative
(d) Story
Answer:
(c) Narrative

(v) With reference to AI Model Life Cycle, which of the following is true for Building the Model?
(a) This is arguably the most important part of your AI project.
(b) Phrase that characterizes this project stage: “garbage in, garbage out”.
(c)This stage involves the planning and motivational aspects of your project.
(d) It is essentially an iterative process comprising all the steps relevant to building the AI or machine learning model.
Answer:
(d) It is essentially an iterative process comprising all the steps relevant to building the AI or machine learning model.

(vi) RMSE stands for- _____.
(a) Root Median Squared Error
(b) Radian Mean Squared Error
(c) Root Mean Search Error
(d) Root Mean Squared Error
Answer:
(d) Root Mean Squared Error

Question 5.
Answer any 5 out of the given 6 questions (5 × 1 = 5)

(i) Good stories don’t just emerge from data itself; they need to be unraveled fromrelationship.
(a) Data
(b) Numbers
(c) Charts
(d) Computer
Answer:
(a) Data

(ii) The train-test procedure is appropriate when there is a sufficiently———data sets available.
(a) Comparative
(b) Large
(c) Small
(d) Equal
Answer:
(b) Large

(iii) During the third step of AI Model Life Cycle, the volume of test data can be large, which presents_______.
(a) Complexities
(b) Accuracy
(c) Efficiency
(d) Redundancy
Answer:
(a) Complexities

(iv) In ______ we run our modeling process on different subsets of the data to get multiple measures of model quality.
(a) Train-Test Split
(b) Regression
(c) Cross-validation
(d) Machine learning
Answer:
(c) Cross-validation

(v) The machine learning life cycle is the _____ process that AI or machine learning projects follow.
(a) Irreversible
(b) Cyclic
(d) One-time
(d) Static
Answer:
(b) Cyclic

(vi) Data Modeling focuses on developing models that are either descriptive or-
(a) Inclusive
(b) Predictive
(c) Selective
(d) Reactive
Answer:
(b) Predictive

Section-B

Subjective Type Questions

Answer any 3 out of the given 5 questions on Employability Skills. Answer each question in 2030 words: (3 × 2 = 6)

Question 6.
Differentiate between ‘sentence’ and ‘phrase’ with the help of suitable example.
Answer:
In linguistics, a sentence is a grammatically complete unit of language that typically expresses a complete thought or idea. It consists of a subject and a predicate, and it can stand alone as a complete statement.

Example of a sentence:

“The cat chased the mouse.”

A phrase, on the other hand, is a group of related words within a sentence that function together as a single part of speech but does not contain both a subject and a predicate. It can’t stand alone as a complete sentence.

Example of a phrase:

“The cat”

In this example, “The cat” is a noun phrase because it consists of a noun (“cat”) modified by a determiner (“the”). It doesn’t express a complete thought on its own but acts as a part of a larger sentence.

Question 7.
Briefly explain the following terms:
(a) Personality
(b) Personality disorders
(a) Personality refers to the unique set of psychological traits, patterns of behavior, thoughts, and emotions that characterize an individual’s consistent and enduring way of interacting with the world. It encompasses various aspects of an individual’s identity, including their attitudes, values, beliefs, motivations, and interpersonal relationships. Personality traits are relatively stable over time and across different situations, influencing how individuals perceive and respond to their environment. Psychologists often study personality through various theoretical frameworks and assessment methods to understand individual differences and predict behavior.

(b) Personality disorders-Personality disorders are a group of mental health conditions characterized by pervasive and inflexible patterns of behavior, thoughts, and emotions that deviate significantly from societal norms and cause impairment in functioning and interpersonal relationships. These patterns are typically longstanding and deeply ingrained, causing distress and problems in various areas of life, such as work, social interactions, and personal development.

There are several types of personality disorders, each with its own unique set of symptoms and characteristics. Some common examples include borderline personality disorder, antisocial personality disorder, narcissistic personality disorder, and obsessive-compulsive personality disorder.

Individuals with personality disorders often struggle with understanding and regulating their emotions, maintaining stable relationships, and adapting to changing circumstances. Treatment for personality disorders typically involves psychotherapy, such as cognitive-behavioral therapy or dialectical behavior therapy, aimed at helping individuals develop insight into their patterns of behavior and learn healthier ways of coping and relating to others. In some cases, medication may also be prescribed to manage symptoms such as depression or anxiety that co-occur with the personality disorder.

Question 8.
Mr. Chowdhary wants to explain the working of a product to his clients. To make an impact on their audience, either he can use homemade charts or make a digital presentation using a computer and presentation software. Which out of the two options is more advantageous and why? Give any three points to support your answer.
Answer:
Using a digital presentation with computer software is more advantageous for Mr. Chowdhary to explain the working of a product to his clients. Here are three points to support this answer:
Visual Appeal: Digital presentations allow for the integration of multimedia elements such as images, videos, and animations, which can significantly enhance the visual appeal and engagement of the presentation. Homemade charts may be limited in their ability to convey complex information effectively, whereas digital presentations can offer dynamic visuals that better illustrate the product’s features and functionalities.

Interactivity: Presentation software often includes features that enable interactivity, such as clickable links, navigation buttons, and interactive elements. This interactivity can facilitate a more engaging and immersive experience for the audience, allowing them to explore the product’s details at their own pace and delve deeper into specific areas of interest. Homemade charts, on the other hand, may lack this level of interactivity and may not be as engaging or interactive for the audience.

Professionalism and Versatility: Digital presentations are perceived as more professional and polished compared to homemade charts. Presentation software offers a wide range of design templates, layouts, and customization options that allow Mr. Chowdhary to create visually appealing and professional-looking presentations tailored to his audience and branding requirements. Additionally, digital presentations can be easily updated, edited, and shared electronically, making them more versatile and convenient for Mr. Chowdhary to use in various settings and contexts.

Question 9.
What do you mean by interpersonal skills? Why is it importan’t for an entrepreneur to possess it? Briefly discuss.
Answer:
Interpersonal skills, also known as people skills or social skills, refer to the ability to communicate, interact, and build positive relationships with others effectively. These skills encompass a range of behaviors and qualities, including verbal and nonverbal communication, active listening, empathy, conflict resolution, teamwork, and emotional intelligence.

For an entrepreneur, possessing strong interpersonal skills is essential for several reasons:
1. Building Relationships: Entrepreneurs need to interact with various stakeholders, including customers, employees, investors, suppliers, and business partners. Effective interpersonal skills enable entrepreneurs to establish and maintain positive relationships with these individuals, fostering trust, loyalty, and collaboration, which are crucial for the success of their ventures.

Communication: Clear and effective communication is vital for conveying ideas, sharing information, articulating vision and goals, negotiating deals, and resolving conflicts. Entrepreneurs with strong interpersonal skills can communicate persuasively, listen actively, and adapt their communication style to different audiences, ensuring mutual understanding and alignment of objectives.

Leadership and Teamwork: Entrepreneurs often lead teams of employees or collaborators to achieve common objectives. Strong interpersonal skills enable entrepreneurs to inspire, motivate, and empower their team members, fostering a supportive and collaborative work environment conducive to creativity, innovation, and productivity.

Customer Relations: Understanding customers’ needs, preferences, and feedback is critical for developing products or services that meet market demands and deliver value. Entrepreneurs with strong interpersonal skills can engage with customers effectively, gather valuable insights, build rapport, and address concerns, leading to improved customer satisfaction and loyalty.

Networking and Opportunities: Networking plays a crucial role in entrepreneurship, providing access to resources, expertise, partnerships, and opportunities for growth and expansion. Entrepreneurs with strong interpersonal skills can network effectively, establish valuable connections, and leverage these relationships to open doors, attract investment, secure partnerships, and access new markets.

Question 10.
Mention any four advantages of Green jobs.
Answer:
Environmental Benefits: Green jobs contribute to sustainable development and environmental conservation by promoting practices that minimize resource consumption, reduce pollution, and mitigate climate change. These jobs focus on renewable energy, energy efficiency, waste management, conservation, and other environmentally friendly practices, helping to protect ecosystems, preserve natural resources, and combat environmental degradation.

Economic Growth: Green jobs stimulate economic growth and create employment opportunities in emerging sectors of the green economy. Investments in renewable energy, clean technologies, green infrastructure, and sustainable practices drive innovation, entrepreneurship, and productivity, leading to job creation, business development, and economic diversification. Green industries also offer long-term stability and resilience, reducing dependency on fossil fuels and volatile markets.

Health and Well-being: Green jobs contribute to improved public health and well-being by reducing exposure to hazardous substances, pollutants, and environmental risks. Green initiatives such as energy-efficient buildings, clean transportation, and sustainable agriculture promote healthier living environments, enhance air and water quality, and reduce the prevalence of respiratory diseases, allergies, and other health issues associated with pollution and environmental degradation.

Social Equity: Green jobs offer opportunities for inclusive and equitable economic development, benefiting marginalized communities, disadvantaged populations, and vulnerable groups disproportionately affected by environmental injustices and economic disparities. By promoting inclusive hiring practices, workforce development, and community engagement, green initiatives can help address social inequalities, promote social cohesion, and empower underrepresented individuals and communities to participate in the green economy and share in its benefits.

Answer any 4 out of the given 6 questions in 20-30 words each. (4 × 2 = 8)

Question 11.
What is training set?
Answer:
In the context of machine learning and data science, a training set refers to a subset of data that is used to train a machine learning model. It consists of a collection of input data points along with their corresponding output labels or target values. The training set serves as the basis for the model to learn the patterns, relationships, and underlying structure present in the data, enabling it to make predictions or classifications on new, unseen data.

The training process involves feeding the input data from the training set into the machine learning model and adjusting the model’s parameters or weights iteratively based on the errors or discrepancies between the predicted outputs and the actual labels. Through this iterative process of optimization, the model gradually learns to generalize from the training data and make accurate predictions on unseen data.

It’s important for the training set to be representative of the overall data distribution and to cover a diverse range of examples to ensure that the model learns robust and generalizable patterns. Additionally, the training set should be sufficiently large to capture the complexity of the underlying data and prevent overfitting, where the model memorizes the training data rather than learning meaningful patterns.

Once the training process is complete, the performance of the trained model is typically evaluated using a separate validation set or test set to assess its accuracy, generalization ability, and performance on unseen data.

Question 12.
Name the two categories of loss functions.
Answer:
The two categories of loss functions in machine learning are:
Regression Loss Functions: These loss functions are used when the machine learning task involves predicting continuous numerical values. Regression loss functions measure the discrepancy between the predicted values generated by the model and the actual target values in the dataset. Common examples of regression loss functions include Mean Squared Error (MSE), Mean Absolute Error (MAE), and Huber loss.

Classification Loss Functions: These loss functions are used when the machine learning task involves predicting discrete class labels or probabilities. Classification loss functions measure the difference between the predicted class probabilities or scores assigned by the model and the true class labels in the dataset. Common examples of classification loss functions include CrossEntropy Loss (also known as Log Loss), Hinge Loss (used in Support Vector Machines), and Binary Cross-Entropy Loss.

Question 13.
“Stories create engaging experiences that transport the audience to another space and time” Justify this statement.
Answer:
Emotional Connection: Stories have the ability to evoke emotions and resonate with the audience on a deep, personal level. By weaving together characters, plotlines, and settings, stories can tap into universal themes and human experiences, allowing audiences to empathize with the characters and become emotionally invested in their journey.

Imagination and Creativity: Stories stimulate the imagination and engage the audience’s senses, transporting them to different worlds, cultures, and time periods. Through vivid descriptions, rich imagery, and compelling narratives, stories spark curiosity and creativity, inviting audiences to explore new perspectives and possibilities beyond their own lived experiences.

Suspension of Disbelief: Stories suspend the audience’s disbelief and create a sense of immersion, allowing them to temporarily escape from reality and enter the fictional universe of the story. Whether it’s a fantastical adventure, a historical drama, or a science fiction epic, stories have the power to captivate the imagination and make the audience feel like they are part of the narrative unfolding before them.

Shared Experience: Stories create shared experiences that bring people together and foster a sense of community and connection. Whether it’s through books, films, theater, or oral storytelling traditions, stories have been passed down through generations, uniting people across cultures, languages, and generations through a common thread of shared narratives and shared emotions.

Overall, stories have a profound impact on the human psyche, offering a means of escape, exploration, and connection that transcends the boundaries of time and space. Through storytelling, audiences are transported to another world where they can laugh, cry, learn, and grow alongside the characters, creating memorable and meaningful experiences that linger long after the story ends.

Question 14.
What is a Capstone project? Give any two examples.
Answer:
A Capstone project is a culminating academic project that integrates and applies the knowledge and skills acquired throughout a course of study or program. It serves as a comprehensive demonstration of the student’s mastery of the subject matter and their ability to solve real-world problems, conduct independent research, and produce high-quality work. Capstone projects are often undertaken in the final year or semester of an academic program and may take various forms depending on the discipline and requirements of the institution.

Examples of Capstone projects:
Business and Management: In a business or management program, a Capstone project might involve developing a comprehensive business plan for a startup venture or an existing company. Students would conduct market research, analyze industry trends, develop financial projections, and create a strategic roadmap for launching or expanding the business. The project may culminate in a formal presentation to a panel of industry experts or potential investors.

Engineering and Technology: In an engineering or technology program, a Capstone project could involve designing and prototyping a new product or system to address a specific engineering challenge or societal need. For example, students might design a renewable energy system, develop a mobile application to improve healthcare delivery, or build a prototype for a sustainable transportation solution. The project would typically involve research, design, testing, and iteration, with the goal of producing a functional prototype or proof-of-concept demonstration.

These are just two examples of the many types of Capstone projects that students may undertake across various academic disciplines. The specific nature and scope of a Capstone project can vary widely depending on factors such as the academic program, student interests, faculty expertise, and industry partnerships.

Question 15.
Name the two techniques that can be used to validate AI model quality.
Answer:
Two techniques commonly used to validate the quality of an AI model are:
Cross-Validation: Cross-validation is a technique used to assess how well a predictive model generalizes to new, unseen data. It involves splitting the available dataset into multiple subsets, typically referred to as folds. The model is trained on a portion of the data (training set) and evaluated on the remaining data (validation set). This process is repeated multiple times, with each fold serving as both the training and validation set in turn. Cross-validation helps to estimate the model’s performance more accurately by reducing the variance introduced by a single train-test split.

Holdout Validation: Holdout validation is a simple technique where the available dataset is divided into two subsets: a training set and a validation set (or test set). The model is trained on the training set and then evaluated on the validation set to assess its performance. Holdout validation provides a straightforward way to estimate how well the model generalizes to new data. However, the performance estimate may be sensitive to the particular split of the data, especially with smaller datasets.

Question 16.
Name any two open frameworks and two development tools that can be used to build an AI model.
Answer:
Two open frameworks commonly used for building AI models are:
TensorFlow: TensorFlow is an open-source machine learning framework developed by Google Brain. It provides a comprehensive ecosystem of tools, libraries, and resources for building and deploying machine learning models across a range of domains, including deep learning, neural networks, and reinforcement learning. TensorFlow offers flexibility, scalability, and support for both research and production-level applications.

PyTorch: PyTorch is an open-source machine learning library developed by Facebook’s AI Research lab (FAIR). It is known for its dynamic computation graph, which allows for more flexibility and intuitive model development compared to static graph frameworks like, TensorFlow. PyTorch is widely used for research, prototyping, and production deployment of deep learning models.

Two development tools commonly used for building AI models are:

Jupyter Notebook: Jupyter Notebook is an open-source web application that allows users to create and share documents containing live code, equations, visualizations, and narrative text. It supports various programming languages, including Python, R, and Julia, making it a popular choice for interactive data analysis, prototyping, and collaborative research in AI and machine learning projects.

scikit-learn: scikit-learn is an open-source machine learning library for Python that provides simple and efficient tools for data mining and analysis. It includes a wide range of algorithms for classification, regiression, clustering, dimensionality reduction, and model evaluation, as well as utilities for data preprocessing, feature extraction, and model selection. scikit-learn is widely used for building and deploying machine learning models in both research and production environments.

Answer any 3 out of the given 5 questions in 50-80 words each. (3 × 4 = 12)

Question 17.
List any four importance of Data Storytelling.
Answer:
Data stopytelling is a powerful technique that enables organizations to effectively communicate insights and findings derived from data. Here are four key reasons why data storytelling is important:
Enhances Understanding: Data storytelling helps simplify complex data and analytical findings into compelling narratives that are easy to understand and interpret by non-technical audiences. By presenting data in the form of a story, with context, characters, and a plot, it enables stakeholders to grasp the significance of the data and its implications more intuitively, leading to better decision-making.

Drives Engagement: Data storytelling captivates the audience’s attention and fosters emotional engagement by weaving together data, visuals, and narrative elements into a cohesive and persuasive storyline. By appealing to both the rational and emotional aspects of human cognition, data storytelling makes data more relatable, memorable, and impactful, encouraging active participation and buy-in from stakeholders.

Facilitates Action: Effective data storytelling goes beyond merely presenting information-it motivates and inspires action by framing data-driven insights in the context of real-world challenges, opportunities, and goals. By highlighting the implications of the data and suggesting actionable recommendations or next steps, data storytelling empowers decision-makers to translate insights into tangible outcomes and drive positive change within their organizations.

Builds Trust: Data storytelling builds trust and credibility by providing transparency and context around the data analysis process, assumptions, and limitations. By openly acknowledging uncertainties and potential biases in the data, and presenting findings in a clear, honest, and compelling manner, data storytellers can earn the trust of their audience and foster a culture of data-driven decision-making within their organizations.

Question 18.
What is Design Thinking? List its main stages.
Answer:
Design Thinking is a human-centered approach to problem-solving and innovation that emphasizes empathy, creativity, and collaboration to develop: innovative solutions to complex challenges. It involves understanding the needs and perspectives of end-users, generating creative ideas, prototyping and testing solutions iteratively, and refining designs based on feedback. Design Thinking is widely used in various fields, including product design, service design, business strategy, and social innovation.

The main stages of Design Thinking typically include:
Empathize: In this stage, designers seek to understand the needs, motivations, and behaviors of the people they are designing for. This involves conducting research, observing users in their natural environment, and engaging in empathy-building activities to gain insights into users’ experiences and pain points.

Define: In this stage, designers synthesize the research findings from the Empathize stage to define the problem’statement or design challenge they are addressing. This involves reframing the problem from the perspective of the user and identifying specific needs, goals, and constraints that the design solution should address.

Ideate: In this stage, designers brainstorm and generate a wide range of creative ideas and potential solutions to address the problem defined in the previous stage. This involves encouraging divergent thinking, suspending judgment, and exploring unconventional ideas through techniques such as brainstorming, mind mapping, and rapid prototyping.

Prototype: In this stage, designers create tangible representations of their ideas and concepts to test and refine them. Prototypes can take various forms, from rough sketches and mockups to interactive prototypes and physical models. The goal is to quickly and cheaply explore different design possibilities and gather feedback from users to inform further iteration.

Test: In this stage, designers test the prototypes with real users to gather feedback, evaluate usability, and validate assumptions. This involves conducting user tests, usability studies, and observational research to identify strengths and weaknesses in the design and iteratively refine the prototypes based on user insights.

Iterate: Design Thinking is an iterative process, and the stages described above are often repeated multiple times as designers refine and improve their solutions based on feedback and insights gained through testing. Iteration allows designers to continuously learn and evolve their designs to better meet the needs of users and address emerging challenges and opportunities.

By following these stages of Design Thinking, designers can develop innovative solutions that are user-centered, feasible, and viable, ultimately leading to more successful outcomes and greater impact.

Question 19.
Explain the ‘Design/Building the Model” step of the AI Model lifecycle in detail.
Answer:
The “Design/Building the Model” step in the AI Model lifecycle involves the process of designing and constructing the machine learning model based on the defined problem statement or objective. This step is crucial as it lays the foundation for the entire model development process and directly influences the model’s performance, accuracy, and effectiveness in solving the problem at hand. Here is a detailed explanation of this step:

Problem Definition: The first step in designing the model is to clearly define the problem statement or objective that the model aims to address. This involves understanding the business context, identifying key stakeholders, and specifying the goals, constraints, and requirements of the project. The problem definition serves as a guiding framework for designing the model and determining the appropriate approach and techniques to be used.

Data Collection and Preparation: Once the problem is defined, the next step is to collect and preprocess the relevant data needed to train and evaluate the model. This involves gathering data from various sources, cleaning and preprocessing the data to handle missing values, outliers, and inconsistencies, and transforming the data into a suitable format for model training. Data preparation is critical as the quality and completeness of the data directly impact the performance and generalization ability of the model.

Feature Engineering: Feature engineering involves selecting, creating, and transforming the input features (variables) in the dataset to enhance the predictive power of the model. This may include selecting relevant features, encoding categorical variables, scaling numerical features, and creating new features through transformations, aggregations, or interactions. Effective feature engineering plays a crucial role in improving the model’s performance and robustness.

Model Selection and Architecture Design: In this step, the appropriate machine learning algorithm or model architecture is selected based on the nature of the problem, the characteristics of the data, and the desired outcome. This may involve choosing between different types of models such as regression, classification, clustering, or deep learning models, as well as selecting the specific parameters and hyperparameters of the chosen model. The model architecture is designed to optimize performance metrics such as accuracy, precision, recall, or AUC (Area Under the ROC Curve).

Model Training: Once the model architecture is defined, the next step is to train the model using the prepared dataset. This involves feeding the training data into the model, adjusting the model parameters iteratively through optimization algorithms such as gradient descent, and minimizing the loss function to learn the underlying patterns and relationships in the data. The model training process aims to optimize the model’s performance on the training data while avoiding overfitting, where the model memorizes the training data and fails to generalize to new, unseen data.

Model Evaluation: After the model is trained, it is evaluated using a separate validation dataset to assess its performance, generalization ability, and robustness. This involves measuring various performance metrics such as accuracy, precision, recall, F1-score, or ROC-AUC on the validation data and comparing them against predefined thresholds or benchmarks. Model evaluation helps identify any issues or deficiencies in the model’s performance and guides further iterations or improvements to the model design.

Model Deployment: Once the model has been designed, trained, and evaluated, it is ready for deployment in real-world applications. Model deployment involves integrating the trained model into the production environment, deploying it on the appropriate infrastructure (e.g., cloud, edge devices), and implementing mechanisms for monitoring, maintenance, and updates. Successful deployment of the model enables it to generate predictions or insights in real-time and deliver value to end-users or stakeholders.

Overall, the “Design/Building the Model” step in the AI Model lifecycle is a systematic and iterative process that involves defining the problem, collecting and preparing data, engineering features, selecting and designing the model architecture, training and evaluating the model, and deploying it for practical use. Each sub-step in this process plays a crucial role in ensuring the effectiveness, accuracy, and reliability of the AI model in solving real-world problems and delivering actionable insights.

Question 20.
Expand and explain the term MSE. Give the mathematical formula to calculate MSE. Why use MSE? Briefly discuss.
Answer:
MSE stands for Mean Squared Error, which is a commonly used metric in statistics and machine learning to evaluate the performance of regressiort models. It measures the average squared difference between the predicted values generated by a model and the actual observed values in a dataset.

The mathematical formula to calculate MSE is as follows:

Where:

• n is the total number of data points in the dataset.
• y i represents the actual (observed) value of the target variable for the i th data point.
• ^ y^ i represents the predicted value of the target variable for the i th data point.

To calculate MSE, you take the squared difference between each actual and predicted value, sum these squared differences across all data points, and then divide by the total number of data points.

MSE is commonly used for several reasons:
Sensitivity to Errors: Squaring the errors in the MSE formula amplifies larger errors, making MSE particularly sensitive to outliers or large deviations between actual and predicted values. This property makes it useful for detecting and penalizing models that perform poorly on specific data points, helping to identify areas where the model needs improvement.

Differentiability: MSE is a differentiable function, which means it can be easily optimized using gradient-based optimization algorithms like gradient descent. This makes MSE suitable for use as a loss function during the training of machine learning models, allowing the model parameters to be updated iteratively to minimize the MSE and improve model performance.

Mathematically Intuitive: MSE has a clear and intuitive interpretation, representing the average squared deviation between predicted and actual values. This makes it easy to understand and interpret, both for practitioners and stakeholders involved in model evaluation and decisionmaking.

Widely Used: MSE is a widely used and accepted metric in the fields of statistics and machine learning, making it easy to compare model performance across different studies, datasets, and applications. Its ubiquity also makes it a standard choice for model evaluation and selection in many machine learning projects.

Overall, MSE provides a simple yet effective measure of the accuracy and reliability of regression models, allowing practitioners to assess model performance, identify areas for improvement, and make informed decisions based on the quality of predictions generated by the model.

Question 21.
(a) Why Storytelling is so powerful and cross-cultural? Explain
(b) Which of the following is a better data story? Give reasons.

image 1.

image 2.

Answer:
(a) Stories are a powerful communication tool in a multicultural workplace, because they enable listeners to make connections between what is said and their own experiences, facilitating understanding of important meanings, beliefs, and behaviors from different cultures.

Cross-cultural communication means being fully aware of the differences and having respect for the values, customs, beliefs, and behaviours that shape the way people see in the world and interact with others.

The Key Elements of Effective Cross-Cultural Communication

Awareness: Understanding of how the societal norms, beliefs, rituals and etiquettes of individuals from different cultures might impact communication.

Language: Being able to communicate effectively in the same language as your audience or making use of interpreters to avoid misunderstandings.

Listening: Being present and listening to what someone is saying, while also paying attention to non-verbal forms of communication like facial expressions, hand gestures, posture, eye contact, tone of voice, etc.

Respect: Being mindful of and respecting the differences in cultural backgrounds and making efforts to avoid using offensive language or performing actions that might be deemed insensitive.

Openness: Being open and adaptable to new ways of communicating and expressing oneself, with a willingness to learn about other ways of doing things.
Answer:
(b) Because Image 2’s data story on the observations drawn from the image is well articulated, it has a superior data story than Image 1.

The post CBSE Class 12 AI Question Paper 2024 with Solutions appeared first on Learn CBSE.

NCERT Books for Class 4 All Subjects

NCERT Books for Class 4 हिंदी

रिमझिम

• पाठ 1: मन के भोले-भाले बादल
• पाठ 2: जैसा सवाल वैसा जवाब
• पाठ 3: किरमिच की गेंद
• पाठ 4: पापा जब बच्चे थे
• पाठ 5: दोस्त की पोशाक
• पाठ 6: नाव बनाओ नाव बनाओ
• पाठ 7: दान का हिसाब
• पाठ 8: कौन?
• पाठ 9: स्वतंत्रता की ओर
• पाठ 10: थप्प रोटी थप्प दाल
• पाठ 11: पढ़क्कू की सूझ
• पाठ 12: सुनीता की पहिया कुर्सी
• पाठ 13: हुदहुद
• पाठ 14: मुफ़्त ही मुफ़्त

NCERT Books for Class 4 English

Marigold

• Unit 1:
  • Wake Up!
  • Neha’s Alarm Clock
• Unit 2:
  • Noses
  • The Little Fir Tree
• Unit 3:
  • Run!
  • Nasruddin’s Aim
• Unit 4:
  • Why?
  • Alice in Wonderland
• Unit 5:
  • Don’t be Afraid of the Dark
  • Helen Keller
• Unit 6:
  • The Donkey
  • I had a Little Pony
  • The Milkman’s Cow
• Unit 7:
  • Hiawatha
  • The Scholar’s Mother Tongue
• Unit 8:
  • A Watering Rhyme
  • The Giving Tree
• Unit 9:
  • Books
  • Going to Buy a Book
• Unit 10:
  • The Naughty Boy
  • Pinocchio

NCERT Books for Class 4 Maths – English Medium

Maths Mela

• Chapter 1 Shapes Around Us
• Chapter 2 Hide and Seek
• Chapter 3 Pattern Around Us
• Chapter 4 Thousands Around Us
• Chapter 5 Sharing and Measuring
• Chapter 6 Measuring Length
• Chapter 7 The Cleanest Village
• Chapter 8 Weigh it, Pour it
• Chapter 9 Equal Groups
• Chapter 10 Elephants, Tigers, and Leopards
• Chapter 11 Fun with Symmetry
• Chapter 12 Ticking Clocks and Turning Calendar
• Chapter 13 The Transport Museum
• Chapter 14 Data Handling

Maths Magic

• Chapter 1: Building with Bricks
• Chapter 2: Long and Short
• Chapter 3: A Trip to Bhopal
• Chapter 4: Tick-Tick-Tick
• Chapter 5: The Way The World Looks
• Chapter 6: The Junk Seller
• Chapter 7: Jugs and Mugs
• Chapter 8: Carts and Wheels
• Chapter 9: Halves and Quarters
• Chapter 10: Play with Patterns
• Chapter 11: Tables and Shares
• Chapter 12: How Heavy? How Light?
• Chapter 13: Fields and Fences
• Chapter 14: Smart Charts

NCERT Books for Class 4 Maths – Hindi Medium

गणित का जादू

• अध्याय 1: ईंटों से बनी इमारत
• अध्याय 2: लंबा और छोटा
• अध्याय 3: भोपाल की सैर
• अध्याय 4: टिक टिक टिक
• अध्याय 5: दुनिया कुछ ऐसी दिखती है
• अध्याय 6: कबाड़ीवाली
• अध्याय 7: जग मग, जग मग
• अध्याय 8: गाड़ियाँ और पहिए
• अध्याय 9: आधा और चौथाई
• अध्याय 10: पैटर्न
• अध्याय 11: पहाड़े और बँटवारे
• अध्याय 12: कितना भारी? कितना हल्का?
• अध्याय 13: खेत और बाड़
• अध्याय 14: स्मार्ट चार्ट

NCERT Books for Class 4 EVS – English Medium

Environmental Studies – Looking Around

• Chapter 1. Going to School
• Chapter 2. Ear to Ear
• Chapter 3. A Day with Nandu
• Chapter 4. The Story of Amrita
• Chapter 5. Anita and the Honeybees
• Chapter 6. Omana’s Journey
• Chapter 7. From the Window
• Chapter 8. Reaching Grandmother’s House
• Chapter 9. Changing Families
• Chapter 10. Hu Tu Tu, Hu Tu Tu
• Chapter 11. The Valley of Flowers
• Chapter 12. Changing Times
• Chapter 13. A River’s Tale
• Chapter 14. Basva’s Farm
• Chapter 15. From Market to Home
• Chapter 16. A busy Month
• Chapter 17. Nandita in Mumbai
• Chapter 18. Too Much Water, Too Little Water
• Chapter 19. Abdul in the Garden
• Chapter 20. Eating Together
• Chapter 21. Food and Fun
• Chapter 22. The World in my Home
• Chapter 23. Pochampalli
• Chapter 24. Home and Abroad
• Chapter 25. Spicy Riddles
• Chapter 26. Defence Officer: Wahida
• Chapter 27. Chuskit Goes to School

NCERT Books for Class 4 EVS – Hindi Medium

पर्यावरण अध्ययन – आस-पास

• पाठ 1: चलों, चलें स्कूल!
• पाठ 2: कान-कान में
• पाठ 3: नंदू हाथी
• पाठ 4: अमृता की कहानी
• पाठ 5: अनीता की मधुमक्खियाँ
• पाठ 6: ओमना का सफ़र
• पाठ 7: खिड़की से
• पाठ 8: नानी के घर तक
• पाठ 9: बदलते परिवार
• पाठ 10: हु तू तू , हु तू तू
• पाठ 11: फुलवारी
• पाठ 12: कैसे-कैसे बदले घर
• पाठ 13: पहाड़ों से समुंदर तक
• पाठ 14: बसवा का खेत
• पाठ 15: मंडी से घर तक
• पाठ 16: चूँ-चूँ करती आई चिड़िया
• पाठ 17: नंदिता मुंबई में
• पाठ 18: पानी कहीं ज़्यादा, कहीं कम
• पाठ 19: जड़ों का जाल
• पाठ 20: मिलकर खाएँ
• पाठ 21: खाना-खिलाना
• पाठ 22: दुनिया मेरे घर में
• पाठ 23: पोचमपल्ली
• पाठ 24: दूर देश की बात
• पाठ 25: चटपटी पहेलियाँ!
• पाठ 26: फ़ौजी वहीदा
• पाठ 27: कोशिश हुई कामयाब

The post NCERT Books for Class 4 appeared first on Learn CBSE.

NCERT Books for Class 6 All Subjects

Ganita Prakash Class 6 Maths NCERT Book

• Class 6 Maths Index
• Chapter 1 Patterns in Mathematics
• Chapter 2 Lines and Angles
• Chapter 3 Number Play
• Chapter 4 Data Handling and Presentation
• Chapter 5 Prime Time
• Chapter 6 Perimeter and Area
• Chapter 7 Fractions
• Chapter 8 Playing with Constructions
• Chapter 9 Symmetry
• Chapter 10 The Other Side of Zero

Ganita Prakash Class 6 Maths NCERT Book in Hindi Medium

• Class 6 Maths HM Index
• Chapter 1 गणित में पैटर्न
• Chapter 2 रेखाएँ और कोण
• Chapter 3 संख्याओं का खेल
• Chapter 4 आकँड़ों का प्रबंधन और प्रस्तुतिकरण
• Chapter 5 अभाज्य समय
• Chapter 6 परिमाप और क्षेत्रफल
• Chapter 7 भिन्न
• Chapter 8 रचनाओं के साथ खेलना
• Chapter 9 सममिति
• Chapter 10 शून्य के दूसरी ओर

Also Read Class 6

• NCERT Solutions for Class 6
• NCERT Solutions for Class 6 Maths (Ganita Prakash)
• NCERT Solutions for Class 6 Science (Curiosity)
• NCERT Solutions for Class 6 Social Science (Exploring Society: India and Beyond)
• NCERT Solutions for Class 6 English (Poorvi)
• NCERT Solutions for Class 6 Hindi (Malhar मल्हार)
• NCERT Solutions for Class 6 Sanskrit (Deepakam दीपकम्)

Curiosity Class 6 Science NCERT Book

• Class 6 Science Index
• Chapter 1 The Wonderful World of Science
• Chapter 2 Diversity in the Living World
• Chapter 3 Mindful Eating: A Path to a Healthy Body
• Chapter 4 Exploring Magnets
• Chapter 5 Measurement of Length and Motion
• Chapter 6 Materials Around Us
• Chapter 7 Temperature and its Measurement
• Chapter 8 A Journey through States of Water
• Chapter 9 Methods of Separation in Everyday Life
• Chapter 10 Living Creatures: Exploring their Characteristics
• Chapter 11 Nature’s Treasures
• Chapter 12 Beyond Earth

Jigyasa Class 6 Science NCERT Book

• Class 6 Science HM Index
• Chapter 1 विज्ञान का अनूठा संसार
• Chapter 2 सजीव जगत में विविधता
• Chapter 3 उचित आहार – स्वस्थ शरीर का आधार
• Chapter 4 चुंबकों को जानें
• Chapter 5 लंबाई एवं गति का मापन
• Chapter 6 हमारे आस-पास की सामग्री
• Chapter 7 ताप एवं उसका मापन
• Chapter 8 जल की विविध अवस्थाओं की यात्रा
• Chapter 9 दैनिक जीवन में पृथक्करण विधियाँ
• Chapter 10 सजीव – विशेषताओं का अन्वेषण
• Chapter 11 प्रकृति की अमूल्य संपदा
• Chapter 12 पृथ्वी से परे

Exploring Society: India and Beyond Class 6 Social Science NCERT Book

• Class 6 Social Science Index

Theme A – India and the World: Land and the People

• Chapter 1 Locating Places on the Earth
• Chapter 2 Oceans and Continents
• Chapter 3 Landforms and Life

Theme B – Tapestry of the Past

• Chapter 4 Timeline and Sources of History
• Chapter 5 India, That is Bharat
• Chapter 6 The Beginnings of Indian Civilisation

Theme C – Our Cultural Heritage and Knowledge Traditions

• Chapter 7 India’s Cultural Roots
• Chapter 8 Unity in Diversity, or ‘Many in the One’

Theme D – Governance and Democracy

• Chapter 9 Family and Community
• Chapter 10 Grassroots Democracy Part 1: Governance
• Chapter 11 Grassroots Democracy Part 2: Local Government in Rural Areas
• Chapter 12 Grassroots Democracy Part 3: Local Government in Urban Areas

Theme E – Economic Life Around Us

• Chapter 13 The Value of Work
• Chapter 14 Economic Activities Around Us

Samaj Ka Aadhyan: Bharat or Uske Aage Class 6 Social Science NCERT Book

• Class 6 Social Science HM Index

विषय (क) – भारत एवं विश्व: भूभाग एवं उनके निवासी

• Chapter 1 पृथ्वी पर स्थानों की स्थिति
• Chapter 2 महासागर एवं महाद्वीप
• Chapter 3 स्थलरूप एवं जीवन

विषय (ख) – अतीत के चित्रपट

• Chapter 4 इतिहास की समय-रेखा एवं उसके स्रोत
• Chapter 5 इंडिया, अर्थात भारत
• Chapter 6 भारतीय सभ्यता का प्रारंभ

विषय (ग) – हमारी सांस्कृतिक विरासत एवं ज्ञान परंपर

• Chapter 7 भारत की सांस्कृतिक जड़ें
• Chapter 8 विविधता में एकता या ‘एक में अनेक’

विषय (घ) – शासन और लोकतंत्र

• Chapter 9 परिवार और समुदाय
• Chapter 10 आधारभूत लोकतंत्र – भाग 1: शासन
• Chapter 11 आधारभूत लोकतंत्र – भाग 2: ग्रामीण क्षेत्रों में स्थानीय सरकार
• Chapter 12 आधारभूत लोकतंत्र – भाग 3: नगरीय क्षेत्रों में स्थानीय सरकार

विषय (ङ) – हमारे आस-पास का आर्थिक जीवन

• Chapter 13 कार्य का महत्व
• Chapter 14 हमारे आस-पास की आर्थिक गतिविधियाँ

Malhar Class 6 Hindi NCERT Book

• Class 6 Hindi Index
• Chapter 1 मातृभूमि (कविता)
• Chapter 2 गोल (संस्मरण)
• Chapter 3 पहली बूँद (कविता)
• Chapter 4 हार की जीत (कहानी)
• Chapter 5 रहीम के दोहे (दोहे)
• Chapter 6 मेरी माँ (आत्मकथा)
• Chapter 7 जलाते चलो (कविता)
• Chapter 8 सत्रिया और बिहू नृत्य (निबंध)
• Chapter 9 मैया मैं नहिं माखन खायो (पद)
• Chapter 10 परीक्षा (कहानी)
• Chapter 11 चेतक की वीरता (कविता)
• Chapter 12 हिंद महासागर में छोटा-सा हिंदुस्तान (यात्रा वृतांत)
• Chapter 13 पेड़ की बात (निबंध)

Poorvi Class 6 English NCERT Book

• Class 6 English Index
• Unit 1 Fables and Folk Tales
• Unit 2 Friendship
• Unit 3 Nurturing Nature
• Unit 4 Sports and Wellness
• Unit 5 Culture and Tradition

Deepakam Class 6 Sanskrit NCERT Book

• Class 6 Sanskrit Index
• Chapter 1 वयं वर्णमालां पठामः
• Chapter 2 एषः कः ? एषा का ? एतत् किम् ?
• Chapter 3 अहं च त्वं च
• Chapter 4 अहं प्रातः उत्तिष्ठामि
• Chapter 5 शूराः वयं धीराः वयम्
• Chapter 6 सः एव महान् चित्रकारः
• Chapter 7 अतिथिदेवो भव
• Chapter 8 बुद्धिः सर्वार्थसाधिका
• Chapter 9 यो जानाति सः पण्डितः
• Chapter 10 त्वम् आपणं गच्छ
• Chapter 11 पृथिव्यां त्रीणि रत्नानि
• Chapter 12 आलस्यं हि मनुष्याणां शरीरस्थो महान् रिपुः
• Chapter 13 सङ्ख्यागणना ननु सरला
• Chapter 14 माधवस्य प्रियम् अङ्गम्
• Chapter 15 वृक्षाः सत्पुरुषाः इव

Khayal Class 6 Urdu NCERT Book

• Class 6 Urdu Index
• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11
• Chapter 12
• Chapter 13
• Chapter 14

Kriti – I Textbook for Arts Class 6

• Class 6 Arts Index

Visual Arts

• Chapter 1 Objects and Still Life
• Chapter 2 Changing the Typical Picture
• Chapter 3 Portraying People
• Chapter 4 Paper Crafts
• Chapter 5 Seals to Prints

Music

• Chapter 6 Music and your Emotions
• Chapter 7 Musical Instruments
• Chapter 8 Taal or Talam and Raga or Ragam in Indian Music
• Chapter 9 Melodies of Diversity
• Chapter 10 Songwriting
• Chapter 11 Music and Society

Dance

• Chapter 12 My Body in Motion
• Chapter 13 Breaking Barriers with Dance
• Chapter 14 Harmony in Motion
• Chapter 15 Dances of Our Land

Theatre

• Chapter 16 Emotions Unveiled!
• Chapter 17 Let’s Design
• Chapter 18 In the Company of Theatre
• Chapter 19 Stories of Shadows and Strings – Puppetry
• Chapter 20 The Grand Finale
• Chapter 21 Integration of All Art Forms
• Chapter 22 Assessment

Kaushal Bodh Vocational Education Activity Book for Class 6

• Class 6 Vocational Education Index

Part 1 Work with Life Forms

• Project 1 School Kitchen Garden
• Project 2 Biodiversity Register

Part 2 Work with Machines and Materials

• Project 3 Maker Skills
• Project 4 Animation and Games

Part 3 Work in Human Services

• Project 5 School Museum
• Project 6 Cooking without Fire

Kaushal Bodh Vocational Education Activity Book for Class 6 in Hindi Medium

भाग 1 – जीव रूपों के साथ कार्य करना

• Chapter 1 विद्यालयी रसोई उद्यान (किचन गार्डन)
• Chapter 2 जैव विविधता विवरणिका

भाग 2 – मशीनों और उपकरणों के साथ कार्य करना

• Chapter 3 निर्माता कौशल (मेकर स्किल्स)
• Chapter 4 एनिमेशन और खेल (गेम्स)

भाग 3 – मानव सेवाओं में कार्य करना

• Chapter 5 विद्यालयी संग्रहालय
• Chapter 6 आग के बिना खाना बनाना

The post NCERT Books for Class 6 All Subjects appeared first on Learn CBSE.

Class 7 Maths Chapter 1 Notes Large Numbers Around Us

Class 7 Maths Notes Chapter 1 – Class 7 Large Numbers Around Us Notes

→ We came across large numbers — lakhs, crores, and arabs; millions and billions.

→ We learnt how to read and write these numbers in the Indian and American/International naming systems.

→ 1 lakh is 1 followed by 5 zeroes: 1,00,000

→ 1 crore is 1 followed by 7 zeroes: 1,00,00,000

→ 1 million is 1 followed by 6 zeroes: 1,000,000 (which is also ten lakhs)

→ 1 arab is 1 followed by 9 zeroes: 1,000,000,000 (which is also 100 crore or 1 billion)

→ We generally round up or round down large numbers. It is often enough to know roughly how big or small something is.

→ To get a sense of large numbers or quantities, we can check how many times bigger they are compared to numbers or quantities that are more familiar.

→ We saw how to factorise numbers and regroup them to simplify multiplications.

→ We carried out interesting thought experiments, such as “Would one be able to watch 1000 movies in a year?”

A Lakh Varieties! Class 7 Notes

Eshwarappa is a farmer in Chintamani, a town in Karnataka. He visits the market regularly to buy seeds for his rice field. During one such visit, he overheard a conversation between Ramanna and Lakshmamma. Ramanna said, “Earlier, our country had about a lakh varieties of rice. Farmers used to preserve different varieties of seeds and use them to grow rice. Now, we only have a handful of varieties. Also, farmers have to come to the market to buy seeds”.

Lakshmamma said, “There is a seed bank near my house. So far, they have collected about a hundred indigenous varieties of rice seeds from different places. You can also buy seeds from there.” You may have heard the word ‘lakh’ before. Do you know how big one lakh is? Let us find out.

Eshwarappa shared this incident with his daughter, Roxie, and son Estu. Estu was surprised to know that there were about one lakh varieties of rice in this country. He wondered, “One lakh! So far, I have only tasted 3 varieties. If we tried a new variety each day, would we even come close to tasting all the varieties in a lifetime of 100 years?” What do you think? Guess. But how much is one lakh?

Roxie and Estu found that if they ate one variety of rice a day, they would come nowhere close to a lakh in a lifetime! Roxie suggests, “What if we ate 2 varieties of rice every day? Would we then be able to eat 1 lakh varieties of rice in 100 years?”

What if a person ate 3 varieties of rice every day? Will they be able to taste all the lakh varieties in a 100-year lifetime? Find out.
Estu said, “We know how many days there are in a year — 365, if we ignore leap years. If we live for y years, the number of days in our lifetime will be 365 × y.”

Getting a Feel of Large Numbers
You may have come across interesting facts like these:

• The world’s tallest statue is the ‘Statue of Unity’ in Gujarat, depicting Sardar Vallabhbhai Patel. Its height is about 180 metres.
• Kunchikal waterfall in Karnataka is said to drop from a height of about 450 metres.

It is not always easy to get a sense of how big these measurements are. But we can get a better sense of their size when we compare them with something familiar.

Is One Lakh a Very Large Number?
Eshwarappa asked Roxie and Estu, “Is a lakh big or small?”

Roxie feels that 1 lakh is a large number:

• “We had one lakh varieties of rice — that is a lot.”
• “Living 1 lakh days would mean living for 274 years—that is a long time!”
• “If 1 lakh people stood shoulder to shoulder in a line, they could stretch as far as 38 kilometres.”

Estu, however, thinks it is not that big:

• “Do you know that the cricket stadium in Ahmedabad has a seating capacity of more than 1 lakh? One lakh people in such a small area!”
• “Most humans have 80,000 to 1,20,000 hairs on their heads. Imagine, 1 lakh hairs fit in such a tiny space!”
• “I heard that there are some species of fish where a female fish can lay almost one lakh eggs at once, very comfortably. Some even lay tens of lakhs of eggs at a time.”

Reading and Writing Numbers
We have already been using commas for 5-digit numbers like 45,830 in the Indian place value system. As numbers grow bigger, using commas helps in reading the numbers easily. We use a comma in between the digits representing the “ten thousands” place and the “one lakh” place, as you have seen just before (1,00,000). The number name of 12,78,830 is twelve lakh seventy eight thousand eight hundred thirty. Similarly, the number 15,75,000 in words is fifteen lakh seventy-five thousand.

Land of Tens Class 7 Notes

Systematic Sippy is a different kind of calculator. It has the following buttons: +1, +10, +100, +1000, +10000, +100000. It wants to be used as minimally as possible.

What if we press the +10,00,000 button ten times? What number will come up? How many zeroes will it have? What should we call it?
The number will be 100 lakhs, which is also called a crore. 1 crore is written as 1,00,00,000—it is 1 followed by seven zeroes.

Of Crores and Crores! Class 7 Notes

The table below shows some numbers according to both the Indian system and the American system (also called the International system) of naming numerals and placing commas. Observe the placement of commas in both systems.

Notice that in the Indian system, commas are placed to group the digits in a 3-2-2-2… pattern from right to left (thousands, lakhs, crores, etc.). In the American system, the digits are grouped uniformly in a 3-3-3-3… pattern from right to left (thousands, millions, billions, etc.).

The Indian system of naming numbers is also followed in Bhutan, Nepal, Sri Lanka, Pakistan, Bangladesh, the Maldives, Afghanistan, and Myanmar. The words lakh and crore originate from the Sanskrit words lakṣha (लक्ष and koṭi (कोटि‍). The American system is also used in many countries.
Observe the number of zeroes in 1 lakh and 1 crore.

• 1 lakh, written in numbers, would be 1 followed by 5 zeroes.
• 1 crore, written in numbers, would be 1 followed by 7 zeroes.

A lakh is a hundred times a thousand, a crore is a hundred times a lakh, and an arab is a hundred times a crore (i.e., a hundred thousand is 1 lakh, 100 lakhs is 1 crore, and 100 crores is 1 arab).

Exact and Approximate Values Class 7 Notes

What do you think of this conversation? Have you read or heard such headlines or statements?
Very often, exact numbers are not required and just an approximation is sufficient. For example, according to the 2011 census, the population of Chintamani town is 76,068. Instead, saying that the population is about 75,000 is enough to give an idea of how big the quantity is.

There are situations where it makes sense to round up a number (rounding up is when the approximated number is more than the actual number).
For example, if a school has 732 people, including students, teachers, and staff, the principal might order 750 sweets instead of 700 sweets.

There are situations where it is better to round down (rounding down is when the approximated number is less than the actual number).
For example, if the cost of an item is ₹470, the shopkeeper may say that the cost is around ₹450 instead of saying it is around ₹500.

Nearest Neighbours
With large numbers it is useful to know the nearest thousand, lakh or crore. For example, the nearest neighbours of the numbers 6,72,85,183 are shown in the table below.

Patterns in Products Class 7 Notes

Roxie and Estu are playing with multiplication. They encounter an interesting technique for multiplying a number by 10, 100, 1000, and so on.

A Multiplication Shortcut
Roxie evaluated 116 × 5 as follows:
116 × 5 = 116 × \(\frac{10}{2}\)
= 58 × 10
= 580

Estu evaluated 824 × 25 as follows:
824 × 25 = 824 × \(\frac{100}{4}\) = 20600

Fascinating Facts about Large Numbers
Some interesting facts about large numbers are hidden below. Calculate the product or quotient to uncover the facts. Once you find the product or quotient, read the number in both Indian and American naming systems. Share your thoughts and questions about the fact with the class after you discover each number.

1250 × 380 is the number of kirtanas composed by Purandaradasa according to legends. Purandaradasa was a composer and singer in the 15th century. His kirtanas spanned social reform, bhakti, and spirituality. He systematised methods for teaching Carnatic music, which are followed to the present day.

2100 × 70,000 is the approximate distance in kilometers, between the Earth and the Sun. This distance keeps varying throughout the year. The farthest distance is about 152 million kilometers.

6400 × 62,500 is the average number of litres of water the Amazon River discharges into the Atlantic Ocean every second. The river’s flow into the Atlantic is so much that drinkable freshwater is found even 160 kilometers into the open sea.

As you did before, divide the given numbers to uncover interesting facts about division. Share your thoughts and questions with the class after you uncover each number.

13,95,000 ÷ 150 is the distance (in kms) of the longest single-train journey in the world. The train runs in Russia between Moscow and Vladivostok. The duration of this journey is about 7 days. The longest train route in India is from Dibrugarh in Assam to Kanyakumari in Tamil Nadu; it covers 4219 kms in about 76 hours.

Adult blue whales can weigh more than 10,50,00,000 ÷ 700 kilograms. A newborn blue whale weighs around 2,700 kg, which is similar to the weight of an adult hippopotamus. The heart of a blue whale was recorded to be nearly 700 kg. The tongue of a blue whale weighs as much as an elephant. Blue whales can eat up to 3500 kg of krill every day. The largest known land animal, Argentinosaurus, is estimated to weigh 90,000 kg.

52,00,00,00,000 ÷ 130 was the weight, in tonnes, of global plastic waste generated in the year 2021.

Large Number Fact
In a single gram of healthy soil, there can be 100 million to 1 billion bacteria and 1 lakh to 1 million fungi, which can support plants’ growth and health. Share such large-number facts you know/come across with your class.

Did You Ever Wonder…? Class 7 Notes

Estu is amused by all these interesting facts about large numbers. While thinking about these, he came up with an unusual question, “Could the entire population of Mumbai fit into 1 lakh buses?” What do you think? How can we find out?

Let us assume a bus can accommodate 50 people. Then 1 lakh buses can accommodate 1 lakh × 50 = 50 lakh people.

The population of Mumbai is more than 1 crore 24 lakhs. So, the entire population of Mumbai cannot fit in 1 lakh buses.

Class 7 Maths Notes

The post Large Numbers Around Us Class 7 Notes Maths Chapter 1 appeared first on Learn CBSE.

Class 7 Maths Chapter 2 Notes Arithmetic Expressions

Class 7 Maths Notes Chapter 2 – Class 7 Arithmetic Expressions Notes

→ We have been reading and evaluating simple expressions for quite some time now. Here we started by revising the meaning of some simple expressions and their values.

→ We learnt how to compare certain expressions through reasoning instead of bluntly evaluating them.

→ To help read and evaluate complex expressions without confusion, we use terms and brackets.

→ When an expression is written as a sum of terms, changing the order of the terms or grouping the terms does not change the value of the expression. This is because of the “commutative property of addition” and the “associative property of addition”, respectively.

→ To evaluate expressions within brackets, we saw that when we remove brackets preceded by a negative sign, the terms within the brackets change their sign.

→ We also learnt about the “distributive property” — multiplying a number by an expression inside brackets is equal to multiplying the number by each term in the bracket.

Simple Expressions Class 7 Notes

You may have seen mathematical phrases like 13 + 2, 20 – 4, 12 × 5, and 18 ÷ 3. Such phrases are called arithmetic expressions.

Every arithmetic expression has a value, which is the number it evaluates to. For example, the value of the expression 13 + 2 is 15. This expression can be read as ‘13 plus 2’ or ‘the sum of 13 and 2’.

We use the equality sign ‘=’ to denote the relationship between an arithmetic expression and its value.
For example: 13 + 2 = 15.

Example 1.
Mallika spends ₹25 every day on lunch at school. Write the expression for the total amount she spends on lunch in a week from Monday to Friday.
Solution:
The expression for the total amount is 5 × 25.
5 × 25 is “5 times 25” or “the product of 5 and 25”.

Different expressions can have the same value. Here are multiple ways to express the number 12, using two numbers and any of the four operations +, –, ×, and ÷: 10 + 2, 15 – 3, 3 × 4, 24 ÷ 2.

Comparing Expressions
As we compare numbers using ‘=’, ‘<’, and ‘>’ signs, we can also compare expressions. We compare expressions based on their values and write the ‘equal to’, ‘greater than’, or ‘less than’ sign accordingly. For example, 10 + 2 > 7 + 1 because the value of 10 + 2 = 12 is greater than the value of 7 + 1 = 8. Similarly, 13 – 2 < 4 × 3.

Example 2.
Which is greater? 1023 + 125 or 1022 + 128?
Solution:
Imagining a situation could help us answer this without finding the values. Raja had 1023 marbles and got 125 more today. Now he has 1023 + 125 marbles. Joy had 1022 marbles and got 128 more today. Now he has 1022 + 128 marbles. Who has more?

This situation can be represented as shown in the picture on the right. To begin with, Raja had 1 more marble than Joy. But Joy got 3 more marbles than Raja today. We can see that Joy has (two) more marbles than Raja now. That is, 1023 + 125 < 1022 + 128.

Example 3.
Which is greater? 113 – 25 or 112 – 24?
Solution:
Imagine a situation, Raja had 113 marbles and lost 25 of them. He has 113 – 25 marbles. Joy had 112 marbles and lost 24 today. He has 112 – 24 marbles. Who has more marbles left with them?

Raja had 1 marble more than Joy. But he also lost 1 marble more than Joy did. Therefore, they have an equal number of marbles now. That is, 113 – 25 = 112 – 24.

Reading and Evaluating Complex Expressions Class 7 Notes

Sometimes, when an expression is not accompanied by a context, there can be more than one way of evaluating its value. In such cases, we need some tools and rules to specify how exactly the expression has to be evaluated.
To give an example with language, look at the following sentences:
(a) Sentence: “Shalini sat next to a friend with toys”.
Meaning: The friend has toys, and Shalini sat next to her.

(b) Sentence: “Shalini sat next to a friend, with toys”.
Meaning: Shalini has the toys, and she sat with them next to her friend.

This sentence without the punctuation could have been interpreted in two different ways. The appropriate use of a comma specifies how the sentence has to be understood. Let us see an expression that can be evaluated in more than one way.

Example 4.
Mallesh brought 30 marbles to the playground. Arun brought 5 bags of marbles, with 4 marbles in each bag. How many marbles did Mallesh and Arun bring to the playground?
Solution:
Mallesh summarized this by writing the mathematical expression 30 + 5 × 4.
Without knowing the context behind this expression, Purna found the value of this expression to be 140.
He added 30 and 5 first, to get 35, and then multiplied 35 by 4 to get 140.
Mallesh found the value of this expression to be 50.
He multiplied 5 and 4 first to get 20 and added 20 to 30 to get 50.

In this case, Mallesh is right. But why did Purna get it wrong?
Just looking at the expression 30 + 5 × 4, it is not clear whether we should do the addition fist or multiplication. Just as punctuation marks are used to resolve confusions in language, brackets and the notion of terms are used in mathematics to resolve confusions in evaluating expressions.

Brackets in Expressions
In the expression to fid the number of marbles — 30 + 5 × 4 — we had to first multiply 5 and 4, and then add this product to 30. This order of operations is clarifid by the use of brackets as follows: 30 + (5 × 4).

When evaluating an expression having brackets, we need to fi4st fid the values of the expressions inside the brackets before performing other operations. So, in the above expression, we first find the value of 5 × 4, and then do the addition. Thus, this expression describes the number of marbles: 30 + (5 × 4 ) = 30 + 20 = 50.

Example 5.
Irfan bought a pack of biscuits for ₹15 and a packet of toor dal for ₹56. He gave the shopkeeper ₹100. Write an expression that can help us calculate the change Irfan will get back from the shopkeeper.
Solution:
Irfan spent ₹15 on a biscuit packet and ₹56 on toor dal.
So, the total cost in rupees is 15 + 56. He gave ₹100 to the shopkeeper.
So, he should get back 100 minus the total cost.

Can we write that expression as 100 – 15 + 56?
Can we first subtract 15 from 100 and then add 56 to the result?
We will get 141. It is absurd that he gets more money than he paid the shopkeeper!
We can use brackets in this case: 100 – (15 + 56).
Evaluating the expression within the brackets first, we get 100 minus 71, which is 29. So, Irfan will get back ₹29.

Terms in Expressions
Suppose we have the expression 30 + 5 × 4 without any brackets. Does it have no meaning?
When expressions have multiple operations, and the order of operations is not specified by the brackets, we use the notion of terms to determine the order.
Terms are the parts of an expression separated by a ‘+’ sign. For example, in 12+7, the terms are 12 and 7, as marked below.
12 + 7 = 12 + 7

We will keep marking each term of an expression as above. Note that this way of marking the terms is not a usual practice. This will be done until you become familiar with this concept.

Now, what are the terms in 83 – 14? We know that subtracting a number is the same as adding the inverse of the number. Recall that the inverse of a given number has the sign opposite to it. For example, the inverse of 14 is –14, and the inverse of –14 is 14. Thus, subtracting 14 from 83 is the same as adding –14 to 83. That is, 83 – 14 = 83 + – 14. Thus, the terms of the expression 83 – 14 are 83 and –14.

All subtractions in an expression are converted to additions in this manner to identify the terms. Here are some more examples of expressions and their terms:
–18 – 3 = –18 + –3
6 × 5 + 3 = 6 × 5 + 3
2 – 10 + 4 × 6 = 2 + –10 + 4 × 6
Note that 6 × 5, 4 × 6 are single terms as they do not have any ‘+’ sign.

Now we will see how terms are used to determine the order of operations to find the value of an expression. We will start with expressions having only additions (with all the subtractions suitably converted into additions).

Swapping and Grouping
Let us consider a simple expression having only two terms.

Example 6.
Madhu is flying a drone from a terrace. The drone goes 6 m up and then 4 m down. Write an expression to show how high the final position of the drone is from the terrace.
Solution:
The drone is 6 – 4 = 2 m above the terrace.
Writing it as the sum of terms: 6 + –4 = 2
Will the sum change if we swap the terms?
–4 + 6 = 2
It doesn’t in this case.
We already know that swapping the terms does not change the sum when both terms are positive numbers.

Thus, in an expression having two terms, swapping them does not change the value.

Now consider an expression having three terms: (–7) + 10 + (–11).
Let us add these terms in the following two diffrent orders:

(adding the fist two terms and then adding their sum to the third term)

(adding the last two terms and then adding their sum to the first term)

What do you see? The sums are the same in both cases. Again, we know that while adding positive numbers, grouping them in any of the above two ways gives the same sum.

Thus, grouping the terms of an expression in either of the following ways gives the same value.

Let us consider the expression (–7) + 10 + (–11) again. What happens when we change the order and add -7 and -11 first, and then add this sum to 10? Will we get the same sum as before? We see that adding the terms of the expression (–7) + 10 + (–11) in any order gives the same sum of –8.

Thus, the addition of terms in any order gives the same value. Therefore, in an expression having only additions, it does not matter in what order the terms are added: they all give the same value.

Now let us consider expressions having multiplication and division also, without the order of operations specifid by the brackets. The values of such expressions are found by fist evaluating the terms. Once all the terms are evaluated, they are added.

For example, the expression 30 + 5 × 4 is evaluated as follows:
30 + 5 × 4 = 30 + 5 × 4 = 30 + 20 = 50
The expression 5 × (3 + 2) + 78 + 3 is evaluated as follows:
5 × (3 + 2) + 78 + 3 = 5 × (3 + 2) + 7 × 8 + 3
Where (3 + 2) is first evaluated, and this sum is multiplied by 5 (= 25).
The expression 7 × 8 is evaluated (= 56). This simplifies to 25 + 56 + 3 = 84.

In mathematics, we use the phrase commutative property of addition instead of saying “swapping terms does not change the sum”. Similarly, “grouping does not change the sum” is called the associative property of addition.

Swapping the Order of Things in Everyday Life
Manasa is going outside to play. Her mother says, “Wear your hat and shoes!” Which one should she wear fist? She can wear her hat first and then her shoes. Or she can wear her shoes first and then her hat.

Manasa will look exactly the same in both cases. Imagine a diffrent situation: Manasa’s mother says “Wear your socks and shoes!” Now the order matters. She should wear socks and then shoes. If she wears shoes and then socks, Manasa will feel very uncomfortable and look very different.

More Expressions and Their Terms

Example 7.
Amu, Charan, Madhu, and John went to a hotel and ordered four dosas. Each dosa costs ₹23, and they wish to thank the waiter by tipping ₹5. Write an expression describing the total cost.
Solution:
Cost of 4 dosas = 4 × 23
Can the total amount with tip be written as 4 × 23 + 5?
Evaluating it, we get 4 × 23 + 5 = 4 × 23 + 5 = 92 + 5 = 97
Thus, 4 × 23 + 5 is a correct way of writing the expression.

Example 8.
Children in a class are playing “Fire in the mountain, run, run, run!”. Whenever the teacher calls out a number, students are supposed to arrange themselves in groups of that number. Whoever is not part of the announced group size is out.
Solution:

Ruby wanted to rest and sat on one side. The other 33 students were playing the game in
the class. The teacher called out ‘5’. Once children settled, Ruby wrote 6 × 5 + 3 (understood as 3 more than 6 × 5)

Example 9.
Raghu bought 100 kg of rice from the wholesale market and packed them into 2 kg packets. He already had four 2 kg packets. Write an expression for the number of 2 kg packets of rice he has now and identify the terms.
Solution:
He had 4 packets.
The number of new 2 kg packets of rice is 100 ÷ 2, which we also write as \(\frac{100}{2}\).
The number of 2 kg packets he has now is 4 + \(\frac{100}{2}\).
The terms are 4 + \(\frac{100}{2}\)

Example 10.
Kannan has to pay ₹432 to a shopkeeper using coins of ₹1 and ₹5, and notes of ₹10, ₹20, ₹50 and ₹100. How can he do it?
Solution:
There is more than one possibility.
For example, 432 = 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1
Meaning: 4 notes of ₹100, 1 note of ₹20, 1 note of ₹10 and 2 notes of ₹1
432 = 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1
Meaning: 8 notes of ₹50, 1 note of ₹10, 4 notes of ₹5 and 2 notes of ₹1

Example 11.
Here are two pictures. Which of these two arrangements matches the expression 5 × 2 + 3?

Solution:
Let us write this expression as a sum of terms.
5 × 2 + 3 = 10 + 3 = 13
This expression, 5 × 2 + 3, can be understood as 3 more than 5 × 2, which describes the arrangement on the left.

What is the expression for the arrangement in the right,t making use of the number of yellow and blue squares?
Do you recall the use of brackets? We need to use brackets for this: 2 × (5 + 3)
Notice that this arrangement can also be described using 5 + 3 + 5 + 3 OR 5 × 2 + 3 × 2

Removing Brackets — I
Let us find the value of this expression, 200 – (40 + 3).
We first evaluate the expression inside the bracket to 43 and then subtract it from 200. But it is simpler to first subtract 40 from 200:
200 – 40 = 160.
And then subtract 3 from 160:
160 – 3 = 157.
What we did here was 200 – 40 – 3.
Notice that we did not do 200 – 40 + 3.
So, 200 – (40 + 3) = 200 – 40 – 3.

Example 12.
We also saw this earlier in the case of Irfan purchasing a biscuit packet (₹15) and a toor dal packet (₹56). When he paid ₹100, the change he got in rupees is:
100 – (15 + 56) = 29.
The change could also have been calculated as follows:
(a) First, subtract the cost of the biscuit packet (15) from 100:
100 – 15 = 85.
This is the amount the shopkeeper owes Irfan if he had purchased only the biscuits. As he has purchased toor dal also, its cost is taken from this remaining amount of 85.
(b) So, to find the change, we need to subtract the cost of toor dal from 85.
85 – 56 = 29.
What we have done here is 100 – 15 – 56.
So, 100 – (15 + 56) = 100 – 15 – 56.

Notice how, upon removing the brackets preceded by a negative sign, the signs of the terms inside the brackets change.
Observe the signs of 40 and 3 in the first example, and those of 15 and 56 in the second.

Example 13.
Consider the expression 500 – (250 – 100). Is it possible to write this expression without the brackets?
Solution:
To evaluate this expression, we need to subtract 250 – 100 = 150 from 500:
500 – (250 – 100) = 500 – 150 = 350.
If we were to directly subtract 250 from 500, then we would have subtracted 100 more than what we needed to.
So, we should add back that 100 to 500 – 250 to make the expression take the same value as 500 – (250 – 100).
This sequence of operations is 500 – 250 + 100.
Thus, 500 – (250 – 100) = 500 – 250 + 100.
Check that 500 – (250 – 100) is not equal to 500 – 250 – 100.

Notice again that when the brackets preceded by a negative sign are removed, the signs of the terms inside the brackets change. In this case, the signs of 250 and –100 change to –250 and 100.

Example 14.
Hira has a rare coin collection. She has 28 coins in one bag and 35 coins in another. She gifts her friend 10 coins from the second bag. Write an expression for the number of coins left with Hira.
Solution:
This can be expressed by 28 + (35 – 10).
We know that this is the same as 28 + (35 + (–10)).
Since the terms can be added in any order, this expression can simply be written as 28 + 35 + (–10), or 28 + 35 – 10.
Thus, 28 + (35 – 10) = 28 + 35 – 10 = 53.

When the brackets are NOT preceded by a negative sign, the terms within them do not change their signs upon removing the brackets. Notice the sign of the terms 35 and –10 in the above expression.

Tinker the Terms I
What happens to the value of an expression if we increase or decrease the value of one of its terms? Some expressions are given in the following three columns. In each column, one or more terms are changed from the first expression. Go through the example (in the first column) and fill the blanks, doing as little computation as possible.

Removing Brackets II

Example 15.
Lhamo and Norbu went to a hotel. Each of them ordered a vegetable cutlet and a rasgulla. A vegetable cutlet costs ₹43 and a rasgulla costs ₹24. Write an expression for the amount they will have to pay.
Solution:
As each of them had one vegetable cutlet and one rasagulla, each of their shares can be represented by 43 + 24.

What about the total amount they have to pay? Can it be described by the expression: 2 × 43 + 24?
Writing it as the sum of terms gives: 2 × 43 + 24
This expression means 24 more than 2 × 43. But, we want an expression which means twice or double of 43 + 24.
We can make use of brackets to write such an expression: 2 × (43 + 24).

So, we can say that together they have to pay 2 × (43 + 24). This is also the same as paying for two vegetable cutlets and two rasgullas: 2 × 43 + 2 × 24.
Therefore, 2 × (43 + 24) = 2 × 43 + 2 × 24.

Example 16.
In the Republic Day parade, Boy Scouts and Girl Guides are marching together. The scouts march in 4 rows with 5 scouts in each row. The guides march in 3 rows with 5 guides in each row (see the figure below). How many scouts and guides are marching in this parade?

Solution:
The number of Boy Scouts marching is 4 × 5.
The number of Girl Guides marching is 3 × 5.
The total number of scouts and guides will be 4 × 5 + 3 × 5.
This can also be found by first finding the total number of rows, i.e., 4 + 3,
and then multiplying their sum by the number of children in each row.
Thus, the number of boys and girls can be found by (4 + 3) × 5.
Therefore, 4 × 5 + 3 × 5 = (4 + 3) × 5.
Computing these expressions, we get
4 × 5 + 3 × 5 = 4 × 5 + 3 × 5 = 20 + 15 = 35
(4 + 3) × 5 = 7 × 5 = 35

5 × 4 + 3 ≠ 5 × (4 + 3). Can you explain why?
Is 5 × (4 + 3) = 5 × (3 + 4) = (3 + 4) × 5?
The observations that we have made in the previous two examples can be seen in a general way as follows.
Consider 10 × 98 + 3 × 98. This means taking the sum of 10 times 98 and 3 times 98.

This is the same as 10 + 3 = 13 times 98.
Thus, 10 × 98 +3 × 98 = (10 + 3) × 98.
Writing this equality the other way, we get (10 + 3) 98 = 10 × 98 + 3 × 98.
Swapping the numbers in the products above, this property can be seen in the following form:
98 × 10 + 9 × 83 = 98 (10 + 3), and 98 (10 + 3) = 98 × 10 + 98 × 3.
Similarly, let us consider the expression 14 × 10 – 6 × 10. This means subtracting 6 times 10 from 14 times 10.

This is 14 – 6 = 8 times 10.
Thus, 14 × 10 – 6 × 10 = (14 – 6) × 10,
or (14 – 6) × 10 = 14 × 10 – 6 × 10
This property can be nicely summed up as follows:
The multiple of a sum (difference) is the same as the sum (difference) of the multiples.

Tinker the Terms II
Let us understand what happens when we change the numbers occurring in a product.

Example 17.
Given 53 × 18 = 954. Find out 63 × 18.
Solution:
As 63 × 18 means 63 times 18,
63 × 18 = (53 + 10) × 18
= 53 ×18 + 10 × 18
= 954 + 180
= 1134

Example 18.
Find an effective way of evaluating 97 × 25.
Solution:
97 × 25 means 97 times 25.
We can write it as (100 – 3) × 25
We know that this is the same as the difference of 100 times 25 and 3 times 25:
97 × 25 = 100 × 25 – 3 × 25

Class 7 Maths Notes

The post Arithmetic Expressions Class 7 Notes Maths Chapter 2 appeared first on Learn CBSE.

Class 7 Maths Chapter 3 Notes A Peek Beyond the Point

Class 7 Maths Notes Chapter 3 – Class 7 A Peek Beyond the Point Notes

→ We can split a unit into smaller parts to get more exact/accurate measurements.

→ We extended the Indian place value system and saw that

• 1 unit = 10 one-tenths
• 1 tenth = 10 one-hundredths
• 1 hundredth = 10 one-thousandths
• 10 one-hundredths = 1 tenth
• 100 one-hundredths = 1 unit

→ A decimal point (‘.’) is used in the Indian place value system to separate the whole number part of a number from its fractional part.

→We also learnt how to compare decimal numbers, locate them on the number line, and perform addition and subtraction.

The Need for Smaller Units Class 7 Notes

Sonu’s mother was fixing a toy. She was trying to join two pieces with the help of a screw. Sonu was watching his mother with great curiosity. His mother was unable to enter the pieces. Sonu asked why. His mother said that the screw was not of the right size.

She brought another screw from the box and was able to fix the toy. The two screws looked the same to Sonu. But when he observed them closely, he saw they were of slightly different lengths.

Sonu was fascinated by how such a small difference in lengths could matter so much. He was curious to know the difference in lengths. He was also curious to know how little the diffrence was because the screws looked nearly the same.

In the following fiure, screws are placed above a scale. Measure them and write their length in the space provided.

What is the meaning of 2\(\frac{7}{10}\) cm (the length of the fist screw)?
As seen on the ruler, the unit length between two consecutive numbers is divided into 10 equal parts.
To get the length 2\(\frac{7}{10}\) cm, we go from 0 to 2 and then take seven parts of \(\frac{1}{10}\).
The length of the screw is 2 cm and \(\frac{7}{10}\) cm.
Similarly, we can make sense of the length 3\(\frac{2}{10}\) cm.
We read 2\(\frac{7}{10}\) cm as two and seven-tenth centimeters, and 3\(\frac{2}{10}\) cm as three and two-tenth centimeters.

Write the measurements of the objects shown in the picture:

As seen here, when exact measures are required, we can make use of smaller units of measurement.

A Tenth Part Class 7 Notes

The length of the pencil shown in the figure below is 3\(\frac{4}{10}\) units, which can also be read as 3 units and four one-tenths, i.e., (3 × 1) + (4 × \(\frac{1}{10}\)) units.

This length is the same as 34 one-tenths unit because 10 one-tenths unit makes one unit.
\(34 \times \frac{1}{10}=\frac{34}{10}=\frac{10}{10}+\frac{10}{10}+\frac{10}{10}+\frac{4}{10}\) (34 one-tenths)
= 1 + 1 + 1 + \(\frac{4}{10}\) (3 and 4 one-tenths)
A few numbers with fractional units are shown below, along with how to read them.
4\(\frac{1}{10}\) → ‘four and one-tenth’
\(\frac{4}{10}\) → ‘four one-tenths’ or ‘four-tenths’
\(\frac{41}{10}\) → ‘forty-one one-tenths’ or ‘forty-one tenths’
41\(\frac{1}{10}\) → ‘forty-one and one-tenth’

For the objects shown below, write their lengths in two ways and read them aloud. An example is given for the USB cable. (Note that the unit length used in each diagram is not the same.)
The length of the USB cable is 4 and \(\frac{8}{10}\) units or \(\frac{48}{10}\) units.

Sonu is measuring some of his body parts. The length of Sonu’s lower arm is 2\(\frac{7}{10}\) units, and that of his upper arm is 3\(\frac{6}{10}\) units. What is the total length of his arm?
To get the total length, let us see the lower and upper arm lengths as 2 units and 7 one-tenths, and 3 units and 6 one-tenths, respectively.
So, there are (2 + 3) units and (7 + 6) one-tenths. Together, they make 5 units and 13 one-tenths. But 13 one-tenths is 1 unit and 3 one-tenths.
So, the total length is 6 units and 3 one-tenths.

Or, both lengths can be converted to tenths and then added:
(c) 27 one-tenths and 35 one-tenths is 62 one-tenths
\(\frac{27}{10}+\frac{35}{10}=\frac{62}{10}\)
\(\frac{62}{10}\) is the same as 60 one-tenths (\(\frac{60}{10}\)) and 2 one-tenths (\(\frac{2}{10}\)), which is equal to 6 units and 2 one-tenths, i.e., 6\(\frac{2}{10}\).

The lengths of the body parts of a honeybee are given. Find its total length.

Head: 2\(\frac{3}{10}\) units
Thorax: 5\(\frac{4}{10}\) units
Abdomen: 7\(\frac{5}{10}\) units

The length of Shylaja’s hand is 12\(\frac{4}{10}\) units, and her palm is 6\(\frac{7}{10}\) units, as shown in the picture. What is the length of the longest (middle) finger?

The length of the finger can be found by evaluating (12 + \(\frac{4}{10}\)) – (6 + \(\frac{7}{10}\)). This can be done in different ways. For example,

As in the case of counting numbers, it is convenient to start subtraction from the tenths. We cannot remove 7 one-tenths from 4 one-tenths. So we split a unit from 12 and convert it to 10 one-tenths. Now, the number has 11 units and 14 one-tenths. We subtract 7 one-tenths from 14 one-tenths and then subtract 6 units from 11 units.

A Hundredth Part Class 7 Notes

The length of a sheet of paper was 8\(\frac{9}{10}\) units, which can also be said as 8 units and 9 one-tenths. It is folded in half along its length. What is its length now?

We can say that its length is between 4\(\frac{4}{10}\) units and 4\(\frac{5}{10}\) units. But we cannot state its exact measurement, since there are no markings. Earlier, we split a unit into 10 one-tenths to measure smaller lengths. We can do something similar and split each one-tenth into 10 parts.

What is the length of this smaller part? How many such smaller parts make a unit length?
As shown in the figure below, each one-tenth has 10 smaller parts, and there are 10 one-tenths in a unit; therefore, there will be 100 smaller parts in a unit. Therefore, one part’s length will be \(\frac{1}{100}\) of a unit.

Returning to our question, what is the length of the folded paper?
We can see that it ends at \(4 \frac{4}{10} \frac{5}{100}\), read as 4 units and 4 one-tenths and 5 one-hundredths.

Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill in the lengths in the empty boxes.

The length of the wire in the first picture is given in three different ways. Can you see how they denote the same length?

\(1 \frac{1}{10} \frac{4}{100}\) → One and one-tenth and four-hundredths
1\(\frac{14}{100}\) → One and fourteen-hundredths
\(\frac{114}{100}\) → One Hundred and Fourteen-hundredths

For the lengths shown below, write the measurements and read out the measures in words.

In each group, identify the longest and the shortest lengths. Mark each length on the scale.

What will be the sum of \(15 \frac{3}{10} \frac{4}{100}\) and \(2 \frac{6}{10} \frac{8}{100}\)?
This can be solved in different ways. Some are shown below.
(a) Method 1

(b) Method 2

Observe the addition done below for 483 + 268. Do you see any similarities between the methods shown above?
(400 + 80 + 3) + (200 + 60 + 8)
= (400 + 200) + (80 + 60) + (3 + 8)
= 600 + 140 + 11
= 600 + 150 + 1
= 700 + 50 + 1
= 751
One can also find the sum \(15 \frac{3}{10} \frac{4}{100}+2 \frac{6}{10} \frac{8}{100}\) by converting to hundredths, as follows.

What is the diffrence: 25\(\frac{9}{10}\) – \(6 \frac{4}{10} \frac{7}{100}\)?
One way to solve this is as follows:

Solve this by converting to hundredths.
What is the diffrence \(15 \frac{3}{10} \frac{4}{100}-2 \frac{6}{10} \frac{8}{100}\)?
One way to solve this is as follows:

Observe the subtraction done below for 653 – 268. Do you see any similarities with the methods shown above?
(600 + 50 + 3) – (200 + 60 + 8)
= (600 – 200) + (50 – 60) + (3 – 8)
= (600 – 200) + (40 – 60) + (13 – 8)
= (600 – 200) + (40 – 60) + 5
= (500 – 200) + (140 – 60) + 5
= 300 + 80 + 5
= 385

Decimal Place Value Class 7 Notes

You may have noticed that whenever we need to measure something more accurately, we split a part into 10 (smaller) equal parts ― we split a unit into 10 one-tenths and then split each one-tenth into 10, and then we use these smaller parts to measure.

Can we not split a unit into 4 equal parts, 5 equal parts, 8 equal parts, or any other number of equal parts instead?
Yes, we can. The example below compares how the same length is represented when the unit is split into 10 equal parts and when the unit is split into 4 equal parts.

If an even more precise measure is needed, each quarter can be further split into four equal parts. Each part then measures \(\frac{1}{16}\) of a unit, i.e., 16 such parts make 1 unit.

Then why split a unit into 10 parts every time?
The reason is the special role that 10 plays in the Indian place value system. For a whole number written in the Indian place value system — for example, 281 — the place value of 2 is hundreds (100), that of 8 is tens (10), and that of 4 is ones (1). Each place value is 10 times bigger than the one immediately to its right. Equivalently, each place value is 10 times smaller than the one immediately to its left:
10 ones make 1 ten,
10 tens make 1 hundred,
10 hundreds make 1 thousand, and so on.

To extend this system of writing numbers to quantities smaller than one, we divide one into 10 equal parts. What does this give? It gives one-tenth. Further dividing it into 10 parts gives one-hundredth, and so on.

Can we extend this further? What will the fraction be when \(\frac{1}{100}\) is split into 10 equal parts?
It will be \(\frac{1}{1000}\), i.e., a thousand such parts make up a unit.

Just as when we extend to the left of 10,000, we get bigger place values at each step, we can also extend to the right of \(\frac{1}{1000}\), getting smaller place values at each step.

This way of writing numbers is called the “decimal system” since it is based on the number 10; “decem” means ten in Latin, which in turn is cognate to the Sanskrit daśha meaning 10, with similar words for 10 occurring across many Indian languages including Odia, Konkani, Marathi, Gujarati, Hindi, Kashmiri, Bodo, and Assamese. We shall learn about other ways of writing numbers in later grades.

How Big?
We already know that a hundred 10s make 1000, and a hundred 100s make 10000.

Notation, Writing, and Reading of Numbers
We have been writing numbers in a particular way, say 456, instead of writing them as 4 × 100 (4 hundreds) + 5 × 10 (5 tens) + 6 × 1 (6 ones). Similarly, can we skip writing tenths and hundredths?
Can the quantity 4\(\frac{2}{10}\) be written as 42 (skipping the \(\frac{1}{10}\) in 2 × \(\frac{1}{10}\))?
If yes, how would we know if 42 means 4 tens and 2 units or it means 4 units and 2 tenths?
Similarly, 705 could mean:

• 7 hundreds, 0 tens, and 5 ones (700 + 0 + 5)
• 7 tens and 0 units and 5 tenths (70 + 0 + \(\frac{5}{10}\))
• 7 units and 0 tenths and 5 hundredths (7 + \(\frac{0}{10}\) + \(\frac{5}{100}\))

Since these are different quantities, we need to have distinct ways of writing them.

To identify the place value where integers end and the fractional parts start, we use a point or period (‘.’) as a separator, called a decimal point. The above quantities in decimal notation are then:

These numbers, when shown through place value, are as follows:

Thus, decimal notation is a natural extension of the Indian place value system to numbers also having fractional parts. Just as 705 means 7 × 100 + 5 × 1, the number 70.5 means 7 × 10 + 5 × \(\frac{1}{10}\), and 7.05 means 7 × 1 + 5 × \(\frac{1}{100}\).

We have seen how to write numbers using the decimal point (‘.’). But how do we read/say these numbers?
We know that 705 is read as seven hundred and five.
70.5 is read as seventy point five, short for seventy and five-tenths.
7.05 is read as seven point zero five, short for seven and five hundredths.
0.274 is read as zero point two seven four. We don’t read it as zero point two hundred and seventy four, as 0.274 means 2 one-tenths and 7 one-hundredths and 4 one-thousandths.

In the chapter on large numbers, we learned how to write 23 hundred.
23 hundreds = 23 × 100 = 2000 + 300 = 2300.

Similarly, 23 tens would be:
23 tens = 23 × 10 = 200 + 30 = 230.

How can we write 234 tenths in decimal form?
234 tenths = \(\frac{234}{10}\)
= \(\frac{200}{10}+\frac{30}{10}+\frac{4}{10}\)
= 20 + 3 + \(\frac{4}{10}\)
= 23.4

Units of Measurement Class 7 Notes

Length Conversion
We have been using a scale to measure length for a few years. We already know that 1 cm = 10 mm (millimeters).

How many cm is 1 mm?
1 mm = \(\frac{1}{10}\) cm = 0.1 cm (i.e., one-tenth of a cm).

How many cm is (a) 5 mm? (b) 12 mm?
(a) 5 mm = \(\frac{5}{10}\) cm = 0.5 cm
(b) 12 mm = 10 mm + 2 mm
= 1 cm + \(\frac{2}{10}\) cm
= 1.2 cm

How many mm is 5.6 cm?
Since each cm has 10 mm, 5.6 cm (5 cm + 0.6 cm) is 56 mm.

The illustration below shows how small some things are! Try taking an approximate measurement of each.

→ The three blue stripes represent the typical relative sizes of pen strokes: fine stroke, medium stroke, and bold stroke.

→ A human hair is about 0.1 mm in thickness.

→ The thickness of a newspaper can range from 0.05 to 0.08 mm.

→ Mustard seeds have a thickness of 1 – 2 mm.

→ The smallest ant species discovered so far, Carabera Bruni, has a total length of 0.8 – 1 mm. They are found in Sri Lanka and China.

→ The smallest land snail species discovered so far, Acmella Nana, has a shell diameter of 0.7 mm. They are found in Malaysia.

We also know that 1 m = 100 cm. Based on this, we can say that
1 cm = \(\frac{1}{100}\) m = 0.01 m.

How many m is (a) 10 cm? (b) 15 cm?
(a) 10 cm = \(\frac{1}{10}\) m = 0.1 m
Since each cm is one-hundredth of a meter, 15 cm can be written as
(b) 15 cm = \(\frac{15}{100}\) m
= \(\frac{10}{100}\) m + \(\frac{5}{100}\) m
= \(\frac{1}{10}\) m + \(\frac{5}{100}\) m
= 0.15 m

Here, we have some more interesting facts about small things in nature!

→ The egg of a hummingbird typically is 1.3 cm long and 0.9 cm wide.

→ The Philippine Goby is about 0.9 cm long. It can be found in the Philippines and other Southeast Asian countries.

→ The smallest known jellyfish, Irukandji, has a bell size of 0.5 – 2.5 cm. Its tentacles can be as long as 1 m. They are found in Australia. Its venom can be fatal to humans.

→ The Wolf octopus, also known as the Star-sucker Pygmy Octopus, is the smallest known octopus in the world. Their typical size is around 1 – 2.5 cm, and they weigh less than 1 g. They are found in the Pacific Ocean.

Weight Conversion
Let us look at kilograms (kg). We know that 1 kg = 1000 gram (g). We can say that
1 g = \(\frac{1}{1000}\) kg = 0.001 kg.

How many kilograms is 5 g?
5 g = \(\frac{5}{1000}\) kg = 0.005 kg.

How many kilograms is 10 g?
10 g = \(\frac{10}{1000}\) = \(\frac{1}{100}\) kg = 0.010 kg.
As each gram is one-thousandth of a kg, 254 g can be written as
254 g = \(\frac{254}{1000}\) kg
= (\(\frac{200}{1000}+\frac{50}{1000}+\frac{4}{1000}\)) kg
= (\(\frac{2}{10}+\frac{5}{100}+\frac{4}{1000}\)) kg
= 0.254 kg.

Look at the picture below showing different quantities of rice. Starting from the 1 g heap, subsequent heaps can be found that are 10 times heavier than the previous heap/packets. The combined weight of rice in this picture is 11.111 kg.

Also, 1 gram = 1000 milligrams (mg).
So, 1 mg = \(\frac{1}{1000}\) g = 0.001 g.

Rupee ─ Paise Conversion
You may have heard of ‘paisa’. 100 paise is equal to 1 rupee. As we have coins and notes for rupees, coins for paise were also used commonly until recently. There were coins for 1 paisa, 2 paise, 3 paise, 5 paise, 10 paise, 20 paise, 25 paise, and 50 paise. All denominations of 25 paise and less were removed from use in the year 2011. But we still see paisa in bills, account statements, etc.

1 rupee = 100 paise
1 paisa = \(\frac{1}{100}\) rupee = 0.01 rupee
As each paisa is one-hundredth of a rupee,
75 paise = \(\frac{75}{100}\) rupee
= (\(\frac{70}{100}+\frac{5}{100}\)) rupee
= (\(\frac{7}{10}+\frac{5}{100}\)) rupee
= 0.75 rupee.

During the 1970s, a masala dosa cost just 50 paise, one could buy a banana for 20-25 paise, a handful of peppermints were available for 2 paise or 3 paise, and a kg of rice cost ₹ 2.45.3

Locating and Comparing Decimals

Let us consider the decimal number 1.4. It is equal to 1 unit and 4 tenths. This means that the unit between 1 and 2 is divided into 10 equal parts, and 4 such parts are taken. Hence, 1.4 lies between 1 and 2. Draw the number line and divide the unit between 1 and 2 into 10 equal parts. Take the fourth part, and we have 1.4 on the number line.

There is Zero Dilemma!
Sonu says that 0.2 can also be written as 0.20, 0.200; Zara thinks that putting zeros on the right side may alter the value of the decimal number. What do you think?
We can figure this out by looking at the quantities these numbers represent using place value.

We can see that 0.2, 0.20, and 0.200 are all equal as they represent the same quantity, i.e., 2 tenths. But 0.2, 0.02, and 0.002 are different.

In the number line shown below, what decimal numbers do the boxes labelled ‘a’, ‘b’, and ‘c’ denote?

The box with ‘b’ corresponds to the decimal number 7.5; are you able to see how? There are 5 units between 5 and 10, divided into 10 equal parts. Hence, every 2 divisions make a unit, and so every division is \(\frac{1}{2}\) unit. What numbers do ‘a’ ‘c’ denote?

Which is larger: 6.456 or 6.465?
To answer this, we can use the number line to locate both decimal numbers and show which is larger. This can also be done by comparing the corresponding digits at each place value, as we do with whole numbers. This comparison is visualised step by step below. Note that the visualisation below is not to scale.

We start by comparing the most significant digits (digits with the highest place value) of the two numbers. If the digits are the same, we compare the next smaller place value. We keep going till we find a position where the digits are not equal. The number with the larger digit at this position is the greater of the two.

Closest Decimals
Consider the decimal numbers 0.9, 1.1, 1.01, and 1.11. Identify the decimal number that is closest to 1.
Let us compare the decimal numbers. Arranging these in ascending order, we get 0.9 < 1 < 1.01 < 1.1 < 1.11.
Among the neighbours of 1, 1.01 is \(\frac{1}{100}\) away from 1, whereas 0.9 is \(\frac{10}{100}\) away from 1.
Therefore, 1.01 is closest to 1.

Addition and Subtraction of Decimals Class 7 Notes

Priya requires 2.7 m of cloth for her skirt, and Shylaja requires 3.5m for her kurti. What is the total quantity of cloth needed?
We have to find the sum of 2.7 m + 3.5 m.
Earlier, we saw how to add 2\(\frac{7}{10}\) + 3\(\frac{5}{10}\) (also shown below). Can you carry out the same addition using decimal notation? It is shown below. Share your observations.
The total quantity of cloth needed is 6.2 m.

How much longer is Shylaja’s cloth compared to Priya’s?
We have to find the difference between 3.5 m – 2.7 m.
Again, observe how the diffrences 3\(\frac{5}{10}\) – 2\(\frac{7}{10}\) and 3.5 m – 2.7 m are computed.

As you can see, the standard procedure for adding and subtracting whole numbers can be used to add and subtract decimals.

A detailed view of the underlying place value calculation is shown below for the sum 75.345 + 86.691. Its compact form is shown next to it.

Decimal Sequences
Observe this sequence of decimal numbers and identify the change after each term.
4.4, 4.8. 5.2, 5.6, 6.0, ……
We can see that 0.4 is being added to a term to get the next term.

Estimating Sums and Differences
Sonu has observed sums and differences of decimal numbers and says, “If we add two decimal numbers, then the sum will always be greater than the sum of their whole number parts. Also, the sum will always be less than 2 more than the sum of their whole number parts.”

Let us use an example to understand what his claim means: If the two numbers to be added are 25.936 and 8.202, the claim is that their sum will be greater than 25 + 8 (whole number parts) and will be less than 25 + 1 + 8 + 1.

More on the Decimal System Class 7 Notes

Decimal and Measurement Disasters
Decimal point and unit conversion mistakes may seem minor sometimes, but they can lead to serious problems. Here are some actual incidents in which such errors caused major issues.

→ In 2013, the finance office of the Amsterdam City Council (Netherlands) mistakenly sent out €188 million in housing benefits instead of the intended €1.8 million due to a programming error that processed payments in euro cents instead of euros. (1 euro-cent = \(\frac{1}{100}\) euro).

→ In 1983, a decimal error nearly caused a disaster for an Air Canada Boeing 767. The ground staff miscalculated the fuel, loading 22,300 pounds instead of kilograms—about half of what was needed (1 pound ~ 0.453 kg). The plane ran out of fuel mid-air, forcing the pilots to make an emergency landing at an abandoned airfield. Fortunately, everyone survived.

Several incidents have occurred due to the incorrect reading of decimal numbers while giving medication. For example, reading 0.05 mg as 0.5 mg can lead to using a medicine 10 times more than the prescribed quantity. It is therefore important to pay attention to units and the location of the decimal point.

Deceptive Decimal Notation
Sarayu gets a message: “The bus will reach the station 4.5 hours post noon.” When will the bus reach the station: 4:05 p.m., 4:50 p.m., 4:25 p.m.? None of these! Here, 0.5 hours means splitting an hour into 10 equal parts and taking 5 parts out of it. Each part will be 6 minutes (60 minutes/10) long. 5 such parts make 30 minutes. So, the bus will reach the station at 4:30.

Here is a short story of a decimal mishap: A girl measures the width of an opening as 2 ft 5 inches but conveys to the carpenter to make a door 2.5 ft wide. The carpenter makes a door of width 2 ft 6 inches (since 1 ft = 12 inches, 0.5 ft = 6 inches), and it wouldn’t close fully

If you watch cricket, you might have noticed decimal-looking numbers like ‘Overs left: 5.5’. Does this mean 5 overs and 5 balls or 5 overs and 3 balls? Here, 5.5 overs means 5\(\frac{5}{6}\) overs (as 1 over = 6 balls), i.e., 5 overs and 5 balls.

A Pinch of History – Decimal Notation Over Time
Decimal fractions (i.e., fractions with denominators like \(\frac{1}{10}\), \(\frac{1}{100}\), \(\frac{1}{1000}\), and so on) are used in the works of several ancient Indian astronomers and mathematicians, including in the important 8th century works of Shridharacharya on arithmetic and algebra. Decimal notation, in essentially its modern form, was described in detail in Kitab al-Fusul fi al-Hisab al Hindi (The Book of Chapters on Indian Arithmetic) by Abul Hassan al-Uqlidisi, an Arab mathematician, in around 950 CE. He represented the number 0.059375 as 0.059375.

In the 15th century, to separate whole numbers from fractional parts, several different notations were used:

• a vertical mark on the last digit of the whole number part (as shown above),
• Use of different colours and
• numerical superscript giving the number of fractional decimal places (0.36 would be written as 362 ).

In the 16th century, John Napier, a Scottish mathematician, and Christopher Clavius, a German mathematician, used the point/period (‘.’) to separate the whole number and the fractional parts, while François Viète, a French mathematician, used the comma (‘,’) instead.

Currently, several countries use the comma to separate the integer part and the fractional part. In these countries, the number 1,000.5 is written as 1000,5 (space as a thousand separator). But the decimal point has endured as the most popular notation for writing numbers having fractional parts in the Indian place value system.

Class 7 Maths Notes

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Class 7 Maths Chapter 4 Notes Expressions using Letter-Numbers

Class 7 Maths Notes Chapter 4 – Class 7 Expressions using Letter-Numbers Notes

→ Algebraic expressions are used in formulas to model patterns and mathematical relationships between quantities, and to make predictions.

→ Algebraic expressions use not only numbers but also letter-numbers.

→ The rules for manipulating arithmetic expressions also apply to algebraic expressions. These rules can be used to reduce algebraic expressions to their simplest forms.

→ Algebraic expressions can be described in ordinary language, and vice versa. Patterns or relationships that are easily written using algebra can often be long and complex in ordinary language. This is one of the advantages of algebra.

The Notion of Letter-Numbers Class 7 Notes

In this chapter we shall look at a concise way of expressing mathematical relations and patterns. We shall see how this helps us in thinking about these relationships and patterns, and in explaining why they may hold true.

Example 1.
Shabnam is 3 years older than Aftab. When Aftab is 10 years old, Shabnam’s age will be 13 years. Now Aftab’s age is 18 years, what will Shabnam’s age be? Given Aftab’s age, how will you find out Shabnam’s age?
Easy: We add 3 to Aftab’s age to get Shabnam’s age.

Can we write this as an expression?
Shabnam’s age is 3 years more than Aftab’s. In short, this can be written as:
Shabnam’s age = Aftab’s age + 3.

Such mathematical relations are generally represented in a shorthand form. In the relation above, instead of writing the phrase ‘Aftab’s Age’, the convention is to use a convenient symbol. Usually, letters or short phrases are used for this purpose.

Let us say we use the letter a to denote Aftab’s age (we could have used any other letter), and s to denote Shabnam’s age. Then the expression to find Shabnam’s age will be a + 3, which can be written as s = a + 3.

If a is 23 (Aftab’s age in years), then what is Shabnam’s age?
Replacing a by 23 in the expression a + 3, we get s = 23 + 3 = 26 years.
Letters such as a and s that are used to represent numbers are called letter-numbers.
Mathematical expressions containing letter-numbers, such as the expression a + 3, are called algebraic expressions.

Given the age of Shabnam, write an expression to find Aftab’s age.
We know that Aftab is 3 years younger than Shabnam. So, Aftab’s age will be 3 less than Shabnam’s. This can be described as
Aftab’s age = Shabnam’s age – 3.
If we again use the letter a to denote Aftab’s age and the letter s to denote Shabnam’s age, then the algebraic expression would be: a = s – 3, meaning 3 less than s.

Example 2.
Parthiv is making matchstick patterns. He repeatedly places Ls next to each other. Each L has two matchsticks as shown in the Figure.

How many matchsticks are needed to make 5 Ls? It will be 5 × 2.
How many matchsticks are needed to make 7 Ls? It will be 7 × 2.
How many matchsticks are needed to make 45 Ls? It will be 45 × 2.
Now, what is the relation between the number of Ls and the number of sticks?
First, let us describe the relationship or the pattern here.
Every L needs 2 matchsticks. So the number of matchsticks needed will be 2 times the number of L’s.
This can be written as:
Number of matchsticks = 2 × Number of L’s
Now, we can use any letter to denote the number of L’s. Let’s use n.
The algebraic expression for the number of matchsticks will be: 2 × n.
This expression tells us how many matchsticks are needed to make n L’s.
To find the number of matchsticks, we just replace n by the number of Ls.

Example 3.
Ketaki prepares and supplies coconut-jaggery laddus. The price of a coconut is ₹35, and the price of 1 kg of jaggery is ₹60. How much should she pay if she buys 10 coconuts and 5 kg of jaggery?
Cost of 10 coconuts = 10 × ₹35
Cost of 5 kg jaggery = 5 × ₹60
Total cost = 10 × ₹35 + 5 × ₹60 = ₹350 + ₹300 = ₹650.

Write an algebraic expression to find the total amount to be paid for a given number of coconuts and quantity of jaggery.
Let us identify the relationships and then write the expressions.

Here, ‘c’ represents the number of coconuts and ‘j’ represents the number of kgs of jaggery.
The total amount to be paid will be:
Cost of coconuts + Cost of jaggery.
The corresponding algebraic expression can be written as c × 35 + j × 60

Use this expression (or formula) to find the total amount to be paid for 7 coconuts and 4 kg of jaggery.
Notice that for different values of ‘c’ and ‘j’, the value of the expression also changes.
Writing this expression as a sum of terms, we get c × 35 + j × 60

Example 4.
We are familiar with calculating the perimeters of simple shapes. Write expressions for perimeters.
The perimeter of a square is 4 times the length of its side. This can be written as the expression: 4 × q, where q stands for the sidelength.

You must have realised how the use of letter-numbers and algebraic expressions allows us to express general mathematical relations in a concise way. Mathematical relations expressed this way are often called formulas.

Revisiting Arithmetic Expressions Class 7 Notes

We learnt to write expressions as sums of terms and it became easy for us to read arithmetic expressions. Many times, they could have been read in multiple ways, and it was confusing. We used swapping (adding two numbers in any order) and grouping (adding numbers by grouping them conveniently) to find easy ways of evaluating expressions. Swapping and grouping terms does not change the value of the expression. We also learnt to use brackets in expressions, including brackets with a negative sign outside. We learnt the distributive property (multiple of a sum is the same as sum of multiples). Let us revise these concepts and find the values of the following expressions:

1. 23 – 10 × 2
2. 83 + 28 – 13 + 32
3. 34 – 14 + 20
4. 42 + 15 – (8 – 7)
5. 68 – (18 + 13)
6. 7 × 4 + 9 × 6
7. 20 + 8 × (16 – 6)

Let us evaluate the first expression, 23 – 10 × 2. First, we shall write the terms of the expression. Notice that one of the terms is a number, while the other one has to be converted to a number before adding the two terms.
23 – 10 × 2 = 23 + –10 × 2 = 23 + –20 = 3

Let us now evaluate the second one. All the terms of this expression are numbers. If we notice the terms, we find that it will be easier to evaluate if we swap and group the terms.

Let us now look at the fifth expression. It has brackets with a negative sign outside. This can be evaluated in two ways—by solving the bracket first (like the solution on the left side) or by removing the brackets appropriately (as on the right side).

Now, find the values of the other arithmetic expressions. Algebraic expressions also take numerical values when the letter numbers they contain are replaced by numbers.
In Example 1, to find Shabnam’s age when Aftab is 23 years old, we replaced the letter-number a in the expression a + 3 by 23, and it took the value 26.

Omission of the Multiplication Symbol in Algebraic Expressions Class 7 Notes

Look at this number sequence:
4, 8, 12, 16, 20, 24, 28, …….
How can we describe this sequence or pattern?
Easy: These numbers appear in the multiplication table of 4 (multiples of 4 in increasing order).
What is the third term of this sequence? It is 4 × 3.
What is the 29th term of this sequence? It is 4 × 29.

Find an algebraic expression to get the nth term of this sequence.
Note that here ‘n’ is a letter-number that denotes a position in the sequence.
As it is the sequence of multiples of 4, it can be seen that the nth term will be 4 times n: 4 × n
As a standard practice, we shorten 4 × n to 4n by skipping the multiplication sign. We write the number first, followed by the letter(s).
Find the value of the expression 7k when k = 4. The value is 7 × 4 = 28.
Find the value that the expression 5m + 3 takes when m = 2.
As 5m stands for 5 × m, the value of the expression when m = 2 is 5 × 2 + 3 = 13.

Mind the Mistake, Mend the Mistake
Some simplifications are shown below, where the letter-numbers are replaced by numbers and the value of the expression is obtained.

1. Observe each of them and identify if there is a mistake.
2. If you think there is a mistake, try to explain what might have gone wrong.
3. Then, correct it and give the value of the expression.

Simplification of Algebraic Expressions Class 7 Notes

Earlier, we found expressions to find the perimeters of different regular figures in terms of their sides. Let us now find an expression to find the perimeter of a rectangle.

As in the previous cases, we will first describe how to get the perimeter when the length and the breadth of the rectangle are known:
Find the sum of length + breadth + length + breadth.
Let us use the letter-numbers l and b in place of length and breadth, respectively.
Let p denote the perimeter of the rectangle. Then we have p = l + b + l + b.
As we know, these represent numbers, and so the terms of an expression can be added in any order.
Hence, the above expression can be written as l + l + b + b.
As l + l = 2 × l = 2l and b + b = 2 × b = 2b, we have p = 2l + 2b.

Notice that the initial expression (l + b + l + b) and the final expression (2l + 2b) that we got for the perimeter look different. However, they are equal since the expression was obtained from the initial one by applying the same rules and operations we do for numbers; they are equal in the sense that they both take the same values when letter numbers are replaced by numbers.
For example, if we assign l = 3, b = 4, we get
l + b + l + b = 3 + 4 + 3 + 4 = 14, and 2l + 2b = 2 × 3 + 2 × 4 = 14.
We call the expression 2l + 2b the simplified form of l + b + l + b.
Let us see some more examples of simplification.

Example 5.
Here is a table showing the number of pencils and erasers sold in a shop. The price per pencil is c, and the price per eraser is d. Find the total money earned by the shopkeeper during these three days.

Let us first find the money earned by the sale of pencils.
The money earned by selling pencils on Day 1 is 5c. Similarly, the money earned by selling pencils on Day 2 is 3c, and on Day 3 it is 10c.
The total money earned by the sale of pencils is 5c + 3c + 10c.

Can we simplify this expression further and reduce the number of terms?
The expression means 5 times c is added to 3 times c is added to 10 times c. So in total, the letter-number c is added (5 + 3 + 10) times. This is what we have seen as the distributive property of numbers. Thus, 5 × c + 3 × c + 10 × c = (5 + 3 + 10) × c
(5 + 3 + 10) × c can be simplifid to 18 × c = 18c

Write the expression for the total money earned by selling erasers. Then, simplify the expression.
The expression for the total money earned by selling pencils and
erasers during these three days is 18c + 11d.

Can the expression 18c + 11d be simplified further?
There is no way of further simplifying this expression as it contains diffrent letter-numbers. It is in its simplest form. In this problem, we saw the expression 5c + 3c + 10c getting simplified to the expression 18c.

Example 6.
A big rectangle is split into two smaller rectangles as shown. Write an expression describing the area of the bigger rectangle.

The areas of the smaller rectangles are 4v sq. units and 3v sq. units.
The area of the bigger rectangle can be found in two ways: (i) by directly using its side lengths v and (4 + 3), or (ii) by adding the areas of the smaller rectangles.
The first way gives 7v, and the second way gives 4v + 3v. We know that they are equal: 4v + 3v = 7v, and this is the required expression for the area of the bigger rectangle.
As earlier, a big rectangle is split into two smaller rectangles as shown below. Write an expression to find the area of the rectangle AEFD.

Even in this case, the area of rectangle AEFD can be found in two ways: (i) by directly using the side lengths n and (12 – 4), or (ii) subtracting the area of the rectangle EBCF from that of ABCD.

The first method gives us 8n, and the second method gives us 12n – 4n, and they are equal, since 12n – 4n = 8n. This is the expression for the area of the rectangle AEFD.

Sets of terms such as (5c, c, 10c), (12n, – 4n) that involve the same letter-numbers are called like terms. Sets of terms such as {18c, 11d} are called unlike terms as they have different letter-numbers. As we have seen, like terms can be added together and simplified into a single term.

Example 7.
A shop rents out chairs and tables for a day’s use. To rent them, one has to first pay the following amount per piece.

When the furniture is returned, the shopkeeper pays back some amount as follows.

Write an expression for the total number of rupees paid if x chairs and y tables are rented.
For x chairs and y tables, let us find the total amount paid at the beginning and the amount one gets back after returning the furniture.

Describe the procedure to get these amounts. The total amount in rupees paid at the beginning is 40x + 75y, and the total amount returned is 6x + 10y.
So, the total amount paid = (40x + 75y) – (6x + 10y)

Can we simplify this expression? If yes, how? If not, why not?
Recalling how we open brackets in an arithmetic expression, we get
(40x + 75y) – (6x + 10y) = (40x + 75y) – 6x – 10y
Since the terms can be added in any order, the remaining bracket can be opened and the expression becomes 40x + 75y + – 6x + – 10y
We can group the like terms. This results in 40x + – 6x + 75y + – 10y
= (40 – 6)x + (75 – 10)y
= 34x + 65y.
The expression (40x + 75y) – (6x + 10y) is simplified to 34x + 65y, which is the total amount paid in rupees.

Example 8.
Charu has been through three rounds of a quiz. Her scores in the three rounds are 7p – 3q, 8p – 4q, and 6p – 2q. Here, p represents the score for a correct answer, and q represents the penalty for an incorrect answer. What does each of the expressions mean?
If the score for a correct answer is 4 (p = 4) and the penalty for a wrong answer is 1 (q = 1), find Charu’s score in the first round.
Charu’s score is 7 × 4 – 3 × 1. We can evaluate this expression by writing it as a sum of terms.
7 × 4 – 3 × 1 = 7 × 4 + – 3 × 1 = 28 + – 3 = 25

What are her scores in the second and third rounds? What if there is no penalty? What will be the value of q in that situation? What is her final score after the three rounds?
Her final score will be the sum of the three scores: (7p – 3q) + (8p – 4q) + (6p – 2q).
Since the terms can be added in any order, we can remove the brackets and write 7p + – 3q + 8p + – 4q + 6p + –2q
= 7p + 8p + 6p + – (3q) + – (4q) + – (2q) (by swapping and grouping)
= (7 + 8 + 6)p + – (3 + 4 + 2)q
= 21p + – 9q
= 21p – 9q
Charu’s total score after three rounds is 21p–9q.
Her friend Krishita’s score after three rounds is 23p – 7q.

Example 9.
Simplify the expression 4(x + y) – y.
Using the distributive property, this expression can be simplified to
4(x + y) – y = 4x + 4y – y
= 4x + 4y + – y
= 4x + (4 – 1)y
= 4x + 3y

Example 10.
Are the expressions 5u and 5 + u equal to each other?
The expression 5u means 5 times the number u, and the expression 5 + u means 5 more than the number u. These two being different operations, they should give different values for most values of u.

Are the expressions 10y – 3 and 10(y – 3) equal?
10y – 3, short for 10 × y – 3, means 3 less than 10 times y,
10(y – 3), short for 10 × (y – 3), means 10 times (3 less than y).
Let us compare the values that these expressions take for different values of y.

Example 11.
What is the sum of the numbers in the picture (unknown values are denoted by letter-numbers)?

There are many ways to go about it. Here, we show some of them.

1. Adding row wise gives: (4 × 3) + (r + s) + (r + s) + (4 × 3)
2. Adding like terms together gives: (8 × 3) + (r + r) + (s + s)
3. Adding the upper half and doubling gives: 2 × (4 × 3 + r + s)

The three expressions might seem different.
We can simplify each one and see that they are all the same: 2r + 2s + 24.

Mind the Mistake, Mend the Mistake
Some simplifiations of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form.

• Observe each of them and see if there is a mistake.
• If you think there is a mistake, try to explain what might have gone wrong.
• Then, simplify it correctly.

Pick Patterns and Reveal Relationships Class 7 Notes

In the first section, we got a glimpse of algebraic expressions and how to use them to describe simple patterns and relationships concisely and elegantly. Here, we continue to look for general relationships between quantities in different scenarios, fid patterns, and, interestingly, even explain why these patterns occur. Remember the importance of describing in simple language, or visualising mathematical relationships, before trying to write them as expressions.

Formula Detective
Look at the picture given. In each case, the number machine takes in the 2 numbers at the top of the ‘Y’ as inputs, performs some operations, and produces the result at the bottom. The machine performs the same operations on its inputs in each case.

Find out the formula of this number machine.

The formula for the number machine above is “two times the first number minus the second number”. When written as an algebraic expression, the formula is 2a–b. The expression for the first set of inputs is 2 × 5 – 2 = 8. Check that the formula holds for each set of inputs.

Algebraic Expressions to Describe Patterns

Example 12.
Somjit noticed a repeating pattern along the border of a saree.

Somjit wonders if there is a way to describe all the positions where the
(i) Design A occurs, (ii) Design B occurs, and (iii) Design C occurs.
Let us start with design C. It appears for the first time at position 3, the second time at position 6.

Where would design C appear for the nth time?
We can see that this design appears in positions that are multiples of 3. So the nth occurrence of Design C will be at position 3n.

Similarly, find the formula that gives the position where the other Designs appear for the nth time.
The positions where B occurs are 2, 5, 8, 11, 14, and so on. We can see that the position of the nth appearance of Design B is one less than the position at which Design C appears for the nth time. Thus, the nth occurrence of Design B is at position: 3n – 1
Similarly, the expression describing the position at which the design A appears for the nth time is: 3n – 2.

Given a position number can we fid out the design that appears there? Which Design appears at Position 122?
If the position is a multiple of 3, then clearly we have Design C. As seen earlier, if the position is one less than a multiple of 3, it has Design B, and if it is 2 less than a multiple of 3, then it has Design A.

Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below.

Patterns in a Calendar
Here is the calendar of November 2024. Consider 2 × 2 squares, as marked in the calendar. The numbers in this square show an interesting property.

Let us take the marked 2 × 2 square, and consider the numbers lying on the diagonals: 12 and 20; 13 and 19. Find their sums; 12 + 20, 13 + 19. What do you observe? They are equal.

Let us extend the numbers in the calendar beyond 30, creating endless rows.

Will the diagonal sums be equal in every 2 × 2 square in this endless grid? How can we be sure?
To be sure of this, we cannot check with all 2 × 2 squares, as there are an unlimited number of them.
Let us consider a 2 × 2 square. Its top left number can be any number. Let us call it ‘a’.

Given that we know the top left number, how do we find the other numbers in this 2 × 2 square?

As we have been doing, first let us describe the other numbers in words.

• The number to the right of ‘a will be 1 more than it.
• The number below a will be 7 more than it.
• The number diagonal to ‘a will be 8 more than it.

So the other numbers in the 2 × 2 square can be represented as shown in the grid. Let us find the diagonal sums; a + (a + 8), and (a + 1) + (a + 7).

Let us simplify them. Since the terms can be added in any order, the brackets can be opened.
a + (a + 8) = a + a + 8 = 2a + 8
(a + 1) + (a + 7) = a + 1 + a + 7 = a + a + 1 + 7 = 2a + 8
We see that both diagonal sums are equal to 2a + 8 (8 more than 2 times a).

Verify this expression for diagonal sums by considering any 2 × 2 square and taking its top left number to be ‘a. Thus, we have shown that diagonal sums are equal for any value of a, i.e., for any 2 × 2 square!

Matchstick Patterns
Look at the picture below. It is a pattern using matchsticks. Can you identify what the pattern is?

We can see that Step 1 has 1 triangle, Step 2 has 2 triangles, Step 3 has 3 triangles, and so on.
Can you tell how many matchsticks there will be in the next step, Step 5? It is 11. You can also draw this and see.

How many matchsticks will there be in Step 33, Step 84, and Step 108?
Of course, we can draw and count, but is there a quicker way to find the answers using the pattern present here?

What is the general rule to find the number of matchsticks in the next step?
We can see that at each step, 2 matchsticks are placed to get the next one, i.e., the number of matchsticks increases by 2 every time.

Think of a way to use this to find out the number of matchsticks in Step 33 (without continuing to write the numbers). As each time 2 matchsticks are being added, finding out how many 2s will be added in Step 33 will help. Look at the table below and try to find out.

What could be an expression describing the rule/formula to find out the number of matchsticks at any step?
The pattern is such that in Step 10, nine 2s and an added 3 (3 + 2 × 9) gives the number of matchsticks; in Step 11, ten 2s and an added 3 (3 + 2 × 10) gives the number of matchsticks.
For step y, what is the expression?
It is: one less than y (i.e., y–1) 2s and a 3.
Therefore, the expression is 3 + 2 × (y – 1).
This expression gives the number of matchsticks in Step y.
Now we can find the number of matchsticks at any step quickly.
You might have already noticed that there is a 2 in the first step also, 3 = 1 + 2.
Using this, the expression we get is 2y + 1.

Does the above expression also give the number of matchsticks at each step correctly? Are these expressions the same?
We can check by simplifying the expression 3 + 2 × (y – 1).
3 + 2 × (y – 1) = 3 + 2y – 2 = 2y + 1.
Both expressions are the same. There is a different way to count, or see the pattern. Let us take a look at the picture again.

Matchsticks are placed in two orientations — (a) horizontal ones at the top and bottom, and (b) the ones placed diagonally in the middle.
For example, in step 2, there are 2 matchsticks placed horizontally and 3 matchsticks placed diagonally.

Class 7 Maths Notes

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Class 7 Social Science Chapter 1 Notes Geographical Diversity of India

To us, by its very geography, the country [India] appears to be quite distinct from other countries, and that itself gives it a certain national character.

Fig. Jog Falls in Karnataka. Notice the plateau and the waterfalls. The power of the waterfall is converted into electricity (hydroelectricity; ‘hydro’ means water) through special turbines.

In 1984, Rakesh Sharma, the first Indian astronaut to go into space, spoke with the then Prime Minister of India, Indira Gandhi. When she asked him, “How does India look from space?”, he replied, “Sare jahan se achchha” — better than the whole world. (This is the title of a well-known poem of the early 20th century.)

As you go through this chapter, remember to refer periodically to the physical map.
India is the seventh-largest country in the world, and a part of Asia. Along with its neighbours — Pakistan, Bangladesh, Nepal, Bhutan, Sri Lanka, and Myanmar—it forms the region known as the Indian Subcontinent (subcontinent, since it is part of the continent of Asia). Often, for the purpose of discussion, we refer to five regions — the great mountain zone, the plains of the Ganga and the Indus, the desert region, the southern peninsula, and the islands. In this chapter, we will ‘fly’ over these zones, providing you with a bird’s-eye view of these features and offering glimpses of what things look like in close-up at some of these places. It would be difficult to go into all the details at this stage since, as you will soon see, India is large and diverse.

The Himalayan Mountain range stands as a natural barrier in the north, while the Thar Desert and the Arabian Sea mark its western limits. TO the south, the Indian Ocean and to the east, the Bay of Bengal form a natural boundary. These geographical features create a separation between India and the rest of the continent and have played a crucial role in shaping India’s climate, culture, and history.

Let us now journey together from the Himalayas to the islands in the Indian Ocean, and onward to the east of India. The diverse colours on the map already give us a sense of the geographical diversity. Familiarise yourself with the legend on the map. The different colours indicate the altitudes.

The Himalayas

Look at the length of the Himalayan Range on the map. It is like a massive wall. From the legend, can you guess the altitude at different points of the Himalayas?

Fig. These are satellite images of the Himalayan range. Note that the length of the range is about 2500 km.

In the summer, the snow on the mountains melts and feeds major rivers, such as the Ganga, Indus, and Brahmaputra. These rivers and their tributaries provide water for drinking, farming, and industrial use, connecting with the lives of hundreds of millions of people. Hence, the Himalayas are sometimes called the ‘Water Tower of Asia’. The Himalayas are also important to many cultures and belief systems. The mountains themselves are considered sacred, and temples and monasteries have been built within them, attracting monks and spiritual seekers from around the world who come to pray and meditate.

Don’t Miss Out
The Bhagirathi River, a major tributary of the Ganga, originates from Gaumukh (‘COW’S Mouth’), in Uttarakhand. It is the edge of the Gangotri Glacier. This glacier is one of the largest in the Indian Himalayas. It is considered sacred and attracts many pilgrims. Gaumukh is also a popular trekking destination. Next time you see the Ganga, remember—its journey began there!

Fig. Gaumukh

How the Himalayas were formed … an interesting story

A long, long time ago, India was part of a much bigger landmass called ‘Gondwana’, where its neighbour was Africa! At some point, it broke away and slowly started moving north. About 50 million years ago, it reached the landmass of Eurasia and collided with it. AS India pushed against Eurasia, the land between them crumpled and rose up—just like how a carpet wrinkles when you push it. That’s how the mighty Himalayan mountains were formed!

Fig. India’s journey to Eurasia

Amazingly, India is still pushing into Asia today, very slowly—about five centimetres each year, which is much slower than the rate at which your hair grows. This means the Himalayas are still growing taller, just a tiny bit each year—about five millimetres, but over a millennium, that adds up to five metres!

Fig. Folded layers of Himalayan rock

Don’t Miss Out
The word ‘Himalaya’ is a combination of two Sanskrit words—hima, meaning ‘snow’, and alaya, meaning ‘abode’ or ‘dwelling’ — thus, ‘abode of snow’.

The Himalayas are broadly categorised into three main ranges:

• The Himadri (the Greater Himalayas) are the highest and most rugged part of the range, home to towering peaks like Mount Everest and Kanchenjunga. This region remains snow-covered throughout the year. Life here is tough, and there are not many human settlements.
• The Himachal (the Lower Himalayas) lie south of the Greater Himalayas and have a more moderate climate, allowing rich biodiversity and human habitation. Popular hill stations, such as Nainital (Uttarakhand), Darjeeling (West Bengal), Shimla (Himachal Pradesh), and Mussoorie (Uttarakhand), are located in this region.
• The Shivalik Hills (the Outer Himalayas) form the outermost and lowest range, consisting of rolling hills and dense forests. These foothills are rich in wildlife, serving as a transition zone between the Himalayas and the Gangetic Plains (also called the Northern Plains).

Don’t Miss Out
The traditional house construction method in the western Himalayan region is known as ‘kath-kuni’ or ‘dhajjidewari’ style of houses. A combination of locally available stone and wood is used, which not only keeps the house warm but also resists damage in the event of mild earthquakes.

Fig. Kath-kuni House, Himachal Pradesh

The Great Himalayan National Park in Himachal Pradesh has a wide diversity of flora and fauna. The park has been declared a World Heritage Site by UNESCO. The biodiversity of the park is being preserved by the government as well as the village communities that live inside the park.

1. Great Himalayan National Park Conservation Area.
2. Himalayan monal (male).
3. A monastery in Ladakh.
4. The Beas river in Himachal Pradesh.
5. Snow Leopard
6. A display of produce in a local market in the Himalayas.
7. Rhododendron — a type of sherbet is made out of this flower

The cold desert of India

The word ‘desert’ immediately evokes an image of a hot place. However, there are also cold deserts, and we have one in India. Ladakh is a cold desert where winter temperatures drop below -30°C. There is very little rainfall, and the landscape is rugged, with rocky terrain, deep valleys, and lakes such as Pangong TSO (tso means lake).

The terrain here resembles that of the moon; hence, it is called ‘moonland’. Geologists explain the formation of this terrain, as we saw earlier, by the fact that the mountains were ‘folded’ when the Indian landmass collided with Eurasia. This folded portion was part of an ocean, and SO the rocks in this area are made largely of sand and day. Wind and rain have eroded the mountains into the shapes you see in the photograph.

Fig. Moonland, Ladakh

Yaks are very important for the lives of people in the Himalayas. They are reared for their milk, meat. wool and dung, and are also used for transport.

Fig. Pangong Tso, Ladakh. This lake has salty water unlike most other lakes. The saltiness is a result of the minerals that dissolve from the surrounding mountain areas.

Despite the harsh conditions, Ladakh is home to unique wildlife like snow leopards, ibex, and Tibetan antelopes. The Ladakhi people lead a simple life. The region is known for its ancient monasteries and colourful festivals such as Losar and the Hemis Festival.

The Gangetic Plains

As we move southwards from the Himalayas, we reach the vast and fertile Gangetic Plains. These plains have been an important part of the history and civilisation of India. These plains are nourished by mighty rivers originating from the Himalayas, providing a vital lifeline: water. The Ganga, Indus, and Brahmaputra river systems, along with their extensive network of tributaries, enrich the soil with minerals, making the region highly fertile and ideal for agriculture. The rivers bring with them minerals that enrich the soil, enabling abundant agriculture. The rivers are also a source for generating electricity. A large proportion of India’s population lives in these plains.

The flat land of the Northern Plains has allowed for the development of an elaborate transportation network. Road and railway networks facilitate the movement of people and goods over long distances. AS you will see in the Tapestry of the Past chapters, the Ganga, the Brahmaputra and other rivers have been used for millennia for travel and trade.

Fig. Modes of transport in the Gangetic plains

The Great Indian Desert or Thar Desert

If we move westward on the map, you will notice a yellowish area. This area is the Thar Desert. What do you see? A vast stretch of golden dunes, rugged terrain, and a wide-open sky?

Fig. A traveller among the sand dunes of the Thar Desert.

Fig. Jaisalmer, the ‘Golden City’, located in the middle of the Thar desert in India. The Jaisalmer fort is a UNESCO World Heritage site.

Sand dunes are formed when the wind shifts and shapes the sand into hill-like formations. Sometimes these rise as high as 150 metres.
The Thar is a vast arid region. Most of it lies within India, spanning the states of Rajasthan, Gujarat, Punjab, and Haryana. The desert acts as a natural barrier due to the harsh conditions that deter human and animal movement — including very high daytime temperatures and cold nights, as well as a lack of access to water.

People living in the Thar have adapted their way of life to the place and what it offers; food habits, clothing and lifestyle are suited to these harsh conditions.

Fig. Camel vendor at the Pushkar Mela (at the edge of the Thar Desert)

Fig. (Left) Women fetching water from a source far away from home.
(Right) Rainwater harvesting structure in a hamlet.
Water is scarce in the desert. Women often need to travel long distances every day to fetch water for their families. SO, the traditional method for cleaning utensils is to scour them with sand until they are clean. A little water can be used for a light rinse. The water used for rinsing is reused for purposes like watering a plant. SO, the next time you leave the tap running, remember the people of the Thar Desert. Rajasthan is also famous for its ingenious water conservation methods, including taanka or kunds. These are special water collection systems that store rainwater, often for drinking purposes.

The Aravalli Hills

The Aravallis are among the oldest mountains in the world, some 2.5 billion years old! The range has many peaks and ridges. Although its highest peak, Mount Abu, towers at over 1700 m, most of its hills are between 300 and 900 metres high. Isn’t it fascinating that a drive of about 4 and a half hours can take us from Mount Abu in the Aravallis to Jodhpur in the Thar Desert, a drive into a completely different geography?

Fig. (Left) A part of the Aravallis; just beyond this range, the Thar Desert begins.
(Right) A part of the Aravallis seen from space.

The Aravallis play a vital role in shaping the geography and climate of northwestern India. One of its most important functions is acting as a natural barrier, preventing the Thar Desert from expanding further eastward. You will read more about this later.

The Aravallis, rich in minerals like marble, granite, zinc, and copper, have supported mining and construction activities for centuries. In fact, evidence from its ancient mines at Zawar has shown that over eight centuries ago, Indians were the first in the world to master the delicate process of extraction of zinc. Historic forts, such as Chittorgarh, Kumbhalgarh, and Ranthambore, are located here.

Fig. Kumbhalgarh Fort surrounded by the Aravallis; this location in the hills proved to be an excellent deterrent to the enemy.

The Peninsular Plateau

A plateau is a landform that rises up from the surrounding land and has a more or less flt surface; some of its sides are often steep slopes.

India has several plateaus; the most important one is the triangular peninsular area in the middle and south of the country.

It is also a very old land formation! Since this region is a peninsula, surrounded by water on all three sides by the Arabian Sea, the Bay of Bengal, and the Indian Ocean, it is called a peninsular plateau. The plateau is bordered by two mountain ranges, the Western Ghats and Eastern Ghats. The Western Ghats are taller and run along the western coast like a wall, with many beautiful waterfalls flowing down their steep sides during the monsoon season.

Don’t Miss Out
The Western Ghats have been declared a UNESCO World Heritage Site. They are home to many rivers and have a rich biodiversity. The northern portion of the Western Ghats are also known as the Sahyadri Hills.

The Eastern Ghats are lower and broken into smaller hills along the eastern coast. Between these mountain ranges lies the Deccan Plateau, a vast area of flt highlands.
Rivers like the Godavari, Krishna, and Kaveri flw across the plateau from west to east. These rivers are important for farming and provide water to millions of people.

Fig. Dense forests of Chhattisgarh, home to many tribal communities

This plateau is rich in minerals, forests, and fertile land, making it vital for India’s economy. It tilts a little to the east, So a few of the rivers in this region flow towards the Bay of Bengal. East-flowing rivers like the Godavari, Krishna, and Mahanadi originate here, providing water for farming, industries, and hydroelectric power. There are west-flowing rivers (Narmada, Tapti) too, which drain into the Arabian Sea.

Dense forests on the plateau are home to tribal communities, including the Santhal, Gond, Baiga, Bhil, and Korku. These tribes have distinct languages, traditions, and ways of life closely connected to Nature.

Plateaus are home to many beautiful waterfalls in India, as rivers flow over their uneven and rocky surfaces. These waterfalls not only attract tourists but also help in hydroelectric power generation and provide water for irrigation.

Fig. Powerhouse Waterfalls at Periyakanal, near Munnar, Kerala


Fig. This is an insectivorous plant, meaning it feeds on insects, found in the Western Ghats. It catches small insects in sticky traps and digests them!

Fig. Coal mines in the plateau; an important resource, especially in the production of electricity. Coal is a fossil fuel, the use of which contributes to global warming.

Fig. Mumbai, on the west coast, is India’s fiancial centre

India’s Amazing Coastlines

India’s coastline is dotted with beautiful beaches, rocky cliffs, and lush green forests. Some beaches have golden sand while others have black rocks. Some islands have coral reefs while others are covered in thick jungles. India’s coasts are full of surprises! The Indian coastline is over 7500 km long.

The West Coast of India
The West Coast of India stretches from Gujarat to Kerala, passing through Maharashtra, Goa, and Karnataka. Most rivers here originate in the Western Ghats, flow swiftly, and form estuaries. The coastline is shaped by alluvial deposits from short rivers and features coves, creeks, and estuaries, with the Narmada and Tapti estuaries being the largest.

Fig. The west coast has many important ports and cities. These have been the centres of economic activity for millennia.

The East Coast
The East Coast lies between the Eastern Ghats and the Bay of Bengal, stretching from the Ganga delta to Kanyakumari. It has wide plains and major river deltas, including Mahanadi, Godavari, Krishna, and Kaveri. Important water bodies like Chilika Lake and Pulicat Lake (a lagoon, which is a body of water separated from larger bodies of water by a natural barrier) are found here.

Fig. Satellite view of the East Coast of India.

Deltas are landforms formed at the mouth of a river when it deposits sediments into a larger body of water, such as an ocean, a lake, or another river. Over time, these sediments build up, forming a triangular or fan-shaped area. The Godavari, Krishna, Kaveri and Mahanadi rivers create fertile deltas, making the land ideal for farming.

Indian Islands

The Indian Islands refer to the group of islands scattered across the Indian Ocean, the Arabian Sea, and the Bay of Bengal, which form part of India’s territory. India has two major island groups — Lakshadweep in the Arabian Sea and the Andaman and Nicobar Islands in the Bay of Bengal. These islands have unique wildlife, beautiful beaches, coral reefs, and volcanoes. Several ancient tribes made these islands their home tens of thousands of years ago.

Lakshadweep islands
Lakshadweep is an archipelago (a group of islands) located in the Arabian Sea, close to the Malabar coast of Kerala. It is made up of 36 islands made of coral. Not all islands are inhabited by people. India controls a vast marine area, allowing for fishing, resource exploration, and environmental protection.

Fig. (i) Coral reef in the Lakshadweep Islands
(ii) Coral reef in the Andaman Islands

Andaman and Nicobar islands
This archipelago comprises more than 500 large and small volcanic islands, divided into two distinct groups—the Andaman and the Nicobar Islands. Their location is very important.

Fig. A flating dock (a small port) of the Indian Navy near the Andaman Islands.

Fig. An aerial view of the active volcano on Barren Island, the only one in India

They are like the outposts of India, keeping an eye on the ocean. It is home to a variety of flora and fauna. The Andaman Islands are also significant from a historical point of view—many of our freedom fighters were jailed there under the most severe conditions in a prison complex called ‘Cellular Jail’. It has been preserved to remind us of the tremendous sacrifices that our forefathers made SO we could be free. We will discuss this some more in higher classes.

The Delta in West Bengal and the Sundarbans

As we travel back from the islands towards the eastern side of the Himalayas via the Bay of Bengal, we come to the Sundarbans. This is located in the delta of the Ganga, Brahmaputra (you saw them earlier in the chapter) and their tributaries. This delta has a unique combination of the river, sea and land. About half of it is located in India, and the rest is in Bangladesh. This is also a UNESCO Heritage site. The Sundarbans are home to many species, including the Royal Bengal Tiger.

Fig. Mangroves of the delta in the Sundarbans of West Bengal

Note: Do remember to look at the map and identify where the delta is.

Fig. Left to right, top to bottom: Seven Sisters Waterfalls, Meghalaya, India; The Shad Suk Mynsiem festival is celebrated by the Khasi people as a form of gratitude towards nature; Living roots bridge near Nongriat village, Cherrapunjee, Meghalaya

The hills of the Northeast

Stay on the map as we move towards the hills of the Northeast, our final destination for now. Can you see Garo, Khasi, and Jantia marked on the map? These hills, part of the Meghalaya Plateau, are known for their lush greenery, heavy rainfall, and breathtaking waterfalls. This region experiences one of the highest rainfalls in the world, making it rich in forests, unique wildlife, and fertile land.

Mawlynnong Village, situated in the East Khasi Hills of Meghalaya, is renowned as the ‘cleanest village in Asia’. This picturesque village is famous for its well-maintained cleanliness, bamboo dustbins, and eco-friendly living practices. The village is also known for its living root bridges, which are created by weaving tree roots over the course of many years.

Fig. Living root bridges showcase the craftsmanship of the tribes of the Northeast.

Before we move on …

• India gives its name to the subcontinent it is a part of.
• It has many diverse geographical features, ranging from the snowy Himalayas to the heat of the Thar Desert.
• The plains are watered by a large number of rivers. There is also a peninsular plateau with the Arabian Sea on the west and the Bay of Bengal in the east.
• These diverse geographic features have created a variety of conditions with respect to soil, flora, fauna, life and economic opportunities, and honed a rich culture.
• These geographical features have played an important role in shaping our civilisation.

The post Geographical Diversity of India Class 7 Notes Social Science Chapter 1 appeared first on Learn CBSE.

Class 7 Maths Chapter 5 Notes Parallel and Intersecting Lines

Class 7 Maths Notes Chapter 5 – Class 7 Parallel and Intersecting Lines Notes

→ When two lines intersect, they form four angles. The vertically opposite angles are equal, and the linear pairs add up to 180°.

→ When two lines intersect and the angles formed are 90° (i.e., all four angles are equal), the lines are said to be perpendicular to each other.

→ When two lines never intersect on a plane, they are called parallel lines.

→ When a line t intersects another pair of lines, it is called a transversal and it forms 2 sets of 4 angles. Each of the 4 angles in the first set has a corresponding angle in the second set.

→ When a transversal intersects a pair of parallel lines, the corresponding angles are equal.

→ When a transversal intersects a pair of lines and the corresponding angles are equal, then the pair of lines is parallel.

→ When a transversal intersects a pair of parallel lines, the alternate angles are equal.

→ The interior angles on the same side formed by a transversal intersecting a pair of parallel lines always add up to 180°.

Across the Line Class 7 Notes

Take a piece of square paper and fold it in different ways. Now, on the creases formed by the folds, draw lines using a pencil and a scale. You will notice different lines on the paper. Take any pair of lines and observe their relationship with each other. Do they meet? If they do not meet within the paper, do you think they would meet if they were extended beyond the paper?

In this chapter, we will explore the relationship between lines on a plane surface. The tabletop, your piece of paper, the blackboard, and the bulletin board are all examples of plane surfaces.

Let us observe a pair of lines that meet each other. You will notice that they meet at a point. When a pair of lines meet each other at a point on a plane surface, we say that the lines intersect each other. Let us observe what happens when two lines intersect.

How many angles do they form?

In Fig., where line l intersects line m, we can see that four angles are formed.

In Fig., if ∠a is 120°, can you figure out the measurements of ∠b, ∠c, and ∠d, without drawing and measuring them?
We know that ∠a and ∠b together measure 180°, because when they are combined, they form a straight angle which measures 180°. So, if ∠a is 120°, then ∠b must be 60°.

Similarly, ∠b and ∠c together measure 180°. So, if ∠b is 60°, then ∠c must be 120°. And ∠c and ∠d together measure 180°. So, if ∠c is 120°, then ∠d must be 60°.

Therefore, in Fig., ∠a and ∠c measure 120°, and ∠b and ∠d measure 60°.

When two lines intersect each other and form four angles, labelled a, b, c, and d, as in Fig., then ∠a and ∠c are equal, and ∠b and ∠d are equal!

Is this always true for any pair of intersecting lines?
Check this for different measures of ∠a. Using these measurements, can you reason whether this property holds for any measure of∠a?
We can generalise our reasoning for Fig., without assuming the values of ∠a.
Since straight angles measure 180°, we must have ∠a + ∠b = ∠a + ∠d = 180°.
Hence, ∠b and ∠d are always equal.
Similarly, ∠b + ∠a = ∠b + ∠c = 180°, so ∠a and ∠c must be equal.
Adjacent angles, like ∠a and ∠b, formed by two lines intersecting each other, are called linear pairs. Linear pairs always add up to 180°.

Opposite angles, like ∠b and ∠d, formed by two lines intersecting each other, are called vertically opposite angles. Vertically opposite angles are always equal to each other. From the above reasoning, we conclude that whenever two lines intersect, vertically opposite angles are equal. Such a justification is called a proof in mathematics.

Measurements and Geometry
You might have noticed that when you measure linear pairs, sometimes they may not add up to 180°. Or, when you measure vertically opposite angles, they may be unequal sometimes. What are the reasons for this?
There could be different reasons:

• Measurement errors because of improper use of measuring instruments — in this case, a protractor
• Variation in the thickness of the lines drawn. The “ideal” line in geometry does not have any thickness! But we can’t draw lines without any thickness.

In geometry, we create ideal versions of “lines” and other shapes we see around us, and analyse the relationships between them. For example, we know that the angle formed by a straight line is 180°. So, if another line divides this angle into two parts, both parts should add up to 180°. We arrive at this simply through reasoning and not by measurement. When we measure, it might not be exactly so, for the reasons mentioned above. Still, the measurements come out very close to what we predict, because of which geometry finds widespread application in different disciplines such as physics, art, engineering, and architecture.

Perpendicular Lines Class 7 Notes

Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out the measure of each angle?

If two lines intersect and all four angles are equal, then each angle must be a right angle (90°).
Perpendicular lines are a pair of lines that intersect each other at right angles (90°). In Fig., we can say that lines l and m are perpendicular to each other.

Between Lines Class 7 Notes

Observe Fig. and describe the way the line segments meet or cross each other in each case, with appropriate mathematical words (a point, an endpoint, the midpoint, meet, intersect) and the degree measure of each angle.
For example, line segments FG and FH meet at the endpoint F at an angle of 115.3°.

Are line segments ST and UV likely to meet if they are extended?
Are line segments OP and QR likely to meet if they are extended?
Here are some examples of lines we notice around us.

What is common to the lines in the pictures above? They do not seem likely to intersect each other. Such lines are called parallel lines.

Parallel lines are a pair of lines that lie on the same plane, and do not meet however far we extend them at both ends.

Name some parallel lines you can spot in your classroom.

Parallel lines are often used in artwork and shading.

Parallel and Perpendicular Lines in Paper Folding Class 7 Notes

Here is an activity for you to try.

• Take a square sheet of paper, fold it in the middle, and unfold it.
• Fold the edges towards the centre line and unfold them.
• Fold the top right and bottom left corners onto the creased line to create triangles. Refer to Fig.
• The triangles should not cross the crease lines.
• Are a, b, and c parallel to p, q, and r, respectively? Why or why not?

Notations
In mathematics, we use an arrow mark (>) to show that a set of lines is parallel. If there is more than one set of parallel lines (as in Fig.), the second set is shown with two arrow marks and so on. Perpendicular lines are marked with a square angle between them.

From previous exercises we observed that sometimes it is difficult to be sure whether two lines are parallel. To determine this we use the idea of transversals.

Transversals Class 7 Notes

We saw what happens when two lines intersect in different ways. Let us explore what happens when one line intersects two different lines.

In Fig., line t intersects lines l and m. t is called a transversal. Notice that 8 angles are formed when a line crosses a pair of lines.

What about five different angles — 6, 5, 4, 3, and 2?
In Fig., since ∠1 and ∠3 are vertically opposite angles, they are equal. Are there other pairs of vertically opposite angles? We can see that there are a total of four pairs of vertically opposite angles, and in each pair, the angles are equal to each other. Thus, when a transversal intersects two lines, it forms eight angles with a maximum of four distinct angle measures.

Corresponding Angles Class 7 Notes

In Fig., we notice that the transversal t forms two sets of angles — one with line l and another with line m. There are angles in the first set that correspond to angles in the second set based on their position. ∠1 and ∠5 are called corresponding angles. Similarly, ∠2 and ∠6, ∠3 and ∠7, ∠and 4 and ∠8 are the corresponding angles formed when transversal t intersects lines l and m.

Activity 3
Draw a pair of lines and a transversal such that they form two distinct angles.
Step 1: Draw a line l and a transversal t intersecting it at point X.

Step 2: Measure ∠a formed by lines l and t (let us say it is 60°).

How many distinct angles have formed now?
If one angle is 60°, the other angle of the linear pair should be 120°.
So, we already have two distinct angles.
So, when we draw another line intersecting the transversal t we wish to form only two angles, 60° and 120°.

Step 3: Mark a point Y on line t.

Step 4: Draw a line m through point Y that forms a 60° angle to line t.
This can be done either by copying ∠a with a tracing paper or you can use a protractor to measure the angles.

What do you observe about lines l and m? Do they appear to be parallel to each other?
Yes, they do appear to be parallel to each other.
Angles ∠a and∠b are corresponding angles formed by the transversal t on lines l and m. These corresponding angles are equal to each other. From this, we can observe:

When the corresponding angles formed by a transversal on a pair of lines are equal to each other, then the pair of lines are parallel to each other.

Suppose we have a transversal intersecting two parallel lines. What can be said about the corresponding angles?

Activity 4
Fig. has a pair of parallel lines l and m (what is the notation used in the figure to indicate they are parallel?). Line t is the transversal across these two lines. ∠a and ∠b are corresponding angles. Take a tracing paper and trace ∠a on it. Now, place this tracing paper over ∠b and see if the angles align exactly. You will observe that the angles match. Check the other corresponding angles in the figure using a protractor. Are all the corresponding angles equal to each other?

Corresponding angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.

Activity 5
In Fig., draw a transversal t to the lines l and m such that one pair of corresponding angles is equal. You can measure the angles with a protractor.

Are you finding it hard to draw a transversal such that the corresponding angles are equal?

When a pair of lines is not parallel to each other, the corresponding angles formed by a transversal can never be equal to each other.

Drawing Parallel Lines Class 7 Notes

Can you draw a pair of parallel lines using a ruler and a set square?
Fig. shows how you can do it.
Draw a line l with a scale. By sliding your set square, you can make two lines perpendicular to line l.
Are these two lines parallel to each other?
How are we sure that they are parallel to each other?
What angles are formed between these lines and line l?

Since we used a set square, the angles measure 90°. The position of the lines is different, but they make the same angle with l. If line l is seen as a transversal to the two new lines, then the corresponding angles measure 90°.

As we know, these are corresponding angles and they are equal, we can be sure that the lines are parallel. Draw two more parallel lines using the long side of the set square as shown in Fig.

How do you know these two lines are parallel? Can you check if the corresponding angles are equal?

Making Parallel Lines through Paper Folding
Let us try to do the same with paper folding. For a line l (given as a crease), how do we make a line parallel to l such that it passes through point A?
We know how to fold a piece of paper to get a line perpendicular to l. Now, try to fold a perpendicular to l such that it passes through point A.
Let us call this new crease t. Now, fold a line perpendicular to t passing through A again.
Let us call this line m. The lines l and m are parallel to each other.

Alternate Angles Class 7 Notes

In Fig., ∠d is called the alternate angle of∠f, and ∠c is the alternate angle of ∠e.

You can find the alternate angle of a given angle, say ∠f, by first finding the corresponding angle of∠f, which is ∠b and then finding the vertically opposite angle of ∠b, which is ∠d.

Activity 6
In Fig., if ∠f is 120°, what is the measure of its alternate angle ∠d?
We can find the measure of ∠d if we know ∠b because they are vertically opposite angles. Remember, vertically opposite angles are equal.

What is the measure of ∠b?
It is 120° because it is the corresponding angle of ∠f.
So, ∠d also measures 120°.

∠F = ∠b irrespective of the measure of ∠f. Why?
Because ∠b is the corresponding angle of ∠f.

Similarly, ∠b = ∠d irrespective of the measure of ∠b. Why?
Because ∠d is the vertically opposite angle of ∠b.
So, it must always be the case that ∠f = ∠d.

Using our understanding of corresponding angles without any measurements, we have justified that alternate angles are always equal.

Alternate angles formed by a transversal intersecting a pair of parallel lines are always equal to each other.

Example 1.
In Fig., parallel lines l and m are intersected by the transversal t. If ∠6 is 135°, what are the measures of the other angles?

Solution:
∠6 is 135°, so ∠2 is also 135°, because it is the corresponding angle of ∠6, and the lines l and m are parallel.
∠8 is 135°, because it is the vertically opposite angle of ∠6. ∠4 is 135° because it is the corresponding angle of ∠8.
∠2 is 135° because it is the vertically opposite angle of ∠4.
So, ∠2, ∠4, ∠6, and ∠8 are all 135°.
∠5 and ∠6 are a linear pair; together they measure 180°.
If ∠6 is 135°, then ∠5 = 180 – 135 = 45°
We can similarly find out that ∠1, ∠3, and ∠7 measure 45°.

Example 2.
In Fig., lines l and m are intersected by the transversal t. If ∠a is 120° and ∠f is 70°, are lines l and m parallel to each other?

Solution.
∠a is 120°, so ∠b is 60° because ∠a and ∠b form a linear pair.
∠b is the corresponding angle of ∠f.
If l and m are parallel, ∠b should be equal to ∠f, however, they are not equal.
Therefore, lines l and m are not parallel to each other as the corresponding angles formed by the transversal t are not equal to each other.

Example 3.
In Fig., parallel lines l and m are intersected by the transversal t. If ∠3 is 50°, what is the measure of ∠6?

Solution:
∠3 is 50°; therefore, ∠2 is 130°, because ∠2 and ∠3 form a linear pair, and linear pairs always add up to 180°.
∠2 and ∠6 are corresponding angles, and they need to be equal since lines l and m are parallel.
So, ∠6 is 130°.
Angles ∠3 and ∠6 are called interior angles.

Is there a relation between ∠3 and ∠6?
You could try to find the relationship by taking different values for ∠3 and see what ∠6 is. Once you find a relation, try to justify it or prove that this relation holds always. You will find that the sum of the interior angles on the same side of the transversal always adds up to 180°.

Example 4.
In Fig., line segment AB is parallel to CD and AD is parallel to BC. ∠DAC is 65° and ∠ADC is 60°. What are the measures of angles ∠CAB, ∠ABC, and ∠BCD?
Solution:
Let us observe the parallel lines AB and CD. AD is a transversal of these two lines.

We know that the sum of the interior angles formed by a transversal on a pair of parallel lines adds up to 180°.
So ∠ADC + ∠DAB = 180°
60° + ∠DAB = 180°.
So ∠DAB = 120°.
Can we find ∠CAB from this?
∠DAB = ∠DAC + ∠CAB.
So 120° = 65° + ∠CAB.
So ∠CAB = 55°.

Let us observe the parallel line segments AD and BC. They are intersected by a transversal CD.
So, ∠ADC + ∠BCD = 180°, because they are interior angles on the same side of the transversal.
Since ∠ADC is given as 60°, ∠BCD = 120°.
Similarly, we find ∠ABC = 60°.
Therefore, in Fig., ∠CAB = 55°, ∠ABC = 60°, and ∠BCD = 120°.

Parallel Illusions Class 7 Notes

There do not seem to be any parallel lines here. Or, are there?

What causes these illusions?

Class 7 Maths Notes

The post Parallel and Intersecting Lines Class 7 Notes Maths Chapter 5 appeared first on Learn CBSE.

Class 7 Social Science Chapter 2 Notes Understanding the Weather

A change in the weather is sufficient to create the world and oneself anew. (Marcel Proust, French novelist)

Weather and its Elements

You wake up one winter morning and shiver. You reach for thick clothes to keep yourself warm. In the summer, you choose clothes that keep you cool and comfortable. You are responding to your body’s signals; your body is sensing the weather.

What is weather?

Weather is a state of the Earth’s atmosphere at a particular time and place. But what is an atmosphere? In simple terms, it is the layer of gases that surround some planets—in the case of our Earth, we call these gases ‘air’. The Earth’s atmosphere may Fig. 2.2 be compared to a cake
with several layers. The layer closest to the surface of the Earth is called the ‘troposphere’, and that is where all land-based plants and animals (including humans!) live and breathe. It is also where almost all weather phenomena, which we will explore in this chapter, take place. The troposphere extends to a height of 6 to 18 kilometres from the ground; it is less thick at the poles (where the cold air contracts) and thicker in the tropical zone (where the warmer air expands). You will study more about the other layers in your Science classes.

We use many words to describe the weather—hot, cold, rainy, cloudy, humid, snowy, windy, and so on. They describe the different ways in which we experience the elements of weather.

The elements of weather are:

• Temperature: How hot or cold the atmosphere is. Precipitation: Any form of water, such as rain, snow, sleet or hail, that falls from the sky.
• Atmospheric Pressure: The weight of the air above us, felt on the Earth’s surface.
• Wind: The movement of air, including its speed and direction.
• Humidity: The amount of water vapour in the air.

As you can see, it would be difficult for Krishnan to convey his sense of chillness to Amir unless there is a commonly agreed way to measure the temperature. It is the same with other elements of the weather. In this chapter, we will learn how we measure the weather using common standards.

From early times, humans have closely observed Nature and learnt to read her signals to forecast the weather. Observing birds fling low, ants carrying eggs, squirrels gathering nuts, frogs croaking loudly, or even the opening and closing of pine cones, provided valuable information about coming rain or storms. This knowledge has been passed down from generation to generation.

Even today, in many parts of India, people use traditional ways to predict the weather, especially the arrival of the monsoon.

Observing Nature’s clues

Fig. Ants shifting their eggs to higher ground is a natural behaviour that indicates an expected change in the weather, especially heavy rain.

Fig. A frog croaking in a forest of the Western Ghats, in expectation of rain.

Fig. The opening and closing of pine cones are natural mechanisms driven by environmental humidity. Pine cones close in humid conditions to protect their seeds, and open in dry conditions to release them, ensuring they spread in favourable weather

In the last few centuries, scientists have worked out methods to measure and monitor the elements of the weather with great precision. Based on those inputs, meteorologists try to predict how the weather will behave in a particular region after a few hours or a few days, or even a few weeks. How do they do it? Do they just look up at the sky and guess? No, they’ve got some cool gadgets, a few of which we will now look at.

Weather Instruments

a) Temperature
In your Grade 6 Science textbook, Curiosity, you read about different types of thermometers used for measuring the temperature—the clinical thermometer and the laboratory thermometer. You also learnt about temperature scales. One of them is the Celsius scale; another is the Fahrenheit scale. If, for instance, we have a cool temperature of 15 degrees Celsius (noted as 15°C), it is the same as 59 degrees Fahrenheit (noted as 59°F).

Fig. Snow melts quickly when it’s warm.

Fig. Cloudy weather ―it’s getting cold.

Fig. In winter, coconut oil turns solid.

Fig. Curd takes longer to set in cold weather

There are several types of thermometers. Some simply measure the ambient temperature; others record the maximum and minimum temperatures during a day. Thermometers often use a
coloured liquid which expands when the temperature increases. However, more and more, digital thermometers are preferred as they are more precise and can record more data.

Indeed, temperature recordings can be used to collect some useful statistics, including:

• Range of temperature or the maximum temperature minus the minimum temperature during a particular period of time (usually 24 hours).
• Mean daily temperature or the maximum temperature plus the minimum temperature of the day divided by two.

Don’t Miss Out

• The India Meteorological Department was set up in 1875. Its motto is adityat jayate vrishti, which means, “From the sun arises rain.” The phrase comes from the ancient text Manusmṛti, and the complete sentence reads, “From the sun arises rain, from rain comes food, and from food, living beings originate.”
• Can you think of a reason why rain arises from the sun?

b) Precipitation
If the news says that a particular place received 30 mm of rainfall in a day, what does it mean? How is rainfall measured?
The amount of rainfall is measured with the help of an instrument called a rain gauge (Fig. 2.6). When it rains, the water falls into a funnel and is collected in a cylinder. A scale is attached to the cylinder to measure the depth of rainwater collected. For example, when the height of the water collected is 5 mm, we say that the area received 5 mm of rainfall.

Fig. Rain Gauge

c) Atmospheric pressure
Our bodies are quite aware of temperature and rainfall. But you may also have experienced that the weather sometimes feels ‘heavy’, as before a thunderstorm. This is related to atmospheric pressure, which is the pressure exerted by the weight of the air above and around us.

The atmospheric pressure is higher near the sea coast and lower as we go higher up into the mountains. When you climb a mountain, the air gets thinner than in the plain below. As a result, the air pressure is lower, and there is less oxygen available for your lungs to take in. With less oxygen getting into your blood, your body has to work harder to keep you moving! That’s why people sometimes feel breathless, dizzy or tired at high altitudes.

This does not mean that the atmospheric pressure is always high in the plains below or on the coast. In fact, it sometimes drops dramatically, resulting in what meteorologists call a ‘depression’ or ‘low-pressure system’, which can sometimes develop into a storm or even a cyclone.

The instrument used to measure atmospheric pressure is called a barometer. As with thermometers, there are several types of barometers. The unit they display is generally the millibar (abbreviated as mb). The normal atmospheric pressure at the sea coast is around 1013 mb; a pressure below 1000 mb indicates a depression.

People who journey to places at a high altitude are advised to make pauses on the way to allow the body to acclimatise. Our army personnel serve in high-altitude places like Khar dung la in Ladakh, which is over 5600 metres above sea level. It is hard to imagine how they live and work in places where the oxygen level is so low—the atmospheric pressure there is generally about 650 millibars!

d) Wind
Wind is the movement of air from areas of high pressure to areas of low pressure. Speed and direction are two important factors when we describe the wind.

The wind is an important element of the weather. Its direction and speed help in weather forecasting. Besides, air pilots and sailors need to be aware of wind data, as the wind has a great influence on flying or sailing. Farmers also use the wind direction to predict where rain might come from. Also, a greater wind speed will cause the soil to dry faster.

So, how do we measure this direction and speed? A wind vane (or weather vane) has a rotating arm with a pointer at one end and a tail at the other. When the wind blows, the tail is pushed, and the pointer turns in the direction of the wind. It responds even to a light breeze.

Left: Wind vane on the tarmac. Right: Anemometer

This wind vane on the tarmac is called a ‘wind sock’. It gives pilots an indication of the direction of the wind during take-of and landing. Similar socks are used in industries that release ash or gases.

The simplest instrument to measure the wind direction and speed is the anemometer. It has three or four metal cups that rotate on a vertical shaft when the wind blows — the stronger the wind, the faster the rotation. A meter attached at the bottom counts how many times the anemometer spins in a certain period of time and calculates the wind speed in kilometres per hour (km/h)

e) Humidity
Humidity is the last element of the weather on our list. It refers to the amount of water vapour present in the air. It also depends on factors like temperature, wind, pressure and location.

We can answer these questions more precisely by learning how to measure humidity.
Before we move forward, we need to remember our Science lesson from Grade 6 about the states of water. This will help us to understand how humidity is measured.

• When water evaporates, it causes a cooling effect.
• If the amount of water in the air is already high (more humidity), water evaporates slowly. That is typically the case on a rainy day.

Humidity of the air is measured as relative humidity: air that would contain absolutely no water vapour (which is impossible in natural conditions) is rated at 0%, while air saturated with water vapour will have a humidity of 100%. In practice, dry weather has a relative humidity range between 20% and 40%, while humid weather usually falls between 60% and 80% relative humidity.

But how do we measure such numbers? This is done through an instrument called a hygrometer. Again, there are several types of hygrometers, depending on the principle they are based on. The measurement of humidity is of great importance in many industrial processes, such as food processing. Museums also monitor humidity as they need to maintain a dry environment to preserve their exhibit.

Weather Stations

As you can see, we need several instruments to measure the weather at a particular place and time. A weather station brings all these instruments together, making it easy to measure and track the weather. Readings of all the measurements are taken at regular intervals, which helps in mapping and forecasting the weather.

An automated weather station
An Automated Weather Station (AWS) is a self-operating system that uses various sensors to measure and record weather data, such as temperature, humidity, wind speed and direction, precipitation, and atmospheric pressure. Such stations are widely used in fields like agriculture, aviation, navigation, environmental monitoring, and so on, providing accurate and timely weather information without the need for human intervention.

Don’t Miss Out
In 2023, the National Disaster Management Authority set up an AWS at a glacial lake of Sikkim at an altitude of more than 4800 metres above sea level. The AWS provides early information about upcoming weather conditions.

Fig. AWS at a glacial lake of Sikkim

Predicting the Weather

Meteorologists collect data using these instruments over long periods of time. They study the data and use scientific methods to try and predict the weather. Such predictions are very important nowadays, as climate change makes extreme weather, such as droughts, flods, cyclones, etc., more frequent.


Accurate predictions help us to be ready for such events. They also enable local governments to mobilise resources and prepare for any disasters. For example, if stormy weather is expected at sea, fishermen are warned about venturing out in their boats, or an entire coastal area might have to be evacuated if a cyclone is expected.

Before we move on …
Temperature, humidity, precipitation, wind and atmospheric pressure together define the weather at a particular place. The condition of these elements is measured using special instruments. Data collected from these help us to monitor and predict the weather.

• In different times or situations, one of the elements is dominant—for example, rainfall in July, the temperature in May and December, atmospheric pressure when a cyclone is moving, and wind when a loo (strong, hot and dusty winds that blow in north India in summers) is blowing, or forest fires are spreading.
• Weather is closely linked to climate. We will discuss this in the next chapter.

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Class 7 Social Science Chapter 3 Notes Climates of India

kale varshatu parjanyaha prithivl sasyashalinl deshoyam kshobharahitah brahmanasantu nirbhayah May the rains be timely, may the Earth be lush with vegetation May this country be free from turmoil, may good people be fearless! – Subhashita

Weather, Seasons and the Climate

‘Climate’ is a word people use in everyday conversations. But, quite often, what they really mean is ‘weather’, and not ‘climate’. What’s the difference? ‘Weather’ is what we experience every hour or day: it might be raining, or there could be bright sunshine, a lot of wind, etc. The weather keeps changing. Climate, on the other hand, is the pattern of weather an area or a region experiences over a long period of time — several decades at least. This pattern varies from region to region.

Before we come to the various types of climates, let us briefly stop to introduce seasons. Seasons occur as the Earth revolves around the Sun. Each season lasts for a few months and recurs every year. As we know, there are several seasons in the year — spring, summer, monsoon, autumn and winter—that occur in a cycle. Are seasons related to the weather or to the climate? The answer is: to both.

The weather changes with the season, that is, it can be dry and hot in the summer months, or humid and rainy during the monsoon. Further, the pattern of seasons in a region is closely connected with the climate. There are four main seasons across most regions of the world—spring, summer, autumn and winter. However, India receives rainfall during a specific period of the year — the ‘rainy season’ or monsoon — along with the other four.

Traditionally, in many parts of India, the year is divided into six seasons or ritus— vasanta (spring), grlshma (summer), varsha (rainy season), sharad (autumn), hemanta (pre-winter), shishir (winter). Specific rituals and festivals are associated with these six ritus, such as Vasanta PanchamI or Sharad Purnima.
If we observe the world around us, we will notice that human, plant and animal life are in rhythm with the ritus or seasons. The crops we grow, the food we eat, the clothes we wear, and so on, change with the season. Depending on the region, some trees and shrubs bloom with the onset of vasanta or spring, in some places they shed their leaves or change their colour as sharad or autumn approaches and some animals develop thick fur in the cold winter months.

Usually, the climate remains stable over long periods of time. However, scientists have been recording changes in the climates of the world in the recent decades. Studies show that many of these changes have been caused by human actions.

Let us recapitulate:
Weather is what we experience from day to day—windy, rainy, hot, dry, etc.
Seasons recur every year and the weather of a place is different in every season.
Climate is the long-term pattern in a particular region. There are many types of climates in the world. We will now survey the main types found in India.

Types of Climates in India

We have often seen how India is a land of diversity. This is equally true of its climate:
In the north, the Himalayan mountains have an alpine climate with cold, snowy winters and cool summers (the word ‘alpine’ comes from the Alps, a mountain range of Europe). That’s probably where you will see the thickest clothing in India!
Lower in the Himalayas, and in many hilly areas of India, the climate is often said to be temperate, as the winters are moderately cold and the summers not too hot. That is where we find many ‘hill stations’, much frequented by people seeking relief from the heat in the plains below.

• In the northern plains, the climate is subtropical, with very hot summers and cold winters. This is where most of India’s wheat is grown.
• In the west, the Thar Desert has an arid climate featuring extremely hot days, cool nights, and very little rainfall. People there have had to develop unique ways to collect and save water.
• The western coastal strip receives heavy rainfall during monsoon months, creating a tropical wet climate, which is favourable to the growing of rice and spices.
• The central Deccan Plateau has a semi-arid climate with hot summers, mild winters and moderate rainfall during the rainy season.
• Eastern India and the southern peninsula experience a tropical climate with a mild winter and distinct wet and dry periods controlled by monsoon winds (more on them soon).

Note: You will understand later the meaning of the terms ‘tropical’ and ‘subtropical’, which are related to two special parallels of latitude called the ‘tropics’.

Factors Determining the Climate

What creates those different climates? There are many factors at work. Some are general ones — on the scale of the planet—while others are regional or even local.

(a) Latitude:
Places near the Equator, that is, at low latitudes, are warmer, while those near the poles (high latitudes) are colder. This happens due to the angle at which the sun’s rays hit a particular area. At the Equator, the rays are nearly perpendicular, and so all their energy is focused on a smaller area of the Earth’s surface. In the polar regions, the rays are inclined or oblique, and the energy is distributed over a larger surface. Moreover, they have to pass through more of the Earth’s atmosphere, as the diagram shows, which further dissipates their energy. As a result, the polar regions receive less heat than the equatorial zone. We can see this in India too: Kanniyakumari and the Nicobar Islands being close to the Equator are warm or hot almost throughout the year, whereas places in the north, such as Srinagar, are much cooler.

Fig. At the equator exposure to the sun’s rays is concentrated, but near the poles it is dispersed over a wide area

(b) Altitude
We mentioned hill stations above, which are popular tourist centres because of their cooler climate. India has many including Munnar, Theni, Udhagamandalam (Ooty), Madikeri, Mahabaleshwar, Mount Abu, Shimla, Nainital, Darjeeling, Tawang, Shillong, etc. We know they are located at a higher altitude than the plains below, but how does that explain their cooler temperature? You will later learn the full answer in your Science classes.

To keep things simple for now, the temperature decreases as the altitude increases because:

1. The atmospheric pressure, and therefore the air density, decreases as the altitude increases (as we saw in the chapter ‘Understanding the Weather’), and as the air gets less dense, it gets cooler.
2. The sun heats the surface of the Earth, and so the farther away from the surface, the less hot the air will be. The Himalayas are so high that many peaks maintain a temperature below water’s freezing point, keeping them covered in snow.

c) Proximity to the sea
Temperatures in coastal areas do not vary much; the summers are not too hot and the winters are not too cold. That is because the sea acts as a moderator for the temperature.

This diagram sums up the phenomenon, and your Science textbook explains this further while discussing the heating and cooling of land and water in coastal regions. The result is that those regions tend to be more temperate. As you move inland from the coast, the temperatures get more extreme — summer temperatures will be higher and winter temperatures lower.

For instance, Mumbai and Nagpur are located at a similar latitude, but Mumbai, being near the sea, has cooler summers (around 32°C) and milder winters (around 18°C), while Nagpur, away from the coast, experiences up to 44°C in summer and about 10°C in winter. As you can see, Mumbai’s range of temperature (i.e. the diffrence between the maximum and the minimum) is about 14°C while the range at Nagpur reaches 34°C.

d) Winds
Wind can move masses of warmer or cooler air. States like Punjab, Haryana, Rajasthan and Madhya Pradesh often receive winds blowing from the west. Having travelled over the deserts of Arabia to Afghanistan, they bring dry and hot air that causes severe heat waves in summer. In winter cold winds from across the Himalayas creep into the Himalayan foothills, resulting in cold waves.

Winds affect not only temperature, but also humidity, and in turn, precipitation. We saw dry winds coming from distant deserts; in contrast, winds coming from the sea bring moisture over the land, which may result in rainfall. We will soon see the case of the monsoon winds.

e) Topography
Finally, the topography of a region also plays an important role in determining its climate. For instance, the Himalayas and the Karakoram ranges protect, to some extent, the Indian subcontinent from the winds of the cold deserts of Central Asia. The nearly flat topography of the Thar desert, on the other hand, has nothing to protect it from hot and dry winds. In the next section, we will see the role of the Western Ghats in India’s southwest monsoon.

Putting it all together…
The climate of any region is determined collectively by all the above factors. Describing the climate involves a description of the patterns of temperature, precipitation (rain or snowfall, occurrence of fog or mist) and wind conditions in a region over three decades or more.

A microclimate is a climate localised in a small area, which differs from the climate in the surrounding region. It has a unique pattern of temperature, humidity, precipitation, etc., in a small geographical area.

Fig. Aravallis and urban heat islands

For instance, enclosed valleys and some forests have micro-climates of their own. So do ‘urban heat islands’, that is, some cities that have a large number of buildings and other concrete structures and very little vegetation; all this traps the heat and they are often much warmer than the surrounding region.

Microclimates can influence the local flora and fauna, the crops grown, and impact human health and well-being.

The Monsoons

The monsoon season is central to life in India. During the monsoon months, the rivers fill up, the soil is soaked with water, crops grow and life thrives. Strictly speaking, the word ‘monsoon’, which comes from the Arabic word mausim, meaning ‘season’, refers to seasonal winds over a large area of the Indian Ocean and surrounding regions, including Australia, Africa and South Asia.

There is a yearly pattern to the monsoons. Their mechanism is complex, but based on the simple fact that land heats up or cools down faster than the ocean. Monsoons bring out the fundamental relationship between temperature, pressure and wind movement.

To put it simply, as summer begins, the Asian landmass heats up, creating a powerful low-pressure system over it. Since air always flows from high pressure to low pressure, winds are drawn from the cooler, high-pressure ocean towards the hot land. These ocean winds carry moisture that condenses over the warmer land and falls in the form of heavy monsoon rains. (That is why ‘monsoon’ commonly refers to the seasonal rains rather than the winds.)

The pattern reverses in winter, when the landmass cools down more rapidly than the ocean. Now the land has a high-pressure system while the ocean remains warmer with relatively lower pressure. This causes the winds to blow in the opposite direction — from the land towards the ocean—bringing dry conditions to much of Asia.

Coming to India, the monsoon rains typically advance from the southern tip of India in early June, moving northward over several weeks until they cover the entire subcontinent by mid- July. It is not a smooth progression, though the Western Ghats (remember our brief survey of topography) act as a natural barrier; their western slopes receive much rainfall, while the Deccan plateau to the east receive less, and often with interruptions. This is commonly called the summer or southwest monsoon (‘southwest’ reflecting the direction the winds come from).

As winter approaches, the winds reverse and blow from the land to the ocean, as we just saw. These are dry winds that bring cold weather to south India, but a part of them, passing over the Bay of Bengal, collect some moisture and bring rainfall to parts of east and south India. This is called the winter or northeast monsoon.

Don’t Miss Out
Mawsynram, located in Meghalaya, receives the highest average annual rainfall in the world, about 11,000 mm (which is the same as 11 metres!).

The monsoons have inspired many ragas in both Carnatic and Hindustani classical music. Meghamalhdr and amruthavarshini are names of a couple of them.

Climate and our Lives

Our lives are deeply connected with the climate and dependent on it. The impact of climate is visible in local cultures for instance, and India has many festivals associated with seasons and agricultural activities.

The climate also has a direct impact on the economy. For instance, you may have heard the phrase ‘monsoon failure’, which refers to poor rainfall during the monsoon season; in such a case, the agriculture suffers, people (women, generally) have to walk long distances in search of water, and agricultural labourers are likely to migrate to the cities; food (grains, vegetables and fruits, to begin with) becomes costlier, fuelling inflation. Industrial activity also often depends on a predictable weather and the availability of water. The world over, we can easily detect such connections between the climate and socioeconomic conditions. Those conditions get especially strained when climate disasters strike.

Climates and Disasters

India’s diverse weather patterns can bring about extreme conditions such as cyclones, floods, landslides and other such climate-related disasters. These events affect people’s lives, disrupt agriculture, damage infrastructure and disturb local economies.

a) Cyclones
Every year, the Indian coastline, especially the eastern coast, witness several cyclones. In past years, some of them have been highly destructive, resulting in the loss of human and animal life, damaging property and infrastructure, uprooting trees and causing soil erosion. The India Meteorological Department (IMD) keeps track of coming cyclones and provides information on their formation, evolution, place of landing, etc.

The National Disaster Response Force (NDRF) is specially trained to respond to natural as well as human-made disasters. NDRF battalions are located at 12 different locations in India. The NDRF has played a key role in rescue and evacuation during disasters like cyclones, landslides and floods.

How are cyclones formed? We saw that in some special situations, the atmospheric pressure near the sea becomes lower than the surrounding areas, creating a low-pressure system. This is an invitation to the air from surrounding areas to come into the low-pressure area, and the air from the sea moves in, bringing with it moisture and rain. When the low- pressure system is intense and the wind speeds are high, this may result in a cyclone.

As winds collect moisture, they form clouds and rotate inwards towards the centre of the depression. This centre, which is cloudless, is called ‘the eye of the cyclone’.

Fig. Cyclone Fani

Fig. Eye of the storm

b) Floods
A flood occurs when water overflows into normally dry land. This could be due to heavy rainfall generating huge run-off water that the land cannot absorb, or due to excessive accumulation of water in bodies like rivers and lakes, until the water overflows or their banks are breached. Floods occur frequently during the monsoons. States such as Uttar Pradesh, Bihar, Kerala, Andhra Pradesh and Assam are particularly vulnerable to floods.

In the Himalayan regions, on the other hand, floods occur when glacial lakes overflow. Glacial lakes form a barrier of rocks and ice to hold their water, which often comes from melting glaciers. If the glaciers melt too fast (as is increasingly the case) or if there is too much rainfall, the build-up of pressure can cause the water to break through the barrier—this is called a glacial burst and it often has devastating consequences for people and property.

Don’t Miss Out
In 2013, Uttarakhand experienced a sudden glacial burst caused by continuous heavy rain over several days. Many landslides followed. Areas around one of India’s important sacred sites, the Kedarnath temple, were completely destroyed. Several villages were washed away in the floods, along with many roads and bridges. Altogether about 6,000 people, many of them pilgrims, lost their lives.

Many cities experience flooding when there is heavy rainfall. This may be due to an overburdened drainage system or poorly planned construction encroaching on the waterways and blocking the flow of water. Besides, urban surfaces of concrete or asphalt do not allow water to be absorbed by the earth.

c) Landslides
A landslide is the sudden collapse of rock, soil, or debris, often triggered by heavy rain, earthquakes or volcanic activity. Landslides are common in hilly and mountainous regions such as Himachal Pradesh, Uttarakhand, Sikkim and Arunachal Pradesh, as well as the Western Ghats and hilly regions. These events often occur during the monsoon.

In those regions, the chances of landslides have increased due to human activities such as the cutting down of forests, building infrastructure without following approved methods and the construction of too many buildings that block the natural flow of water.

d) Forest fires
Forest fires are uncontrolled fires that spread rapidly across vegetation, often fuelled by dry climatic conditions, droughts or high winds. Human carelessness is another frequent cause. Forest fires are common in states with large forested or grassland areas such as Uttarakhand, Himachal Pradesh, Madhya Pradesh and Chhattisgarh, as well as mountain ranges such as the Western Ghats. Apart from destroying large areas of forest, fires harm wildlife, degrade the ecosystem, spoil the air quality and displace local communities. The consequences are therefore both environmental and economic.

Climate Change

Climate change refers to significant, long-term changes in the climate. This may be on the scale of the planet or on a regional scale, and it involves shifts in temperature, precipitation and weather events. In past millennia (we can go back millions of years, in fact), natural processes drove climate change. Since the 19th century, however, climate change has been largely driven by human activities, particularly the burning of fossil fuels, deforestation, environmentally harmful industrial practices, and production and patterns of excessive or wasteful consumption.

Why does the burning of fossil fuels affect the climate? In the Earth’s natural carbon cycle, carbon dioxide (C02) and other gases are released gradually into the atmosphere and trap heat from the Sun. This natural ‘greenhouse effect’ warms Earth enough to support life. However, human activities like industry, transportation, and agriculture have released enormous amounts of these ‘greenhouse gases’ in just a few centuries. This sudden increase traps extra heat, causing rapid global warming and disrupting the climate patterns that plants, animals, and human societies have adapted to over thousands of years.

In India, rising temperatures are perceptible in many regions. Early in 2025, for instance, the country’s average temperature was 1 to 3°C above normal, as a result of which the winter was much shorter and milder than usual. This affects not only agricultural production but also many small-scale industries. This is only one example showing how a warmer planet will present us with increasing challenges.

Understanding the relationship between the causes of climate change and disasters can help us to prepare better for these challenges. It also supports the need for more environment- friendly practices and building resilience and adaptation in communities. Governments worldwide, including India’s, attempt to promote measures of climate mitigation, such as cutting down on greenhouse gas emissions, planting trees, boosting renewable energy and improving energy efficiency, promoting sustainable lifestyles, etc. But these often clash with a desire for economic growth and increased consumption.

Before we move on …
India’s diverse climate is shaped by its geography, including mountains, deserts, and plateaus.
Weather is short-term, seasons recur on a yearly basis, and climate reflects long-term patterns over decades.

Factors such as latitude, altitude, proximity to the sea, wind and topography determine the climate.
Monsoons are vital for agriculture, influencing crop cycles and livelihoods.

• Climate is connected with cultural traditions, festivals, agriculture and economic activity.
• Understanding the climate helps prepare for natural disasters like floods and cyclones.
• Climate change leads to extremes of weather or temperature and can have severe consequences on the natural and human worlds.

The post Climates of India Class 7 Notes Social Science Chapter 3 appeared first on Learn CBSE.

Class 7 Social Science Chapter 4 Notes New Beginnings Cities and States

The kingdom shall be protected by fortifying the capital and the towns at the frontiers. The land should not only be capable of sustaining the population but also outsiders in times of calamities…. It should be beautiful, being endowed with cultivable land, mines, timber forests, elephant forests, and good pastures rich in cattle. It should not depend [only on] rain for water. It should have good roads and waterways. It should have a productive economy, with a wide variety of commodities …. – Kautilya, Arthashastra

Let us recall that in the early 2nd millennium BCE (that is, over a few centuries after 2000 BCE), the Indus / Harappan / Sindhu-Sarasvatl civilisation, which we called India’s ‘First Urbanisation’, disintegrated. Some of its cities were abandoned; in others, some people continued living there, but reverting to a rural or village lifestyle. They had to, since all the components of the Harappan urban order had disappeared: elaborate structures, both private and public; crowded streets and busy markets; different communities with specialised occupations (metalsmiths, potters, builders, weavers, craftspeople, and so on); a writing system; a sanitation system; the presence of an administration; and, behind it all, a larger state structure with a ruling class at the head. And for a whole millennium, urban life remained absent from India, though there may have been a few towns here and there in north India.

Fig. The fertile Gangetic plains helped the mahājanapadas to grow and prosper.

Indeed, there were important regional cultures, which we need not study here.
Then, in the 1st millennium BCE, a vibrant new phase of urbanisation began in the Ganga plains, parts of the Indus (or Sindhu) basin and neighbouring regions, gradually spreading to other parts of the Subcontinent. How do we know this?

Mainly from two sources:

1. archaeological excavations that have confirmed the existence of those ancient urban centres, and
2. ancient literature describing them—late Vedic, Buddhist and Jain literatures are full of references to these new urban centres.

This new phase is often called India’s ‘Second Urbanisation’—which, incidentally, has continued right up to today! Let us see how this phase emerged.

Janapadas and Mahdjanapadas

Towards the end of the 2nd millennium BCE, regional cultures gradually reorganised themselves in north India. As people formed clans or groups, probably sharing a common language and common customs, each clan came to be associated with a territory or janapada led by a raja or ruler. CJanapada’ is a Sanskrit word which means ‘where the people (jana) have set foot (pada),’ that is, have settled down.)
The janapadas grew as trade networks expanded and connected them. By the 8th or 7th centuries BCE, some of those early states had merged together; the resulting bigger units were known as mahojanapadas. Although the texts have different lists of them, the more frequent list gives the names of 16 mahdjanapadas, extending from Gandhara in the northwest to Anga in the east and to Ashmaka in central India, close to the Godavari River (see map). There may have been a few more, along with some smaller janapadas continuing independently.

The map (Fig. 4.3) shows the mahajanapadas’ capitals. Most were fairly large well-fortified cities, with a moat running outside the fortifications as further defence. Often, the gateways through the rampart walls would be deliberately kept narrow, so guards may control the movement of people and goods entering or leaving the city. It is fascinating to note that most of those ancient capitals continue to be living cities today—‘modern’ cities that are often 2,500 years old!

Early Democratic Traditions

Each janapada had an assembly or council, called sabha or samiti, where matters concerning the clan would be discussed. (Remember, from the chapter on ‘India’s Cultural Roots’, that the words sabha and samiti first appear in the Vedas, India’s most ancient texts.) We may assume that most of the members were elders in the clan. The raja was not expected to rule independently or arbitrarily;

Fig. Map of the sixteen mahajanapadas. Note that their borders are approximate.

a good ruler was supposed to take the advice from those assemblies, apart from the ministers and administrators. Indeed, according to some texts, an incompetent ruler could be removed by the assembly. Of course, while such mentions are significant, it does not mean that this was an established law; let us remember that the data we have for such remote periods is incomplete.

In their political systems, the mahajanapadas expanded the basic principles of the janapadas. Some were, in effect, monarchies, in the sense that the raja was the ultimate authority, supported by ministers and an assembly of elders. His position was hereditary, in the sense that a raja would usually be the son of the previous one. The king would collect taxes or revenue, maintain law and order, get impressive fortifications built around their capital, and maintain an army to defend the territory or wage war with neighbouring ones, as the case may be. Magadha (located in part of today’s Bihar), Kosala (in part of today’s Uttar Pradesh) and Avanti (in part of today’s Madhya Pradesh) were among the most powerful such states.

Fig. Ruins of a complex at Kauśhāmbī, capital of the Vatsa mahajanapada

However, at least two mahajanapadas, Vajji (or Vrijji) and the neighbouring Malla, had a different system: the sabha or samiti had more power and took important decisions through discussion, and, if necessary, through vote. Surprisingly, this included the selection of the rajal This means that those mahajanapadas, which were called ganas or sanghas, were not monarchies—their functioning might be called democratic, since members of the assembly were the ones to select the ruler and take major decisions. In fact, scholars have often called them ‘early republics’, as they are indeed one of the earliest such systems in the world.

More Innovations

The age of the janapadas and mahajanapadas was an age of profound change, which would impact Indian civilisation until present times. In the chapter ‘India’s Cultural Roots’ in Grade 6, we saw the emergence of several new schools of thought—late Vedic, Buddhist, Jain in particular, and their respective literatures. Those schools disseminated their teachings and literature through scholars, monks and nuns travelling across India or people undertaking pilgrimages. Indian art also underwent a renewal; it will blossom in the age of empires.

Urbanisation does not happen without technologies. Let us remember that the Harappan civilisation mastered copper and bronze metallurgy. Now, in this Second Urbanisation, a major shift in technology involved iron metallurgy. In several regions of India, the techniques of extracting and shaping iron were actually perfected from the early 2nd millennium BCE, but it took a few centuries for iron to become a part of daily life. By the late 2nd millennium BCE, iron tools had become widespread, facilitating agriculture on a bigger scale. Iron also made better weapons than bronze, lighter and sharper—swords, spears, arrows, shields, etc. As it happens, there is some evidence of warfare between neighbouring mahajanapadas—how frequent or how intense it was, of course, is impossible to tell. Such military campaigns, but occasionally alliances too, gave rise to new kingdoms and empires, which we will turn to later in our journey.

Another innovation was the first use of coins in India, made necessary by growing trade. Very soon, coins were exchanged across different regions and even with other parts of the world. The first Indian coins were made of silver, a soft metal into which symbols could be ‘punched’; they are called ‘punch-marked coins’. Later, coins of copper, gold and other metals were also made. Generally, a mahajanapada issued its own coins, but coins from neighbouring regions were used as well as exchanged in trade.

The Varna-Jati System

We saw earlier how human societies grew more complex with the rise of civilisation. Whenever this happens, a society organises itself in several groups based on class, occupation or some other criteria. For instance, there could be different groups concerned with governance, administration, religion, education, trade, town-planning, farming, crafts, arts and all kinds of other professions.

Fig. A panel of the Sanchi stūpa depicting a smithy (or metal workshop), where different workers bring firewood, water, stoke the fire, beat the iron, etc.

Fig. A few punch-marked coins from various ancient cities of north India.

In an ideal society, all those groups would complement each other and work in harmony. But most of the time, these divisions also lead to inequalities: some groups acquire more wealth, power or influence than others. In other words, while equality is an ideal that human societies have often aspired to, very few, if any, have ever achieved it.

In India, the society was organised in a two-fold system. One category was the jati, a group or community of people with a specific professional occupation closely tied to their livelihood. The skills that defined a particular jati—for instance, skill in agriculture, metallurgy, commerce or any craft — was generally transmitted from generation to generation. Often, a jati would get further subdivided into sub-jatis, each of which developed customs and traditions of its own, for instance concerning marriage, rituals or food habits.

Along with the jati, there is another category, that of varna, a concept that emerged from Vedic texts. There were four varnas: Brahmins were engaged in preserving and spreading knowledge, and in the performance of rituals; Kshatriyas were expected to defend the society and the land, and to engage in warfare if necessary; Vaishyas were supposed to increase the society’s wealth through occupations of trade, business or agriculture; finally, Shudras were the artisans, craftspeople, workers or servants.

Don’t Miss Out
You may have heard the English word ‘caste’. It comes from a Portuguese word, casta, as Portuguese travellers to India in the 16th century CE tried to make sense of Indian society. While a few scholars consider ‘caste’ to refer to varnas, most take it to apply to jatis; yet others consider ‘caste’ to refer to the whole varna-jati system.

There is historical evidence, both in texts and inscriptions, that in the early period individuals and communities changed their professional occupations if circumstances demanded. For instance, a long drought or some natural calamity could force a community of farmers to migrate to a city and take up other occupations, or some Brahmins would turn to trade or even military activities. This complex system structured Indian society, organised its activities, including economic ones, and therefore gave it some stability. In time, however, the system became rigid and led to inequalities and discrimination towards the lower jatis or some communities excluded from the varna-jati system. This process will be studied in a higher grade.

The varna-jati system has had a deep impact on Indian society, and generations of scholars have studied its countless aspects. There is a broad agreement that the system was significantly different (more flexible, in particular) in earlier periods and became more rigid with the passage of time, in particular during the British rule in India. Let us also keep in mind that while varna-jati has been an important mechanism at work in Indian society, it is not the only one; there have been many others, some of which we will explore later, especially in the theme ‘Our Cultural Heritage and Knowledge Traditions’.

Developments Elsewhere in India

In this 1st millennium BCE, important communication routes opened up for purposes of trade, pilgrimage, military campaigns, etc. Two routes became widely used and are often mentioned in the literature: the Uttarapatha and the Dakshinapatha. The first connected the northwest regions to the Ganga plains, all the way to eastern India; the second started from KaushambI (near Prayagraj), then a capital of one of the mahajanapadas, and crossed the Vindhya Range of hills to proceed all the way south. We will return to these routes when we explore the formation of empires in India.

Fig. Shishupalgarh (today a suburb of Bhubaneswar, first excavated in 1948): one of the gateways into the city, through the fortifications; the moat, full of water, is visible outside the gateway. Notice the narrowing in the gateway, for control of movement of people and goods.

Many lateral roads also connected with other parts of India, especially the important ports on the western and eastern coasts, which were vibrant centres of trade. In the eastern region, major cities emerged, such as Shishupalgarh (today Sisupalgarh, part of Bhubaneswar), which was the capital of the Kalinga region and followed a strict square ground plan, with imposing fortifications and broad streets.
In the Subcontinent’s southern regions, cities began emerging from about 400 BCE, although recent excavations claim to find some signs of commercial activities going further back. Around this time, three kingdoms emerged—the Cholas, the Cheras and the Pandyas.

Fig. Timeline covering the period from 1900 BCE to 300 BCE.

Fig. Shell and gemstones industries at the site of Kodumanal (near Erode, Tamil Nadu)

Apart from archaeological evidence, the most ancient Tamil literature mentions those kingdoms and several of their kings.

Because the southern regions are rich in resources such as precious and semiprecious stones, gold, and spices, they profitably traded not only with the rest of India but also with kingdoms and empires overseas.

By 300 or 200 BCE, almost the entire Subcontinent, including regions in the Northeast, was one vibrant interconnected land; goods and culture travelled from region to region, and often beyond India to parts of Central and Southeast Asia.

About the same time, the mahajanapadas ceased to exist, leaving the place to fresh developments that were going to reshape India.

Before we move on …

• From the end of the 2nd millennium BCE, janapadas rose in parts of north and central India; they were smaller states with a raja at the head taking counsel from an assembly of elders.
• The 16 mahajanapadas were the first organised states of the 1st millennium BCE; they witnessed the Second Urbanisation of India, which spread in all directions from the Ganga region, all the way to south India. By 300 BCE or so, the mahajanapadas ceased to exist.
• In the same period, a vast network of roads connected north and south, east and west, and eventually all regions of the Subcontinent. People, goods, ideas and teachings travelled along all those roads.

The post New Beginnings Cities and States Class 7 Notes Social Science Chapter 4 appeared first on Learn CBSE.

Class 7 Maths Chapter 6 Notes Number Play

Class 7 Maths Notes Chapter 6 – Class 7 Number Play Notes

→ In the first activity, we saw how to represent information about how a sequence of numbers (e.g., height measures) is arranged without knowing the actual numbers.

→ We learnt the notion of parity — numbers that can be arranged in pairs (even numbers) and numbers that cannot be arranged in pairs (odd numbers).

→ We learnt how to determine the parity of sums and products.

→ While exploring sums in grids, we could determine whether filling a grid is impossible by looking at the row and column sums. We extended this to construct magic squares.

→ We saw how Virahanka numbers were first discovered in history through the Arts. The Virahanka sequence is 1, 2, 3, 5, 8, 13, 21, 34, 55, …

→ We became math detectives through cryptarithms, where letters replace digits.

Numbers Tell Us Things Class 7 Notes

What do the numbers in the figure below tell us?
Remember the children from the Grade 6 mathematics?
Now, they call out numbers using a different rule.

What do you think these numbers mean?
The children rearrange themselves, and each one says a number based on the new arrangement.

Could you figure out what these numbers convey? Observe and try to find out.
The rule is — each child calls out the number of children in front of them who are taller than they are. Check if the number each child says matches this rule in both arrangements.

Picking Parity Class 7 Notes

Kishor has some number cards and is working on a puzzle: There are 5 boxes, and each box should contain exactly 1 number card. The numbers in the boxes should sum to 30. Can you help him find a way to do it?

Can you figure out which 5 cards add to 30? Is it possible?
There are many ways of choosing 5 cards from this collection.
Is there a way to find a solution without checking all possibilities?
Let us find out.

Add a few even numbers together. What kind of number do you get?
Does it matter how many numbers are added?
Any even number can be arranged in pairs without any leftovers.
Some even numbers are shown here, arranged in pairs.

As we see in the figure, adding any number of even numbers will result in a number that can still be arranged in pairs without any leftovers. In other words, the sum will always be an even number.

Now, add a few odd numbers together. What kind of number do you get? Does it matter how many odd numbers are added?
Odd numbers can not be arranged in pairs. An odd number is one more than a collection of pairs. Some odd numbers are shown below:

Can we also think of an odd number as one less than a collection of pairs?

This figure shows that the sum of two odd numbers must always be even! This, along with the other figures her,e are more examples of a proof!

Let us go back to the puzzle Kishor was trying to solve. There are 5 empty boxes. That means he has an odd number of boxes. All the number cards contain odd numbers. They should add to 30, which is an even number. Since adding any 5 odd numbers will never result in an even number, Kishor cannot arrange these cards in the boxes to add up to 30.

Two siblings, Martin and Maria, were born exactly one year apart. Today they are celebrating their birthday. Maria exclaims that the sum of their ages is 112. Is this possible? Why or why not?

As they were born one year apart, their ages will be (two) consecutive numbers. Can their ages be 51 and 52? 51 + 52 = 103. Try some other consecutive numbers and see if their sum is 112.

The counting numbers 1, 2, 3, 4, 5, … alternate between even and odd numbers. In any two consecutive numbers, one will always be even and the other will always be odd!

What would be the resulting sum of an even number and an odd number? We can see that their sum can’t be arranged in pairs and thus will be an odd number.

Since 112 is an even number, and Martin’s and Maria’s ages are consecutive numbers, they cannot add up to 112. We use the word parity to denote the property of being even or odd.

For instance, the parity of the sum of any two consecutive numbers is odd. Similarly, the parity of the sum of any two odd numbers is even.

Small Squares in Grids
In a 3 × 3 grid, there are 9 small squares, which is an odd number. Meanwhile, in a 3 × 4 grid, there are 12 small squares, which is an even number.

Parity of Expressions
Consider the algebraic expression: 3n + 4. For different values of n, the expression has different parity:

Come up with an expression that always has even parity.
Some examples are: 100p and 48w – 2. Try to find more.

Come up with expressions that always have odd parity.

Come up with other expressions, like 3n + 4, which could have either odd or even parity.

The expression 6k + 2 evaluates to 8, 14, 20,… (for k = 1, 2, 3,…) — many even numbers are missing.

Are there expressions that we can use to list all the even numbers?
Hint: All even numbers have a factor of 2.

Are there expressions that we can use to list all odd numbers?

We saw earlier how to express the nth term of the sequence of multiples of 4, where n is the letter-number that denotes a position in the sequence (e.g., first, twenty-third, hundred and seventeenth, etc.).

What would be the nth term for multiples of 2? Or, what is the nth even number?
Let us consider odd numbers.

What is the 100th odd number?
To answer this question, consider the following question:

What is the 100th even number?
This is 2 × 100 = 200.
Does this help in finding the 100th odd number?
Let us compare the sequence of even and odd terms term-by-term.
Even Numbers: 2, 4, 6, 8, 10, 12,…
Odd Numbers: 1, 3, 5, 7, 9, 11,…

We see that at any position, the value in the odd-number sequence is one less than that in the even-number sequence. Thus, the 100th odd number is 200 – 1 = 199.

Write a formula to find the nth odd number.
Let us first describe the method that we have learnt to find the odd number at a given position:
(a) Find the even number at that position. This is 2 times the position number.
(b) Then subtract 1 from the even number.

Writing this in expressions, we get
(a) 2n
(b) 2n – 1
Thus, 2n is the formula that gives the nth even number, and 2n – 1 is the formula that gives the nth odd number.

Some Explorations in Grids Class 7 Notes

Observe this 3 × 3 grid. It is filed following a simple rule — use numbers from 1 – 9 without repeating any of them. There are circled numbers outside the grid.

Are you able to see what the circled numbers represent?
The numbers in the yellow circles are the sums of the corresponding rows and columns.
Fill the grids below based on the rule mentioned above:

You might have realised that it is not possible to find a solution for this grid. Why is this the case?

The smallest sum possible is 6 = 1 + 2 + 3. The largest sum possible is 24 = 9 + 8 + 7. Any number in a circle cannot be less than 6 or greater than 24. The grid has sums 5 and 26. Therefore, this is impossible!

In the earlier grids that we solved, Kishor noticed that the sum of all the numbers in the circles was always 90. Also, Vidya observed that the sum of the circled numbers for all three rows, or for all three columns, was always 45. Check if this is true in the previous grids you have solved.

Why should the row sums and column sums always add to 45?
From this grid, we can see that all the row sums added together will be the same as the sum of the numbers 1 – 9.
This can be seen for column sums as well.
The sum of the numbers 1 – 9 is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.

A square grid of numbers is called a magic square if each row, each column, and each diagonal adds up to the same number. This number is called the magic sum. Diagonals are shown in the picture.

Trying to create a magic square by randomly filling the grid with numbers may be difficult! This is because there are a large number of ways of filling a 3 × 3 grid using the numbers 1 – 9 without repetition. It can be found that there are exactly 3,62,880 such ways. Surprisingly, the number of ways to fill in the grid can be found without listing all of them. We will see in later years how to do this.

Instead, we should proceed systematically to make a magic square. For this, let us ask ourselves some questions.

1. What can the magic sum be? Can it be any number?

Let us focus, for the moment, only on the row sums. We have seen that in a 3 × 3 grid with numbers 1 – 9, the total of row sums will always be 45.
Since in a magic square the row sums are all equal, and they add up to 45, they have to be 15 each. Thus, we have the following observation.

Observation 1: In a magic square made using the numbers 1 – 9, the magic sum must be 15.

2. What are the possible numbers that could occur at the centre of a magic square?

Let us consider the possibilities one by one.
Can the central number be 9? If yes, then 8 must come in one of the other squares.
For example, in this, we must have 8 + 9 + other numbers = 15. But this is not possible!
The same issue will occur no matter where we place 8. So, 9 cannot be at the centre. Can the central number be 1?

If yes, then 2 should come in one of the other squares.
Here, we must have 2 + 1 + other numbers = 15. But this is not possible because we are only using the numbers 1 – 9.
The same issue will occur no matter where we place 1. So, 1 cannot be at the centre, either.

Using such reasoning, find out which other numbers 1 – 9 cannot occur at the centre. This exploration will lead us to the following interesting observation.

Observation 2: The number occurring at the centre of a magic square, filled using 1 – 9, must be 5.

Let us now see where the smallest number 1 and the largest number 9 should come in a magic square.
Our second observation tells us that they will have to come in one of the boundary positions. Let us classify these positions into two categories:

Can 1 occur in a corner position? For example, can it be placed as follows?
If yes, then there should exist three ways of adding 1 to two other numbers to give 15.
We have 1 + 5 + 9 = 1 + 6 + 8 = 15. Is any other combination possible?

Similarly, can 9 can be placed in a corner position?

Observation 3: The numbers 1 and 9 cannot occur in any corner, so they should occur in one of the middle positions.

Can you find the other possible positions for 1 and 9?

Now, we have one full row or column of the magic square!
Try completing it!
[Hint: First fill the row or columns containing 1 and 9]

Generalising a 3 × 3 Magic Square
We can describe how the numbers within the magic square are related to each other, i.e., the structure of the magic square.

Choose any magic square that you have made so far using consecutive numbers. If m is the letter-number of the number in the centre, express how other numbers are related to m, how much more or less than m.
[Hint: Remember how we described a 2 × 2 grid of a calendar month in the Algebraic Expressions chapter].

Once the generalised form is obtained, share your observations with the class.

The First-ever 4 × 4 Magic Square
The first ever recorded 4 × 4 magic square is found in a 10th-century inscription at the Parshvanath Jain temple in Khajuraho, India, and is known as the Chautisa Yantra.

The first ever recorded 4 × 4 magic square, the Chautisa Yantra, at Khajuraho, India
Chautis means 34. Why do you think they called it the Chautisa Yantra?
Every row, column, and diagonal in this magic square adds up to 34.
Can you find other patterns of four numbers in the square that add up to 34?

Magic Squares in History and Culture
The first magic square ever recorded, the Lo Shu Square, dates back over 2000 years to ancient China. The legend tells of a catastrophic flood on the Lo River, during which the gods sent a turtle to save the people. The turtle carried a 3 × 3 grid on its back, with the numbers 1 to 9 arranged in a magical pattern.

Magic squares were studied in different parts of the world at different points in time, including India, Japan, Central Asia, and Europe.

Indian mathematicians have worked extensively on magic squares, describing general methods of constructing them. The work of Indian mathematicians was not just limited to 3 × 3 and 4 × 4 grids, which we considered above, but also extended to 5 × 5 and other larger square grids. We shall learn more about these in later grades.

The occurrence of magic squares is not limited to scholarly mathematical works. They are found in many places in India. The picture to the right is of a 3 × 3 magic square found on a pillar in a temple in Palani, Tamil Nadu. The temple dates back to the 8th century CE.

3 × 3 magic squares can also be found in homes and shops in India. The Navagraha Yantra is one such example shown below.

Notice that a different magic sum is associated with each graha. A picture of a Kubera Yantra is shown below:

Nature’s Favourite Sequence: The Virahanka–Fibonacci Numbers! Class 7 Notes

The sequence 1, 2, 3, 5, 8, 13, 21, 34,… (Virahanka–Fibonacci Numbers) is one of the most celebrated sequences in all of mathematics — it occurs throughout the world of Art, Science, and Mathematics. Even though these numbers are found very frequently in Science, it is remarkable that these numbers were first discovered in the context of Art (specifically, poetry)! The Virahanka–Fibonacci Numbers thus provide a beautiful illustration of the close links between Art, Science, and Mathematics.

Discovery of the Virahanka Numbers
The Virahanka numbers first came up thousands of years ago in the works of Sanskrit and Prakrit linguists in their study of poetry!

In the poetry of many Indian languages, including Prakrit, Sanskrit, Marathi, Malayalam, Tamil, and Telugu, each syllable is classified as either long or short.

A long syllable is pronounced for a longer duration than a short syllable — in fact, for exactly twice as long. When singing such a poem, a short syllable lasts one beat of time, and a long syllable lasts two beats of time.

This leads to numerous mathematical questions, which the ancient poets in these languages considered extensively. Several important mathematical discoveries were made in the process of asking and answering these questions about poetry.

One of these particularly important questions was the following.
How many rhythms are there with 8 beats consisting of short syllables (1 beat) and long syllables (2 beats)? That is, in how many ways can one fill 8 beats with short and long syllables, where a short syllable takes one beat of time and a long syllable takes two beats of time. Here are some possibilities:

• long long long long
• short short short short short short short short
• short long long short long
• long long short short long
  .
  .
  .

Can you find others?
Phrased more mathematically: In how many different ways can one write a number, say 8, as a sum of 1’s and 2’s?
For example, we have:
8 = 2 + 2 + 2 + 2,
8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1,
8 = 1 + 2 + 2 + 1 + 2,
8 = 2 + 2 + 1 + 1 + 2, etc.

Do you see other ways?
Here are all the ways of writing each of the numbers 1, 2, 3, and 4, as the sum of 1’ s and 2’ s.

Try writing the number 5 as a sum of 1s and 2s in all possible ways in your notebook! How many ways did you find? (You should find 8 different ways!)
Can you figure out the answer without listing down all the possibilities? Can you try it for n = 8?

Here is a systematic way to write down all rhythms of short and long syllables having 5 beats. Write a ‘1+’ in front of all rhythms having 4 beats, and then a ‘2+’ in front of all rhythms having 3 beats. This gives us all the rhythms having 5 beats:

Thus, 8 rhythms have 5 beats!

The reason this method works is that every 5-beat rhythm must begin with either a ‘1+’ or a ‘2+’. If it begins with a ‘1+’, then the remaining numbers must give a 4-beat rhythm, and we can write all those down.

If it begins with a 2+, then the remaining number must give a 3-beat rhythm, and we can write all those down. Therefore, the number of 5-beat rhythms is the number of 4-beat rhythms, plus the number of 3-beat rhythms.

How many 6-beat rhythms are there? By the same reasoning, it will be the number of 5-beat rhythms plus the number of 4-beat rhythms, i.e., 8 + 5 = 13. Thus, 13 rhythms have 6 beats.

Use the systematic method to write down all 6-beat rhythms, i.e., write 6 as the sum of 1’s and 2’s in all possible ways. Did you get 13 ways?
This beautiful method for counting all the rhythms of short syllables and long syllables having any given number of beats was first given by the great Prakrit scholar Virahanka around the year 700 CE. He gave his method in the form of a Prakrit poem! For this reason, the sequence 1, 2, 3, 5, 8, 13, 21, 34,… is known as the Virahanka sequence, and the numbers in the sequence are known as the Virahanka numbers. Virahanka was the first known person in history to explicitly consider these important numbers and write down the rule for their formation.

Other scholars in India also considered these numbers in the same poetic context. Virahanka was inspired by the earlier work of the legendary Sanskrit scholar Pingala, who lived around 300 BCE. After Virahanka, these numbers were also written about by Gopala (c. 1135 CE) and then by Hemachandra (c. 1150 CE).

In the West, these numbers have been known as the Fibonacci numbers, after the Italian mathematician who wrote about them in the year 1202 CE — about 500 years after Virahanka. As we can see, Fibonacci was not the first, nor the second, nor even the third person to write about these numbers! Sometimes the term “Virahanka–Fibonacci numbers” is used so that everyone understands what is being referred to.

So, how many rhythms of short and long syllables are there having 8 beats?
We simply take the 8th element of the Virahanka sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, ….. Thus, 34 rhythms have 8 beats.

Write the next number in the sequence after 55.
We have seen that the next number in the sequence is given by adding the two previous numbers. Check that this holds for the numbers given above. The next number is 34 + 55 = 89.

Write the next 3 numbers in the sequence:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ____, ____, ____, …
If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

What is the parity of each number in the sequence? Do you notice any pattern in the sequence of parities?
Today, the Virahanka–Fibonacci numbers form the basis of many mathematical and artistic theories, from poetry to drumming, to visual arts and architecture, to science. Perhaps the most stunning occurrences of these numbers are in nature. For example, the number of petals on a daisy is generally a Virahanka number.

How many petals do you see on each of these flowers?

There are many other remarkable mathematical properties of the Virahanka–Fibonacci numbers that we will see later, in mathematics as well as in other subjects. These numbers truly exemplify the close connections between Art, Science, and Mathematics.

Digits in Disguise Class 7 Notes

You have done arithmetic operations with numbers. How about doing the same with letters?
In the calculations below, digits are replaced by letters. Each letter stands for a particular digit (0 – 9). You have to figure out which digit each letter stands for.

Here, we have a one-digit number that, when added to itself twice, gives a 2-digit sum. The unit’s digit of the sum is the same as the single digit being added.

What could U and T be? Can T be 2? Can it be 3?
Once you explore, you will see that T = 5 and UT = 15.

Let us look at one more example shown on the right.
Here, K2 means that the number is a 2-digit number having the digit ‘2’ in the units place and ‘K’ in the tens place. K2 is added to itself to give a 3-digit sum HMM.

What digit should the letter M correspond to?
Both the tens place and the units place of the sum have the same digit.

What about H? Can it be 2? Can it be 3?
These types of questions can be interesting and fun to solve! Here are some more questions like this for you to try out. Find out what each letter stands for.
Share how you thought about each question with your classmates; you may find some new approaches.

These types of questions are called ‘cryptarithms’ or ‘alphametics’.

Class 7 Maths Notes

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Class 7 Social Science Chapter 5 Notes The Rise of Empires

There cannot be a country without people and there is no kingdom without a country. – Kautilya in Arthashastra

Fig. Rock-cut cave inBarabar Hills, Bihar

Fig. An artistic representation of Pataliputra, located around modern-day Patna

Bhavisha and Dhruv were thrilled; they had just activated their new device, ‘Itihasa’, a time machine to travel to the past! Taking a clue from their history lesson, they decided to visit Pataliputra for their first journey —they knew it was about the same location as modern-day Patna.

Landing on the outskirts of the great city, a little dizzy, they saw a girl talking to a person on horseback dressed in strange clothes. As he left, she turned to them, and they asked her for her name.

“My name is Ira, daughter ofKanhadas, the ironsmith. Welcome to Pataliputra!”
“Glad to meet you, Ira. Our names are Bhavisha and Dhruv.”
“Sshh! Keep your voices down! Do you see those soldiers marching past? They’re heading to battle against a neighbouring kingdom that has been troubling us. Our king avoids war when he can, but he also protects his people when needed. My father helped forge many of the swords they carry, and my uncle is one of the soldiers in the group. I just came to see him off… I don’t know when he’ll return.”
(The group watches as an impressive procession of soldiers crosses a sturdy drawbridge leading out of the city, some on horseback and the army chiefs on elephant back. Then, the three children cross the same drawbridge to enter the city.)
“What kind of bridge is this?” asked Bhavisha. “And is it a river below?”
“This bridge keeps us safe,” Figure explained Ira. “It’s lifted whenever there is a danger of attack on the city. And no, it’s not a river, it’s a moat; once the drawbridge is lifted, it makes it more difficult to approach the fortifications. Canyou see those hills and forests in the distance? They provide us with timber, herbs, and many other valuable resources. Elephants for the army are also captured from the forest and trained for the army.”

“What is the opening in that hill?” asked Dhruv.
“It’s a cave. Our king is getting it carved out for a community of monks. I hope we can visit it when it’s finished!”
(As they move through Pataliputra, they take in its splendour—towering wooden ramparts with watch towers, majestic palaces and buildings, lively streets. Ira gestures towards a bustling market filled with traders from distant lands.)
“You must visit our main market before you leave! Our king welcomes travellers from all over, so you’ll get silk from China, spices and gems from the south, fine clothes from different regions—there’s nothing you won’t find in Pataliputra!”
“What are those people over there doing?” asked Dhruv.
“Oh, these are street acrobats; they build human pyramids, sing and dance, or act in short plays to entertain people. Sometimes, they perform in front of the king!”
“Your king sounds very powerful,” remarked Bhavisha. “Does he rule the region around Pataliputra?”
“Much more than that!” answered Ira. “He rules over a vast land, far, far beyond this city. His authority extends over many villages, towns and kingdoms. My uncle told me that it takes close to two months on horseback to reach the borders of the territory!”
“That sounds bigger than just a kingdom… What do you call it?”
“It is called an Empire,” Ira stated with evident pride.

What is an Empire?

The word ‘empire* comes from the Latin ‘imperium*, which means ‘supreme power*. Simply put, an empire is a collection of smaller kingdoms or territories over which a powerful ruler or group of rulers exert power, often after waging war against the smaller kingdoms. The smaller territories still had their own rulers, but they were all tributaries to the emperor, who ruled the whole territory from a capital, usually a major centre of economic and administrative power.

In ancient Sanskrit texts, words commonly used for ‘emperor* made this clear; they included samraj, meaning ‘the lord of all’ or ‘supreme ruler’; adhiraja or ‘overlord*; and rajadhiraja or ‘king of kings*.
Indian history is full of empires. They rose, expanded, lasted for a while, declined, and disappeared. In fact, the last empire that ruled the Subcontinent existed less than a century ago! But now is not the time to tell that story; we start at the other end of time, so we may understand how empires functioned in the distant past and how they deeply impacted India’s evolution at all levels — political, economic, social, and cultural.

In return for tribute and loyalty, emperors generally allowed regional kings or chiefs to continue to govern their areas.

Fig. To expand into an empire, a kingdom might first wage war against neighbouring territories so as to conquer them.

Fig. Fortified settlements would be built in strategic places, such as the empire’s borders.

Fig. Rulers endeavoured to control rivers and trade networks as that would give them control over precious resources, apart from tax revenue from the trade.

Fig. To expand into an empire, a kingdom might fist wage war against neighbouring territories so as to conquer them.

Fig. With many smaller kingdoms warring for control, the one with access to stronger military power and surplus resources would eventually become the overlord.

Trade, trade routes and guilds

Conducting military campaigns, especially in distant lands, is not as simple as it might seem. Maintaining an army is a costly affair: soldiers need to be fed, clothed, equipped with weapons, and paid; elephants and horses need to be cared for; roads or ships have to be built, and so on. All this requires considerable economic power, control over the workforce, and access to resources.

We can now understand that economic activity—especially production and trade — is one of the keys to maintaining an empire and ensuring people’s welfare and quality of life, which a good ruler should be concerned with. Therefore, establishing and controlling trade routes all over the empire’s territory and beyond is of great importance. That way, the goods traded will grow in quantity and variety, and more trade means more income for the producers and increased tax collections for the ruler.

Returning to the case of ancient India, what would have been the traded goods? There is plenty of evidence on this, at least, both from the literature and archaeological excavations—textiles, spices, agricultural produce, luxury items such as gems and handicraft products, and various animals were among the main items of trade. All this brisk trade was not limited to India; many Indian goods travelled towards distant countries by land or sea.

More often than not, traders were not just isolated individuals carrying out their own business. They soon understood the benefits of joining forces and creating guilds (shrenls). Guilds were powerful associations of traders, craftsmen, moneylenders or agriculturists. As far as evidence shows, a guild had a head (who was usually elected) and executive officers who were supposed to have all kinds of ethical qualities. Two things made traders’ guilds a remarkable institution. First, they brought together people who ended up being collaborators rather than competitors, as they realised that sharing resources and information on markets, supply and demand, workforce, etc., was to everyone’s benefit. Second, as an ancient text put it, “Cultivators, traders, herdsmen, moneylenders, and artisans have authority to lay down rules for their respective classes”; in other words, guilds had the autonomy to create their own internal rules, and the king was not to interfere with them (and why should he, if trade flourished?).

Fig. Some important trade routes from about 500 BCE onward and major cities marked on them. Notice the Uttarapatha and the Dakshinapatha routes.

Guilds spread over large parts of India and endured for centuries. Even after they ceased to exist formally, their spirit continued to influence India’s trade and business activities, sometimes even to this day. The institution of guilds provides an excellent example of the self-organising abilities of Indian society. The ancient village unit, with its various committees and councils, provides another. Indeed, an enlightened ruler would let people organise themselves and refrain from interfering if the local institutions worked satisfactorily.

The Rise of Magadha

The period between the 6th and the 4th century BCE was one of profound change in north India. We briefly visited the sixteen mahajanapadas earlier—those large kingdoms of north and central India with their assembly system. One of them, Magadha (modern-day south Bihar and some adjoining areas), rose in importance and set the stage for the fusion of many kingdoms into India’s first empire. Powerful early kings, such as Ajatashatru, played a crucial role in establishing Magadha as a dominant centre of power.

Don’t Miss Out
Two of the most famed religious figures of the world—Siddhartha Gautama, who became known as the Buddha, and Vardhamanan, better known as Mahavlra—lived in the time of King Ajatashatru. Revisit their teachings in the Grade 6 textbook’s ‘India’s Cultural Roots’ chapter.

Magadha was located in the resource-rich Ganga plains, with fertile land, abundant forests for timber, and elephants. Also, remember how the use of iron transformed other technologies, such as agriculture and warfare. Iron ore and other minerals from the nearby hilly regions proved crucial for the expansion of the kingdom. The use of iron ploughs to till the land increased agricultural produce, and lighter and sharper iron weapons strengthened the capabilities of the army.

Fig. An elaborate panel from the Sanchi Stupa depicting soldiers riding elephants, horses, or on foot, waging battle and laying siege to Kusinara (today Kushinagar), a city of north India, to recover relics of the Buddha (seen carried away on an elephant in the left part of the panel).

The production of surplus food grains allowed more people to focus on the arts and crafts, which were in demand inside and outside the empire’s borders. The Ganga and Son rivers provided a geographical advantage for trade, as they could be used for transportation. The flourishing trade boosted the empire’s income and contributed to Magadha’s rise.

A punch-marked silver coin of Mahapadma Nanda

Around the 5th century BCE, Mahapadma Nanda rose to prominence in Magadha and founded the Nanda dynasty. He successfully unified many smaller kingdoms and extended his empire across parts of eastern and northern India. As the economy thrived, he began issuing coins, demonstrating his economic power. We also learn from Greek accounts that the Nanda dynasty maintained a large army.

From various accounts of the Nanda dynasty, it appears that its last emperor, Dhana Nanda, though very rich, became highly unpopular as he oppressed and exploited his people. This paved the way for the Nanda empire to be conquered and absorbed into what would become one of the largest empires India ever knew—the Maurya empire.

Don’t Miss Out
The famed Sanskrit grammarian Panini lived around the 5th century BCE, during the time of the Nandas. He is known for composing the Ashtadhyayi, an ancient text that lists the rules of Sanskrit grammar in 3,996 short sutras.

Fig. An India post stamp commemorating Pāṇni

The Arrival of the Greeks

While events unfolded in Magadha, located in the eastern part of the subcontinent, what was happening in the northwestern region? This area was home to smaller kingdoms along an ancient route connecting to the Mediterranean. Among them, according to Greek accounts, were the Pauravas, led by their king, Porus.


Alexander, a young and powerful Greek king from Macedonia, campaigned against the Persian Empire to avenge earlier Persian invasions of Greece (during which some Indian soldiers from the Persian-ruled northwest of India fought against Greeks!). Alexander conquered the Persian Empire; the influence of Greek culture spread. His empire now spread over three continents, one of the largest in world history.

Don’t Miss Out
The satraps were governors of provinces of Persian and Greek empires who were left behind by the overlord (like Alexander) to manage the far-off territories. These satraps had significant power and freedom despite being mere officials of the rulers. Can you guess how it was possible for them to exercise such power?

Alexander’s dialogue with the Gymnosophists

Alexander heard of a group of Indian sages whom the Greeks called ‘Gymnosophists’ or ‘naked philosophers’ (probably because they wore very little clothing), who were renowned for their wisdom. Alexander challenged them with tricky questions in the form of riddles, warning that he would put those who gave wrong answers to death. However, the Gymnosophists responded to his questions calmly and intelligently. Alexander was impressed and, in the end, spared them all. Over the centuries, different versions of this story have been told, making it one of the most fascinating encounters in history!

Fig. A Greek coin probably showing Alexander on horseback attacking Porus on his elephant.

According to one account, Alexander asked, “Which is stronger, life or death?” One of the sages replied, “Life, because it endures while death does not.” Alexander then asked, “How can a man be most loved?” “If he is most powerful and yet does not inspire fear,” came the reply, perhaps as a hint to the mighty ruler!
Historians view such exchanges as a meeting of two great traditions — Greek and Indian philosophies.

The Mighty Mauryas

After that brief sojourn to the northwest, let us return to Magadha, where we witnessed the decline of the Nanda empire. Around 321 BCE, just a few years after Alexander left India with his army, a new dynasty and new empire emerged: the Maurya Empire founded by Chandragupta Maurya. It quickly absorbed the Nanda empire’s territories and went on expanding beyond.

As per many accounts, Chandragupta managed this feat with the help of an able mentor named Kautilya, who used his knowledge of politics, governance and economics to create an empire that remains one of the greatest in Indian history.

The story of Kautilya

According to Buddhist texts, Kautilya—sometimes referred to as Chanakya or Vishnugupta—was a teacher at the world-renowned Takshashila (modern-day Taxila) university His legendary tale begins in the court of Dhana Nanda, who as we saw, had become highly unpopular.

Fig. Nanda Empire

Fig. Maurya Empire

Observing this, Kauṭlya advised Dhana Nanda to change his ways or witness the collapse of his empire. Angered, Dhana Nanda insulted Kauṭlya and threw him out of his court. This led to Kauṭlya’s vow to end the ‘evil Nanda’ rule.

The rise of Chandragupta Maurya

There are many stories about the origin and adventures of Chandragupta Maurya, but their common theme is that he overthrew the Nandas and took control of Magadha to establish his rule, with Pataliputra as his capital. Do you remember that Magadha had many advantages because of its geography, an established economic system and a flourishing trade? These, combined with the advice of the master strategist, Kautilya, helped Chandragupta Maurya gradually expand his empire. He defeated the Greek satraps left behind by Alexander in the northwest and integrated the region into an empire that stretched from the northern plains to the Deccan plateau.

Fig. Megasthenes in the court of Chandragupta Maurya
(A 20th-century painting by Asit Kumar Haldar)

After Chandragupta Maurya defeated the Greeks, he maintained a diplomatic relationship with them and hosted in his court a Greek historian and diplomat, Megasthenes, who wrote about his travels in India in his book Indika—the first such written account—unfortunately lost except for some excerpts quoted by later Greek scholars.

Kautilya’s concept of a kingdom

Kautilya had a clear vision of how a kingdom (rajya) should be established, managed and consolidated. In his famous work Arthashastra (literally, ‘the science of governance and economics’), he listed directives in many areas like defence, economics, administration, justice, urban planning, agriculture and people’s welfare. One of his most important political concepts is the saptanga (see fig 5.15) or the seven parts that constitute a kingdom.

According to Kautilya, the saptanga together must create a settled, well-protected, and prosperous kingdom to be maintained both through warfare and through alliances for peace, as the case may be. He emphasised the importance of law and order in society, which necessitated a strong administration. He also detailed many laws to deal with corruption and specified punishments for any activities that went against the wellbeing of the people.

Kautilya’s central philosophy of governance is in tune with Indian values: “In the happiness of his subjects lies the king’s happiness; in their welfare his welfare. He shall not consider as good only that which pleases him but treat as beneficial to him whatever pleases his subjects.” In other words, however powerful a king may be, he must give first place to the people’s interests.

The King Who Chose Peace

Another king of the Maurya dynasty was Ashoka (268— 232 BCE), Chandragupta’s grandson, who came to be credited with major administrative and religious achievements.

Fig. Ashoka visiting the Ramagrama stupa in Nepal (from a panel at the Sanchi stupa)

At the beginning of his reign, Ashoka was quite ambitious. He had inherited a vast empire but further expanded it to cover almost the entirety of the Indian subcontinent, except for the southernmost region, but including present-day Bangladesh and Pakistan and parts of present-day Afghanistan. One encounter, however, is said to have changed the path of his life. According to one of his edicts he once marched on Kalinga (modern-day Odisha), where he waged a ferocious war. Seeing the enormous amount of death and destruction on the battlefield, Ashoka chose to give up violence and, to the greatest extent possible, adopt the path of peace and non-violence that the Buddha taught.

Embracing Buddhist teachings, Ashoka sent emissaries to Sri Lanka, Thailand, Central Asia and beyond to spread the message of the Buddha far and wide.

Historians have sometimes called Ashoka a ‘great communicator’ since he issued in many parts of his empire edicts engraved on rocks or pillars that contained his messages for the people and encouraged them to follow dharma. Most of these edicts were inscribed in Prakrit, which was the popular language in many parts of India and written in the Brahmi script (Brahmi is the mother of all regional scripts of India).

Fig. A few of the many Ashokan edicts across the Subcontinent

In his edicts, Ashoka called himself ‘Devanampiya Piyadasi’; the first word means ‘Beloved of the Gods’; the second, ‘one who regards others with kindness’. And indeed, the language of the edicts makes it clear that he was interested in depicting himself as a benevolent and compassionate ruler. Let us see a few examples of this.
Although some southern kingdoms were not part of the Mauryan kingdom, Ashoka supported their overall wellbeing. He claimed to provide medical care for people and animals even beyond his empire, prohibited hunting and cruelty to animals, and ordered medical treatment for them when necessary. If so, Ashoka was an early contributor to nature conservation and wildlife preservation. He said he had established rest houses and wells at regular intervals along the main roads of his empire and got fruit and shade trees planted. He also claimed to encourage all sects (the different schools of thought present in his time) to accept each other’s best teachings and study them.

Fig. (Left) A reproduction of a part ofAshoka’s rock edict at Girnar, Gujarat. (Right) Detail of the Topra Ashokan pillar atFeroz ShahKotla, Delhi

Although we need not take all of Ashoka’s claims literally, it is clear that in line with Kautilya’s philosophy of governance, he paid attention to the welfare of his subjects and made efforts to reach out to them.

Don’t Miss Out
You read about the word ‘dharma’ (dhamma in Prakrit) in Grade 6. Its essence cannot be easily captured. In simple terms, dharma means moral law or someone’s religious or ethical duties towards family, community or country. At a deeper level, however, dharma extends to living according to the order of the universe or ritam. This includes doing one’s duty truthfully, following rules of righteous conduct and leading a life in harmony with the cosmic order. Dharma is, therefore, duty, law, truth, order and ethics—all of it together!

The Maurya empire continued for half a century after Ashoka’s death. However, his successors were unable to hold the empire together, and many of the smaller kingdoms broke off and became independent. Around 185 BCE, India started on another phase of her journey. Bhavisha and Dhruv will join us on this journey in the next chapter.

Life in the Mauryan period

Cities like Pataliputra were bustling centres of governance and commerce. They had palaces, public buildings, and well-planned streets. With a well-organised taxation system and brisk trade, the treasury remained strong, fuelling the empire’s growth and prosperity. Officials of the administration of the empire, merchants and artisans played key roles in the city life.

Don’t Miss Out
The Sohagaura copper plate inscription, dating back to the 4th-3rd century BCE, is one of India’s earliest known administrative records.

Discovered in Sohgaura, Uttar Pradesh, it is written in Prakrit using the Brahmi script and is believed to have been issued during the reign of Chandragupta Maurya. The inscription mentions the establishment of a granary to store grain as a precaution against famines, highlighting the state’s efforts to ensure food security and support its people during times of crisis. Megasthenes’ account also throws some light on the society of that time. A substantial proportion of the population was engaged in agriculture, which was an important source of revenue for the empire. Two crops were sown in a year, as rain fell in both summer and winter. This ensured that famines were rare and people had ample food. Granaries were well stocked for any eventualities. Even if war raged nearby, farmers were protected from it, and agriculture was not disturbed.

Blacksmiths, potters, carpenters, jewellers and other artisans lived in the cities. The cities were well-planned and had signage on the streets. Communication happened through couriers who carried messages from place to place. The houses were made of wood and could be up to two storeys tall. The streets had vessels of water stored at regular intervals in case of fire.

Later accounts describe the cotton dresses people wore — a lower garment that reached below the knee halfway down to the ankles and an upper garment that they threw over their shoulders. Some wore leather shoes with designs and thick soles to make them look taller.

Fig. The Mauryas were renowned for their highly polished stone pillars, as can be seen in this capital of the Sarnath pillar

Figure has many messages for us, apart from the beauty and perfection of the sculpture, and is a fine example of Mauryan art. This capital (a word which, here, means ‘top portion’ or ‘head’) was the top of a pillar that Ashoka got erected at Sarnath, near Varanasi, where the Buddha gave his first teaching. The four lions symbolise the royal power; on the ring below, four powerful animals (an elephant, a bull, a horse and one more lion) are depicted, along with the dharmachakra or wheel of dharma, which symbolises the Buddha’s teachings.

Some Contributions of the Mauryas

Life and people

Fig. Terracotta fiurine of a dancing girl (notice her elaborate headdress, hairstyle and jewellery).

Fig. Terracotta fiurine of a female deity

Fig. Female deity (yakshī) holding a fl whisk

Fig. Terracotta of Saptamātrikās or seven mother goddesses (a continuing tradition)

Fig. Head of a terracotta horse (notice the elaborate design of the bridle).

Art and architecture

Fig. As one of India’s oldest stone structures, the Great Stupa at Sanchi is among the finest examples of Indian architecture. Note that the original structure was made of bricks and was later enlarged using stone. Ashoka is said to have constructed many such stupas, chaityas, and viharas for worship, study, and meditation.

Fig. Rock sculpture of a life-size elephant at Dhauli (in present-day Odisha, near Bhubaneswar), which symbolises the Buddha — intelligent, powerful, patient, and calm. An edict of Ashoka was engraved on a rock nearby

This image may be familiar to some of you. Indeed, this capital was chosen as India’s national emblem, to which was added the Sanskrit motto satyameva jayate or “truth alone triumphs” (see the national emblem on the left). Besides, the dharmachakra is depicted at the centre of our national flag, as you may also have noticed. The motto comes from the Mundaka Upanishad; in full, it reads satyameva jayate nanritam, that is, “truth alone triumphs, not falsehood”.

Don’t Miss Out
The big, round hemispherical structure in the centre of the stupa is called the anda. It represents the universe and is often built to house sacred relics. People walk around it in a circle as a form of worship (pradakshina).

The Fragile Nature of Empires

You will hear in higher grades about past mighty empires elsewhere in the world, such as the Roman, the Persian, the Ottoman, the Spanish, the Russian, the British empires, and so on. All of them are long gone, but historians keep debating the causes of their decline.

One of those causes, as we saw, is the temptation for some of the empire’s regions to try and become independent. This could happen if, for example, the emperor needed more resources for long military campaigns or in times of drought; local rulers would be burdened with increasing demands for tribute, leading to resentment. Or if a powerful emperor was followed by one perceived to be weak, local kings or chieftains might simply decide to take a chance and stop paying tribute. Also, the larger an empire, the more difficult it is to hold it together, as Alexander experienced; far-off territories are often the first to split away from the empire. Finally, economic crises caused by natural calamities (such as a long drought or floods) could also shake an empire’s structure.

Empires are, therefore, something of a paradox. On the one hand, they can bring about political unity, as the Mauryan empire did to almost the entire Subcontinent, and reduce or eliminate warfare among the smaller kingdoms—indeed, a well-managed empire could lead to greater prosperity than smaller, warring kingdoms. On the other hand, empires have almost always been established through war and have maintained their existence through force and repression. This makes them fragile at their core and unstable over time.

Before we move on …

• An empire is a large territory made up of many smaller kingdoms or territories. Emperors expanded their kingdoms mostly to gain fame, amass power, including military power, and control resources and economic life.
• The first empires of India emerged in regions blessed with abundant natural resources, rivers for irrigation and transport, and the production of a variety of goods for trade.
• Alexander’s campaign in northwest India had a limited political impact but opened the door for Indo-Greek cultural contacts.
• The Mauryas created a vast empire with a legacy that lasted centuries. Their legacy includes strengthening trade routes and economic systems, extensive use of coins for trade, well-designed urban settlements, and an elaborate system of administration. They also promoted art and architecture.
• Ashoka was keen to advertise his achievements and project the image of a benevolent ruler who encouraged his subjects to follow dharma.

Class 7 Social Science Notes

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Class 7 Maths Chapter 7 Notes A Tale of Three Intersecting Lines

Class 7 Maths Notes Chapter 7 – Class 7 A Tale of Three Intersecting Lines Notes

→ Use of a compass simplifies the construction of triangles when the side lengths are given.

→ A set of three lengths where the length of each is smaller than the sum of the other two is said to satisfy the triangle inequality.

→ Sidelengths of a triangle satisfy the triangle inequality, and if a given set of lengths satisfies the triangle inequality, a triangle can be constructed with those sidelengths.

→ Triangles can be constructed when the following measurements are given:

• two of the sides and their included angle.
• two angles and the included side.

→ The sum of the angles of a triangle is always 180°.

→ The altitude of a triangle is a perpendicular line segment from a vertex to its opposite side.

→ Equilateral triangles have sides of equal length. Isosceles triangles have two sides of equal length. Scalene triangles have sides of three different lengths.

→ Triangles are classified based on their angle measures as acute-angled, right-angled, and obtuse-angled triangles.

→ A triangle is the most basic closed shape. As we know, it consists of:

• three corner points, that we call the vertices of the triangle, and
• three line segments or the sides of the triangle that join the pairs of vertices.

Triangles come in various shapes. Some of them are shown below.

Observe the symbol used to denote a triangle and how the triangles are named using their vertices. While naming a triangle, the vertices can come in any order.

The three sides meeting at the corners give rise to three angles that we call the angles of the triangle.
For example, in ∆ABC, these angles are ∠CAB, ∠ABC, ∠BCA, which we simply denote as ∠A, ∠B and ∠C, respectively.

Equilateral Triangles Class 7 Notes

Among all the triangles, the equilateral triangles are the most symmetric ones. These are triangles in which all the sides are of equal lengths. Let us try constructing them.

Construct a triangle in which all the sides are of length 4 cm How did you construct this triangle, and what tools did you use? Can this construction be done only using a marked ruler (and a pencil)?

Constructing this triangle using just a ruler is certainly possible. But this might require several trials. Say we draw the base — let us call it AB — of length 4 cm (see the figure below), and mark the third point C using a ruler such that AC = 4 cm. This may not lead to BC also having a length of 4 cm. If this happens, we will have to keep making attempts to mark C till we get BC to be 4 cm long.

How do we make this construction more efficient?
Recall solving a similar problem in the previous year using a compass (in the Chapter ‘Playing with Constructions’).
We had to mark the top point of a ‘house’ which is 5 cm from two other points. The method we used to get that point can also be used here.
After constructing AB = 4 cm, we can do the following.

Step 1: Using a compass, construct a sufficiently long arc of radius 4 cm from A, as shown in the figure. Point C is somewhere on this arc. How do we mark it?

Step 2: Construct another arc of radius 4 cm from B.

Let C be the point of intersection of the arcs.

Step 3: Join AC and BC to get the required equilateral triangle.

Constructing a Triangle When its Sides are Given Class 7 Notes

How do we construct triangles that are not equilateral?
Construct a triangle of sidelength 4 cm, 5 cm, and 6 cm.
As in the previous case, this triangle can also be constructed using just a marked ruler. But it will involve several trials.

How do we construct this triangle more efficiently?
Choose one of the given lengths to be the base of the triangle: say 4 cm. Draw the base. Let A and B be the base vertices, and call the third vertex C. Let AC = 5 cm and BC = 6 cm.

Like we did in the case of equilateral triangles, let us first get all the points that are at a 5 cm distance from A. These points lie on the circle whose centre is A and has radius 5 cm. Point C must lie somewhere on this circle. How do we find it?

We will make use of the fact that point C is 6 cm away from B.
Construct an arc of radius 6 cm from B.

The required point C is one of the points of intersection of the two circles.

The reason why the point of intersection is the third vertex is the same as for equilateral triangles. This point lies on both circles.
Hence, its distance from A is the radius of the circle centred at A (5 cm), and its distance from B is the same as the radius of the circle centred at B (6 cm).

Let us summarise the steps of construction, noting that constructing full circles is not necessary to get the third vertex (See Figures).

• Step 1: Construct the base AB with one of the side lengths. Let us choose AB = 4 cm (see Fig.).
• Step 2: From A, construct a sufficiently long arc of radius 5 cm (see Fig.).
• Step 3: From B, construct an arc of radius 6 cm such that it intersects the first arc (see Fig.).
• Step 4: The point where both the arcs meet is the required third vertex C. Join AC and BC to get ∆ABC.

Construct
We have seen that triangles having all three equal sides are called equilateral triangles. Those having two equal sides are called isosceles triangles.

Are Triangles Possible for any Lengths?
Can one construct triangles having any given sidelengths? Are there lengths for which it is impossible to construct a triangle? Let us explore this.

Try to find more sets of lengths for which a triangle construction is impossible. See if you can find any pattern in them.
We see that a triangle is possible for some sets of lengths and not possible for others. How do we check if a triangle exists for a given set of lengths? One way is to actually try to construct the triangle and check if it is possible. Is there a more effient way to check this?

Triangle Inequality
Consider the lengths 10 cm, 15 cm, and 30 cm. Does there exist a triangle having these as side lengths?
To tackle this question, let us study a property of triangles. Imagine a small plot of plain land having a tent, a tree, and a pole. Imagine you are at the entrance of the tent and want to go to the tree. Which is the shorter path: (i) the straight-line path to the tree (the red path) or
(ii) the straight-line path from the tent to the pole, followed by the straight-line path from the pole to the tree (the yellow path)?

The direct straight-line path from the tent to the tree is shorter than the roundabout path via the pole. The direct straight-line path is the shortest possible path to the tree from the tent.

Will the direct path between any two points be shorter than the roundabout path via a third point? The answer is yes.

Can this understanding be used to tell something about the existence of a triangle having sidelengths 10 cm, 15 cm and 30 cm?
Let us suppose that there is a triangle for this set of lengths. Remember that at this point we are not sure about the existence of the triangle but we are only supposing that it exists. Let us draw a rough diagram.

Does everything look right with this triangle?
If this triangle were possible, then the direct path between any two vertices should be shorter than the roundabout path via the third vertex. Is this true for our rough diagram?
Let us consider the paths between B and C.
Direct path length = BC = 10 cm

What is the length of the roundabout path via the vertex A? It is the sum of the lengths of line segments BA and AC.
Roundabout path length = BA + AC = 15 cm + 30 cm = 45 cm

Is the direct path length shorter than the roundabout path length? Yes.
Let us now consider the paths between A and B.
Direct path length = AB = 15 cm

Finding the length of the roundabout path via the vertex C, we get
Roundabout path length = AC + CB = 30 cm + 10 cm = 40 cm

Is the direct path length shorter than the roundabout path length? Yes.
Finally, consider paths between C and A.
Direct path length = CA = 30 cm
Roundabout path length = CB + BA = 10 cm + 15 cm = 25 cm

Is the direct path length shorter than the roundabout path length? In this case, the direct path is longer, which is absurd. Can such a triangle exist? No.
Therefore, a triangle having sidelengths 10 cm, 15 cm, and 30 cm cannot exist.

We are thus able to see without construction why a triangle for the set of lengths 10 cm, 15 cm, and 30 cm cannot exist. We have been able to figure this out through spatial intuition and reasoning.

Recall how we used similar intuition and reasoning to discover properties of intersecting and parallel lines. We will continue to do this as we explore geometry.

Given three sidelengths, what do we need to compare to check for the existence of a triangle?
When each length is smaller than the sum of the other two, we say that the lengths satisfy the triangle inequality. For example, the set 3, 4, 5 satisfis the triangle inequality whereas, the set 10, 15, 30 does not satisfy the triangle inequality. We have seen that lengths such as 10, 15, 30 that do not satisfy the triangle inequality cannot be the sidelengths of a triangle.

Does a triangle exist with sidelengths 4 cm, 5 cm and 8 cm?
This satisfies the triangle inequality: 8 < 4 + 5 = 9

Why do we not need to check the other two sides?
This means that all the direct path lengths are less than the roundabout path lengths. Does this confim the existence of a triangle?

If one of the direct path lengths had been longer, we could have concluded that a triangle would surely not exist. But in this case, we can only say that a triangle may or may not exist.

For the triangle to exist, the arcs that we construct to get the third vertex must intersect. Is it possible to determine that this will happen without actually carrying out the construction?

Visualising the Construction of Circles
Let us imagine that we start the construction by constructing the longest side as the base. Let AB be the base of length 8 cm. The next step is the construction of sufficiently long arcs corresponding to the other two lengths: 4 cm and 5 cm.

Instead of just constructing the arcs, let us complete the full circles.
Suppose we construct a circle of radius 4 cm with A as the centre.

Now, suppose that a circle of radius 5 cm is constructed, centred at B.
Can you draw a rough diagram of the resulting figure?
Note that in the figure below, AX = 4 cm and AB = 8 cm. So, what is BX?
Does this length help in visualising the resulting figure?

Since BX = 4 cm, and the radius of the circle centred at B is 5 cm, the circles will intersect each other at two points.

What does this tell us about the existence of a triangle?
The points A and B along with either of the points of intersection of the circles will give us the required triangle. Thus, there exists a triangle having sidelengths 4 cm, 5 cm and 8 cm.

We observe from the previous problems that whenever there is a set of lengths satisfying the triangle inequality (each length < sum of the other two lengths), there is a triangle with those three lengths as sidelengths.

Will triangles always exist when a set of lengths satisfies the triangle inequality? How can we be sure?

We can be sure of the existence of a triangle only if we can show that the circles intersect internally (as in Fig.) whenever the triangle inequality is satisfied. But are there other possibilities when the two circles are constructed? Let us visualise and study them.
The following different cases can be conceived:
Case 1: Circles touch each other

Case 2: Circles do not intersect

Case 3: Circles intersect each other internally

Note that while constructing the circles, we take
(a) the length of the base AB = longest of the given length
(b) The radii of the circles are to be the smaller two lengths.

Which of the above-mentioned cases will lead to the formation of a triangle? Triangles are formed only when the circles intersect each other internally (Case 3).

Let us study each of these cases by finding the relation between the radii (the smaller two lengths) and AB (the longest length).

Case 1: Circles touch each other at a point
For this case to happen,
sum of the two radii = AB
or
sum of the two smaller lengths = longest length

Case 2: Circles do not Intersect Internally
For this case to happen, what should be the relation between the radii and AB?
It can be seen from the figure that,
sum of the two radii < AB
or
sum of the two smaller lengths < longest length

Case 3: Circles intersect each other

AB is composed of one radius and a part of the other. So,
sum of the two radii > AB,
or
sum of the two smaller lengths > longest length

Can we use this analysis to tell if a triangle exists when the lengths satisfy the triangle inequality?
If the given lengths satisfy the triangle inequality, then the sum of the two smaller lengths is greater than the longest length. This means that this will lead to Case 3 where the circles intersect internally, and so a triangle exists.

Conclusion
If a given set of three lengths satisfies the triangle inequality, then a triangle exists having those as side lengths. If the set does not satisfy the triangle inequality, then a triangle with those side lengths does not exist.

Construction of Triangles When Some Sides and Angles are Given Class 7 Notes

We have seen how to construct triangles when their sidelengths are given. Now, we will take up constructions where in place of some sidelengths, angle measures are given.

Two Sides and the Included Angle
How do we construct a triangle if two sides and the angle included between them are given? Here are some examples of measurements showing the included angle.

Construct a triangle ABC with AB = 5 cm, AC = 4 cm and ∠A= 45°.
Let us take one of the given sides, AB, as the base of the triangle.

Step 1: Construct a side AB of length 5 cm.
Step 2: Construct ∠A = 45° by drawing the other arm of the angle.
Step 3: Mark the point C on the other arm such that AC = 4 cm.
Step 4: Join BC to get the required triangle.

Two Angles and the Included Side
In this case, we are given two angles and the side that is a part of both angles, which we call the included side. Here are some examples of such measurements:

Construct a triangle ABC where AB = 5 cm, ∠A = 45°, and ∠B = 80°.
Let us take the given side as the base.

Step 1: Draw the base AB of length 5 cm.
Step 2: Draw ∠A and ∠B of measures 45°, and 80° respectively.
Step 3: The point of intersection of the two new line segments is the third vertex C.

Do triangles always exist?
Do triangles exist for every combination of two angles and their included side? Explore.
As in the case when we are given all three sides, it turns out that there is not always a triangle for every combination of two angles and the included side.

Find examples of measurements of two angles with the included side where a triangle is not possible.
Let us try to visualise such a situation. Once the base is drawn, try to imagine how the other sides should be so that they do not meet. Here are some obvious examples.

If the two angles are greater than or equal to a right angle (90°), then it is clear that a triangle is not possible.

Now we make one of the base angles an acute angle, say 40°. What are the possible values that the other angle should take so that the lines don’t meet?

It is clear that if the line from B is “inclined” sufficiently to the right, then it will not meet the line l.
(a) Try to find a possible ∠B (marked in the figure) for this to happen.
(b) What could be the smallest value of ∠B for the lines not to meet?

The blue line is the line with the least rightward bend that doesn’t meet the line ‘l’.

Visually, it is clear that the line that creates the smallest ∠B has to be the one parallel to 1. Let us call this parallel line m.
Can you tell the actual value of ∠B be in this case?
[Hint: Note that AB is the transversal.]

We have seen that when two lines are parallel, the internal angles on the same side of the transversal add up to 180°. So ∠B = 140°. So, for what values of ∠B, does a triangle not exist? Does the length AB play any part here?

From the discussion above, it can be seen that the length AB does not play any part in deciding the existence of a triangle. We can say that a triangle does not exist when ∠B is greater than or equal to 140°.

Like the triangle inequality, can you form a rule that describes the two angles for which a triangle is possible? Can the sum of the two angles be used for framing this rule?

When the sum of two given angles is less than 180°, a triangle exists with these angles. If the sum is greater than or equal to 180°, there is no triangle with these angles.
Let us take two angles, say 60° and 70°, whose sum is less than 180°.
Let the included side be 5 cm.

In general, once the two angles are fixed, does the third angle depend on the included sidelength? Try with different pairs of angles and lengths.
The measurements might show that the sidelength has no effct (or a very small effect) on the third angle. With this observation, let us see if we can find the third angle without carrying out the construction and measurement.

Consider a triangle ABC with ∠B = 50° and ∠C = 70°. Let us see how we can find ∠A without construction.

We saw that the notion of parallel lines was useful to determine that the sum of any two angles of a triangle is less than 180°. Parallel lines can be used to find the third angle, ∠BAC as well.

Let us suppose we construct a line XY parallel to BC through vertex A.

We can see new angles being formed here: ∠XAB and ∠YAC. What are their values?

Since the line XY is parallel to BC, ∠XAB = ∠B and ∠YAC = ∠C, because they are alternate angles of the transversals AB and AC.

Therefore, ∠XAB = 50°, and ∠YAC = 70°. Can we find ∠BAC from this?
We know that ∠XAB, ∠YAC, and ∠BAC together form 180°.
So, ∠XAB + ∠YAC + ∠BAC = 180°
50° + ∠BAC + 70° = 180°
120° + ∠BAC = 180°
Thus, ∠BAC = 60°
Now construct a triangle (taking BC to be of any suitable length) and verify if this is indeed the case.

Angle Sum Property
What can we say about the sum of the angles of any triangle?
Consider a triangle ABC. To find the sum of its angles, we can use the same method of drawing a line parallel to the base: construct a line through A that is parallel to BC.

We need to find ∠A + ∠B + ∠C.
We know that ∠B = ∠XAB, ∠C = ∠YAC.
So, ∠A + ∠B + ∠C = ∠A + ∠XAB + ∠YAC = 180°, as together they form a straight angle.

Thus, we have proved that the sum of the three angles in any triangle is 180°! This rather surprising result is called the angle sum property of triangles.

Take some time to reflect upon how we figured out the angle sum property. In the beginning, the relationship between the third angle and the other two angles was not at all clear. However, a simple idea of drawing a line parallel to the base through the top vertex (as in Fig.) suddenly made the relationship obvious. This ingenious idea can be found in a very inflential book in the history of mathematics called ‘The Elements’. This book is attributed to the Greek mathematician Euclid, who lived around 300 BCE. This solution is yet another example of how creative thinking plays a key role in mathematics!

There is a convenient way of verifying the angle sum property by folding a triangular cut-out of a paper. Do you see how this shows that the sum of the angles in this triangle is 180°?

Exterior Angles
The angle formed between the extension of a side of a triangle and the other side is called an exterior angle of the triangle. In this figure, ∠ACD is an exterior angle.

Find ∠ACD, if ∠A = 50°, and ∠B = 60°.
From the angle sum property, we know that
50° + 60° + ∠ACB = 180°
110° + ∠ACB = 180°
So, ∠ACB = 70°
So, ∠ACD = 180 ° – 70° = 110°,
Since ∠ACB and ∠ACD together form a straight angle.

Find the exterior angle for different measures of ∠A and ∠B. Do you see any relation between the exterior angle and these two angles?
[Hint: From the angle sum property, we have ∠A + ∠B + ∠ACB = 180°.]
We also have ∠ACD + ∠ACB = 180°, since they form a straight angle. What does this show?

Constructions Related to Altitudes of Triangles Class 7 Notes

There is another set of useful measurements concerning a triangle — the height of each of its vertices concerning the opposite sides.

In the world around us, we talk of the heights of various objects: the height of a person, the height of a tree, the height of a building, etc. What do we mean by the word ‘height’?

Consider a triangle ABC. What is the height of the vertex A from its opposite side BC, and how can it be measured?

Let AD be the line segment from A drawn perpendicular to BC. The length of AD is the height of the vertex A from BC. The line segment AD is said to be one of the ‘altitudes’ of the triangle. The other altitudes are BE and CF in the figure below: the perpendiculars drawn from the other vertices to their respective opposite sides.

Whenever we use the word height of the triangle, we generally refer to the length of the altitude to whatever side we take as base (this altitude is AD in the case of Fig.).
What would the altitude from A to BC be in this triangle?

We extend BC and then drop the perpendicular from A to this line.

Altitudes Using Paper Folding
Cut out a paper triangle. Fix one of the sides as the base. Fold it in such a way that the resulting crease is an altitude from the top vertex to the base. Justify why the crease formed should be perpendicular to the base.

Construction of the Altitudes of a Triangle
Construct an arbitrary triangle. Label the vertices A, B, C, taking BC to be the base.

Construct the altitude from A to BC Constructing the altitude using just a ruler is not accurate. To get a more precise angle of 90°, we use a set square along with a ruler. Can you see how to do this?

Step 1: Keep the ruler aligned with the base. Place the set square on the ruler as shown, such that one of the edges of the right angle touches the ruler.

Step 2: Slide the set square along the ruler till the vertical edge of the set square touches the vertex A.

Step 3: Draw the altitude to BC through A using the vertical edge of the set square.

Does there exist a triangle in which a side is also an altitude?
Visualise such a triangle and draw a rough diagram. We see that this happens in triangles where one of the angles is a right angle.
Triangles having one right angle are called right-angled triangles or simply right triangles.
Altitude from A to BC

Types of Triangles Class 7 Notes

In our study of triangles, we have encountered the following types of triangles: equilateral, isosceles, scalene, and right-angled triangles.

Did you spot any other type of triangle?
The classification of triangles as equilateral and isosceles was based on the equality of sides.

• Equilateral triangles have sides of equal length.
• Isosceles triangles have two sides of equal length.
• Scalene triangles have sides of three different lengths.

Can a similar classification be done based on equality of angles? Is there any relation between these two classifiations? We will answer these questions in a later chapter.
We used angle measures when classifying a triangle as a rightangled triangle.

What are the other types of triangles based on angle measures?
A classification of triangles based on their angle measures is acuteangled, right-angled and obtuse-angled triangles. We have already seen what a right-angled triangle is. It is a triangle with one right angle. Similarly, an obtuse-angled triangle has one obtuse angle.

What could an acute-angled triangle be? Can we define it as a triangle with one acute angle? Why not?
In an acute-angled triangle, all three angles are acute angles.

Class 7 Maths Notes

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