NCERT Solutions for Class 12th Chapter 3 Maths Chapter 3 Matrices Ex 3.3

Get Free NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Matrices Exercise 3.3 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 3 Class 12 Matrices Ex 3.3 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

Topics and Sub Topics in Class 11 Maths Chapter 3 Matrices:

NCERT Solutions for Class 12th Chapter 3 Maths Chapter 3 Matrices Ex 3.3

Ex 3.3 Class 12 Maths Question 1.
Find the transpose of each of the following matrices:
(i)
(ii)
(iii)
Solution:
(i) let A =
∴ transpose of A = A’ =

Ex 3.3 Class 12 Maths Question 2.
If
then verify that:
(i) (A+B)’=A’+B’
(ii) (A-B)’=A’-B’
Solution:

Ex 3.3 Class 12 Maths Question 3.
If
then verify that:
(i) (A+B)’ = A’+B’
(ii) (A-B)’ = A’-B’
Solution:

Ex 3.3 Class 12 Maths Question 4.
If
then find (A+2B)’
Solution:

Ex 3.3 Class 12 Maths Question 5.
For the matrices A and B, verify that (AB)’ = B’A’, where


Solution:

Ex 3.3 Class 12 Maths Question 6.
If (i) ,the verify that A’A=I
If (ii) ,the verify that A’A=I
Solution:
(i)

Ex 3.3 Class 12 Maths Question 7.
(i) Show that the matrix is a symmetric matrix.
(ii) Show that the matrix is a skew-symmetric matrix.
Solution:
(i) For a symmetric matrix a_ij = a_ji
Now,

Ex 3.3 Class 12 Maths Question 8.
For the matrix,
(i) (A+A’) is a symmetric matrix.
(ii) (A-A’) is a skew-symmetric matrix.
Solution:

=>

Ex 3.3 Class 12 Maths Question 9.
Find and ,when

Solution:

Ex 3.3 Class 12 Maths Question 10.
Express the following matrices as the sum of a symmetric and a skew-symmetric matrix.
(i)
(ii)
(iii)
(iv)
Solution:
(i) let
=>

Ex 3.3 Class 12 Maths Question 11.
Choose the correct answer in the following questions:
If A, B are symmetric matrices of same order then AB-BA is a
(a) Skew – symmetric matrix
(b) Symmetric matrix
(c) Zero matrix
(d) Identity matrix
Solution:
Now A’ = B, B’ = B
(AB-BA)’ = (AB)’-(BA)’
= B’A’ – A’B’
= BA-AB
= – (AB – BA)
AB – BA is a skew-symmetric matrix Hence, option (a) is correct.

Ex 3.3 Class 12 Maths Question 12.
If then A+A’ = I, if the
value of α is
(a)
(b)
(c) π
(d)
Solution:
Now

Thus option (b) is correct.

NCERT Solutions for Class 12 Maths Chapter 3 Matrices (आव्यूह) Hindi Medium Ex 3.3

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

The post NCERT Solutions for Class 12th Chapter 3 Maths Chapter 3 Matrices Ex 3.3 appeared first on Learn CBSE.

Mizoram Scholarship: Mizoram Scholarship is open to students residing in Mizoram which is located in southernmost Indian state having a population of 10 Lakh and also an educated agrarian economy. The Samaj Kalyan Vibhag, also mentioned as the Department of Social Welfare under the Mizoram Government performs a key role in presenting this scholarship to all scholars who are pursuing an education at secondary, higher secondary or higher level of education. The scholarships are divided into three parts viz. pre-matric, post-matric and merit cum means. The learners can find a notice about a Mizoram scholarship through Mizoram Scholarship Board (MSB) website, National Scholarship Portal, Mizoram University website, etc.

Get to know here the popular scholarships that you can apply for, the authorities offering the funds for these scholarships, schedule to apply for these scholarships and how to apply for them? Also, know about the eligibility conditions need to be fulfilled. Here, you will get all the highlights and every significant information related to the Mizoram scholarship.

List of Mizoram Scholarship Schemes

The complete list of the scholarship provided by the Mizoram government is provided here with their availability months. Students who want to apply for the scholarship need to check which scholarship is suitable for them and should apply at the given duration of application dates.

Mizoram Scholarships Eligibility Criteria

Students who want to apply for scholarships provided by the Mizoram state government have to check with the eligibility condition mentioned by the board. Only the eligible candidates can apply for the scholarships. Candidates who do not match with the eligibility and have applied for the scholarships could lead to rejection of their application. Kindly check with all the as mentioned below for all the scholarships.

Scholarships for Students

Post-Matric Scholarship for Scheduled Tribes Students

The objective of the scheme is to provide financial assistance to students belonging to Scheduled Tribes pursuing Post-Matriculation recognized courses in recognized institutions. The scheme covers professional/ technical as well as non-professional and non-technical courses at various levels and also correspondence courses including distance and continuing education. The scheme is 90:10 funding from the Ministry of Tribal Affairs and Mizoram state.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 2,50,000/- (Rs. Two lakh Fifty Thousand only) per annum.
2. Candidates must belong to Scheduled Tribes.
3. Permanently settled in Mizoram State. d) Studying Class XI up to Post-Doctoral Fellowship.

Pre-Matric Scholarship for Schedule Tribes studying in Classes IX & X

The objective of the scheme is to support parents of ST children for education of their wards studying in classes IX and X so that the incidence of drop-out, especially in the transition from the elementary to the secondary stage is minimized, and To improve participation of ST children in classes IX and X of the Pre-matric stage, so that they perform better and have a better chance of progressing to the post-matric stage of education. The scheme is 90:10 funding from the Ministry of Tribal Affairs and Mizoram state.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 2,00,000/- (Rs. Two lakh only) per annum.
2. Candidates must belong to Scheduled Tribes.
3. Permanently settled in Mizoram State. d) Studying Class IX and X.

Merit-cum-Means Based Scholarship for Students belonging to minority Community

The objective of the Scheme is to provide financial assistance to the poor and meritorious students belonging to minority communities to enable them to pursue professional and technical courses. The scheme is 100% funding from the Ministry of Minority Affairs.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 2,50,000/- (Rs. Two lakh Fifty Thousand only) per annum.
2. Candidates must belong to Muslim/Christian/Sikh/Buddhist/Jain or Parsi community.
3. Must Secure 50% in the last examination

Post Matric Scholarship for Students belonging to minority Community

The objective of the scheme is to award scholarships to meritorious students belonging to economically weaker sections of minority community so as to provide them better opportunities for higher education increase their rate of attainment in higher education and enhance their employability. The scheme is 100% funding from the Ministry of Minority Affairs.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 2,00,000/- (Rs. Two lakh only) per annum.
2. Candidates must belong to Muslim/Christian/Sikh/Buddhist/Jain or Parsi community.
3. Must Secure 50% in the last examination

Pre Matric Scholarship for Students belonging to minority Community

The objective of the Scheme is to encourage parents from minority communities to send their school-going children to school, lighten their financial burden on school education and sustain their efforts to support their children to complete school education. The funding pattern is 75:25% (Central and State) till 2013, however from 2014 onwards funding is 100% from the Ministry of Minority Affairs.

Eligibility

• Parents/Guardians’ income from all sources should not exceed Rs. 1,00,000/- (Rs. One lakh only) per annum.
• Candidates must belong to Muslim/Christian/Sikh/Buddhist/Jain or Parsi community.
• Must Secure 50% in the last examination.

Post Matric Scholarship for Students belonging to Scheduled Caste Community

The objective of the scheme is to provide financial assistance to students belonging to Scheduled Castes pursuing Post-Matriculation recognized courses in recognized institutions. The scheme covers professional, technical as well as non-professional and non-technical courses at various levels and the scheme also includes correspondence courses including distance and continuing education.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 2,50,000/- (Rs. Two lakh Fifty Thousand only) per annum.
2. Candidates must belong to Scheduled Caste.
3. Permanently settled in Mizoram State. d) Studying Class XI up to Post-Doctoral Fellowship.

Pre Matric Scholarship for Students belonging to Scheduled Caste Community

The objective of the scheme is to support parents of SC children for education of their wards studying in classes IX and X so that the incidence of drop-out, especially in the transition from the elementary to the secondary stage is minimized, and to improve participation of SC children in classes IX and X of the pre-matric stage, so that they perform better and have a better chance of progressing to the post-matric stage of education.

Eligibility

• Parents/Guardians’ income from all sources should not exceed Rs. 2,50,000/- (Rs. Two lakh Fifty Thousand only) per annum.
• Candidates must belong to Scheduled Caste.
• Permanently settled in Mizoram State.
• Studying Class XI up to Post-Doctoral Fellowship.

North Eastern Council (NEC) Merit Scholarship

The objective of the Scheme is to provide financial assistance to permanent residents of any of the States in the North-Eastern States studying various professional & technical at different levels from the fund allotted by the North Eastern Council Secretariat, Shillong.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 4,50,000/- (Rs. Four lakh Fifty Thousand only) per annum.
2. Permanently settled in Mizoram State.
3. Studying technical and professional courses up to Post-Doctoral Fellowship
4. The pass percentage for ST/SC in the last examination should not be less than 60% and 70% for General students.

Scholarship Scheme for Colleges and University Students Scoring 80%

The main objective of scholarships is to assist meritorious students with financial assistance for those who are coming from low-income families. These scholarships will help them to meet a portion of their day to day expenses while doing their higher studies. The scheme is 100% funding from the Ministry of Human Resource Development.

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 8,00,000/- (Rs. Eight lakh only) per annum.
2. Permanently settled in Mizoram State.
3. Students scoring above 80th percentile in Board examination of Class XII pursuing regular courses.

National Merit cum Means Scholarship

The objective of the scheme is to award scholarships to meritorious students of economically weaker sections to arrest their drop out of class VIII and encourage them to continue the study at the secondary stage. Scholarship of Rs. 12000/- per annum (Rs.1000/- per month) per student is awarded by the Ministry of Human Resource Development to selected students every year for study in classes from IX to XII Students studying in recognized Govt / Govt Aided/Local Body/Corporate Schools

Eligibility

1. Parents/Guardians’ income from all sources should not exceed Rs. 1,50,000/- (Rs.One lakh Fifty Thousand only) per annum.
2. The students shall have a minimum of 55 % marks or equivalent grade in Class VII examination for appearing in the selection test for award of scholarship (relax-able by 5% for SC/ST students). The students should be studying as regular students in a Government, Government-aided and local body schools.
3. Mental Ability Test may consist of 90 multiple-choice questions testing verbal and non-verbal meta-cognitive abilities like reasoning and critical thinking. The questions in the test may be on analogy, classification, numerical series, pattern perception, hidden figure, etc.
4. Scholastic Aptitude Test may consist of 90 multiple-choice questions covering subjects namely, science, social studies, and mathematics as taught in classes VII and VIII.

National Scheme of Incentives to Girls for Secondary Education

The objective of the Scheme is to establish an enabling environment to reduce the dropouts and to promote the enrolment of girl child belonging to SC/ST communities in secondary schools and ensure their retention up to 18 years of age. A sum of Rs. 3000 (Rupees three thousand only) would be deposited by the Ministry of Human Resource Development under term /fixed deposit in a public sector bank or in a post office in the name of every eligible girl child. The term/period of the deposit may be counted from the date of deposit to the date on which the girl child attains the age of 18 years. No premature withdrawal will be allowed.

Eligibility

1. The applicant must be an Indian Citizen.
2. The applicant must be female.
3. Applicant must belong to ST/SC category or must have passed class VIII examination
4. Applicant must enroll in class IX in State Government, Government-aided or Local Body schools.

Pre-Matric Scholarship for Students whose Parents engaged in cleaning and prone to health hazards

The scheme aims to provide financial assistance for Pre-Matric Education to children of the following target groups, viz. Scavengers, sweepers having traditional links with scavenging, tanners, and flayers. Under the scheme, 100% central assistance is provided to State Governments from the Ministry of Social Justice & Empowerment, Government of India.

Eligibility

1. The scholarship will be admissible to the children/wards of Indian Nationals who, irrespective of their religion belongs to one of the following categories
2. Persons who are Manual Scavengers
3. Tanners & Flayers
4. Persons engaged in hazardous cleaning
5. Eligible candidates will submit a certificate from District Social Welfare officer, identified officer of Local body, Civic Agency or any such authority as designated by State Government.

Post Matric Merit Scholarship

The objective of the Scheme is to provide financial assistance to permanent residents of Mizoram States securing 60% marks in Board/University Examination. The scheme is 100% funding from the State Plan Fund.

Eligibility

1. The scholarship will be admissible to the children/wards of Indian Nationals.
2. Persons who are Manual Scavengers
3. Tanners & Flayers
4. Persons engaged in hazardous cleaning
5. Eligible candidates will submit a certificate from District Social Welfare officer, identified officer of Local body, Civic Agency or any such authority as designated by State Government.

Scholarship for Cadets of Rashtriya Indian Military College (RIMC), Dehradun

The objective of the Scheme is to provide financial assistance to Mizo students permanently residing in the States of studying at RIMC, Dehradun, to enable them to qualify and joint National Defence Academy and further to the Armed Forces. The scheme is 100% funding from the State Plan Fund

Eligibility:

1. The scholarship will be admissible to Mizo students permanently residing in the state of Mizoram studying at RIMC, Dehradun.
2. The student should not be covered under any other Scholarship scheme.
3. Scholarship for studying in any class will be available for only one year and renewable as per Rules/Condition.
4. If after the award of scholarship the cadet is found medically unfit in any way which according to the appropriate medical authority, renders him unfit for his entry into the NDA, the scholarship awarded shall be discontinued from the date the cadet is declared unfit.

Scholarship for Cadets of Sainik School Imphal

The objective of the Scheme is to provide financial assistance to Mizo students permanently residing in the States of studying at Sainik School Imphal, to enable them to qualify and joint National Defence Academy and further to the Armed Forces.

Eligibility

1. The scholarship will be admissible to Mizo students permanently residing in the state of Mizoram studying at Sainik School, Imphal.
2. The student should not be covered under any other Scholarship scheme.
3. Scholarship for studying in any class will be available for only one year and renewable as per Rules/Condition.
4. If after the award of scholarship the cadet is found medically unfit in any way which according to the appropriate medical authority, renders him unfit for his entry into the NDA, the scholarship awarded shall be discontinued from the date the cadet is declared unfit

Scholarship for Cadets of Sainik School Chhingchhip

The objective of the Scheme is to provide financial assistance to Mizo students permanently residing in the States of studying at Sainik School Chhingchhip, to enable them to qualify and joint National Defence Academy and further to the Armed Forces. The scheme is 100% funding from the State Plan Fund.

Eligibility

1. The student should belong to Mizoram State only.
2. The student should not be covered under any other Scholarship scheme.
3. Not more than one child of the same parents’/guardians will be eligible for the said scheme.
4. The income of the parent/guardian should not exceed Rs. 1,00,000/- (One lakh) per month for Scheduled Caste and Scheduled Tribe and Rs. 50,000/- (Fifty Thousand) for General Category. Income certificates/salary certificates will be submitted at the time of admission for grant of scholarship.
5. Scholarship for studying in any class will be available for only one year and renewable as per Rules/Condition. If a student has to repeat a class, he would not get a scholarship for that class for the second (and subsequent) year.
6. If after the award of scholarship the cadet is found medically unfit in any way which according to the appropriate medical authority, renders him unfit for his entry into the NDA, the scholarship awarded shall be discontinued from the date the cadet is declared unfit.

Mizoram Research Fellowship

The objective of the Scheme is to provide financial assistance to Research workers and teachers preferably below the age of 45 years who have established their reputation for research or who in any field and who have obtained a doctorate degree, have published research work to their credit and have already shown evidence of independent research work.

Eligibility

1. Only the children or works of bonafide permanent residents of Mizoram shall be eligible for the award of fellowship which application will be made in the form as in the Annexure to these Regulations.
2. No student prosecuting higher studies or research any University or institution which is not recognized or established by law shall be eligible for the award of the fellowship.
3. No student found guilty of misconduct or breach of discipline or participating in strikes, demonstrations, picketing and the like for unfair demands from any high-level authority shall eligible for the award of fellowship under these Regulations.
4. No student who fails to secure at least 50% on an average in the last University Examination shall be eligible for the award unless otherwise relaxed by the awarding authority.
5. A student /research fellow in receipt of any scholar or stipend except merit scholarship, from any sources, shall not eligible for an award under these regulations.

Mizoram Scholarships – Application Process

After going through the details of all the scholarship schemes and eligibility criteria, the student can apply for the particular scholarships as per the instructions has given here. They also need to submit the required documents along with the application to the specified departments. Check the table below for complete details.

Note:

• For students studying inside Mizoram: Printout of online application along with documents mentioned above must be submitted to the School/Colleges/Institution for verification through online.
• For students studying outside Mizoram: Hard copy of application along with documents must be submitted to Mizoram Scholarship Board for online and offline verification.

The post Mizoram Scholarship | Complete List, Eligibility, Awards, Application appeared first on Learn CBSE.

NCERT Solutions for Class 12th Chapter 3 Maths Chapter 3 Matrices Ex 3.4

Get Free NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.4 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Matrices Exercise 3.4 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 3 Class 12 Matrices Ex 3.4 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.4 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

Topics and Sub Topics in Class 11 Maths Chapter 3 Matrices:

NCERT Solutions for Class 12th Chapter 3 Maths Chapter 3 Matrices Ex 3.4

Using Elementary transformation, find the inverse each of matrices, if it exists in ques 1 to 17.

Ex 3.4 Class 12 Maths Question 1.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 2.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 3.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 4.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 5.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 6.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 7.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 8.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 9.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Ex 3.4 Class 12 Maths Question 10.

Solution:
Let

Ex 3.4 Class 12 Maths Question 11.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 12.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 13.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 14.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 15.

Solution:
Let

Ex 3.4 Class 12 Maths Question 16.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 17.

Solution:
Let
We know that
A = IA

Ex 3.4 Class 12 Maths Question 18.
Choose the correct answer in the following question:
Matrices A and B will be inverse of each other only if
(a) AB = BA
(b) AB = BA = 0
(c) AB = 0,BA = 1
(d) AB = BA = I
Solution:
Choice (d) is correct
i.e., AB = BA = I

NCERT Solutions for Class 12 Maths Chapter 3 Matrices (आव्यूह) Hindi Medium Ex 3.4

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

The post NCERT Solutions for Class 12th Chapter 3 Maths Chapter 3 Matrices Ex 3.4 appeared first on Learn CBSE.

Ishan Uday Scholarship: Ishan Uday Scholarship is offered by the University Grants Commission (UGC) under MHRD (Ministry of Human Resource Development) is offering for NER 2018-19. This is a special scheme available for the North Eastern domiciled students who have passed the Class12th examination. The objective of this scheme is: to provide equal opportunity for the NE region students to pursue higher education, to increase the Gross Enrolment Ratio (GER) in NE Region, to focus more on professional education in NE Region, Optimum utilization of NER Budgetary Allocation.

It is envisioned to provide ten thousand (10,000) fresh scholarships every year starting from the academic year 2014-15 for general degree courses, technical and professional courses, including medical and para-medical courses.

Ishan Uday Scholarship Eligibility Criteria

• Students with the domicile of NER who have passed Class 12th or equivalent exam from a school located within NER through any recognized Board of Education, including Central Board of Secondary Education (CBSE), Indian Certificate of Secondary Education (ICSE), National Institute of Open Schooling (NIOS) within NER only.
• Students who have secured admission in general degree course, technical and professional courses including medical and paramedical courses (Integrated courses included) in Universities/Colleges/Institutions recognized as under:Universities/Institutions/Colleges included under Section 2(f) and 12 (B) of UGC Act,Deemed to be Universities included under Section 3 of the UGC Act, 1956 and eligible to receive grants-in-aid from UGC, Universities/Institutions/Colleges funded by Central / State Government, Institutes of National Importance,  Institutions recognized by other Statutory councils within as well as outside
• Only first-year Under-graduate students are eligible to apply for the scholarship.
• For integrated degree courses the scholarship will be tenable for the complete duration of the course. For dual degree courses, the scholarship shall be tenable for the first degree only. The university shall clarify it while recommending a candidate for the scholarship.
• The income of the parents of the student should not exceed Rs.4.5 lakh per annum.
• Transgender candidates are eligible to apply under the scheme. The reservation shall be followed as per Government of India norms.

Who are not Eligible for Ishan Uday?

• Students pursuing courses/programs through Open Universities.
• Students already availing scholarship for pursuing the Undergraduate program under any other scheme(s).
• Students gaining admission through ‘management quota’.
• Students pursuing courses, such as Diploma Courses, not leading to the award of a Degree.
• Students whose parents’ income exceeds Rs.4.5 lakh per annum.

Scholarships for Students

Slots of Ishan Uday Scholarship

• UGC will award 10,000 (ten thousand) fresh scholarships per year.
• Allocation of slots for each State shall be done based upon the population (latest census report) of the respective State. If the slots remain vacant in any particular State, the unfilled slots shall be distributed equitably among other States.
• There shall be three Percent horizontal reservation for differently-abled persons.

Ishan Uday Scholarship Rewards

Ishan Uday scholarship for NER expends a total of 10,000 scholarships at the flow of up to INR 7800 per month every year to the chosen students. The scholarship expense is paid to the picked applicants through DBT (Direct Benefit Transfer) mode into their bank accounts. The amount of scholarships designated to each North Eastern State is done based on the population of the respective State (as per the latest Census report). The complete details of the Ishan Uday scholarship reward are highlighted below.

• The scholars who are seeking general degree courses are eligible to avail a scholarship amount of INR 5400 per month till the end of their courses.
• A monthly scholarship amount of INR 7800 is provided to those picked students who are seeking technical/medical/professional/paramedical courses.

How to Apply for Ishan Uday Scholarship?

Students who wish to avail the perks of Ishan Uday scholarship can apply through online mode only for the scholarship via the National Scholarship Portal (NSP). The applicants are suggested to have with them all the required documents while applying for the scholarship scheme. Given here is a complete description of how to apply for this scholarship.

• First, register yourself on NSP Portal as a new candidate by giving correct information.
• The candidates are expected to provide information about their educational qualifications, Aadhaar Number, School/College Enrolment Number, Bank details, and the State of Domicile while filling up the registration form.
• The candidates’ application ID will be generated based on their residence state. This application ID will be further used as ‘Login ID’ on NSP Portal and future references.
• Details such as the applicant’s name, date of birth, email ID, ID proof details and mobile number must be entered carefully.
• After completing the registration form, an OTP will be generated with which the candidate has to log in and fill the scholarship application form.
• Candidates should upload all the required documents and submit the application form.

Ishan Uday Scholarship – Documents Required

• Domicile certificates issued by a competent authority
• Income certificate issued by the competent authority
• Aadhaar card
• Marksheets of the last examination verified by the concerned University/ Institution/ College, in case of renewal of the scholarship
• Affidavit (Undertaking for not availing any other Scholarship in the form of an affidavit on Rs.50/- Stamp paper from the student/parent duly attested by SDM / First Class Magistrate / Gazetted Officer (not below the rank of Tahsildar) copying the prescribed languages

Ishan Uday Scholarship Selection Procedure

The awardees will be selected on the basis of marks scored by them in the Class XII or equivalent exam. The Commission reserves the right to withdraw/cancel the award without assigning any reason. The result shall be published on the UGC website. Award letters can be downloaded from the UGC online application portal. The effective date of release of fellowship shall be 1st April of the election year or the actual date of joining the first year of the undergraduate degree course whichever is later.

Ishan Uday Scholarship – Scholarship Duration and Renewal

• The scholarship awarded under the scheme will be renewed during the term of the first-degree course, subject to good conduct and maintenance of designated attendance.
• Scholars who fail to get promoted to the subsequent class/level would get the scholarship in the coming year, subject to the situation that if the student fails again for the second time, he/she would abandon the scholarship and it would not be recommenced for the remaining period of the course.
• If an awardee is incapable to appear in the annual exam because of illness or on account of any unforeseen incident, the scholarship may be resumed for the next academic year on submission of a medical certificate and other proof to satisfy the Head of the Institution.
• The growth of the scholar is to be observed by the head of the department of the Universities/Colleges/Institutions. The scholar is expected to submit quarterly continuation document duly signed by the head of the Institution of the Universities/Colleges/Institutions to the designated branch of a named bank.
• Any scholar who stops the studies without the prior consent of the UGC will have to repay the whole amount through e-mode (RTGS/NEFT) directly to UGC account.
• An awardee would be allowed to continue/renew his/her scholarship if he/she changes the course of study to another stream (e.g. Law, Fashion Technology, etc.). Scholars pursuing 5 years of courses, would get a scholarship at the designated rates for the duration of the course.
• Scholars changing their college/institute of education would be permitted to continue/renew the scholarship provided the course of study and the institution is recognized, subject to the condition that the scholarship will not be paid twice for the same year of study.
• In case the scholarship is eliminated for any reason, the decision of the UGC will be final. In the event of de-notification/deletion of any institution, the scholarship will remain to be available to the learners already admitted under the scheme, if otherwise eligible, till completion of the course.
• The scholar will be allowed to transfer his/her place of study as well as the course only once in the entire tenure of the scholarship. If the scholar switches from a general degree course to the technical or professional course he will be entitled to the amount of general degree course only, as initially sanctioned. If he/she switches from a technical or professional course to a general degree course he would be entitled to the amount of general degree course only.

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Get Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Determinants Exercise 4.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 4 Class 12 Determinants Ex 4.2 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Determinants chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Using the property of determinants and without expanding in Q 1 to 5, prove that

Ex 4.2 Class 12 Maths Question 1.

Solution:
L.H.S =
(C1 = C3 and C2 = C3)
= 0 + 0
= 0
= R.H.S

Ex 4.2 Class 12 Maths Question 2.

Solution:
L.H.S =

Ex 4.2 Class 12 Maths Question 3.

Solution:

Ex 4.2 Class 12 Maths Question 4.

Solution:
L.H.S =

Ex 4.2 Class 12 Maths Question 5.

Solution:
L.H.S = ∆ =

By using properties of determinants in Q 6 to 14, show that

Ex 4.2 Class 12 Maths Question 6.

Solution:
L.H.S = ∆ = …(i)

Ex 4.2 Class 12 Maths Question 7.

Solution:
L.H.S =

Ex 4.2 Class 12 Maths Question 8.
(a)
(b)
Solution:
(a) L.H.S =

Ex 4.2 Class 12 Maths Question 9.

Solution:
Let ∆ =
Applying R1–>R1 – R2, R2–>R2 – R3

Ex 4.2 Class 12 Maths Question 10.
(a)
(b)
Solution:
(a) L.H.S =

Ex 4.2 Class 12 Maths Question 11.
(a)
(b)
Solution:
(a) L.H.S =
=

Ex 4.2 Class 12 Maths Question 12.

Solution:
L.H.S =

Ex 4.2 Class 12 Maths Question 13.

Solution:
L.H.S =

Ex 4.2 Class 12 Maths Question 14.

Solution:
Let ∆ =

This may be expressed as the sum of 8 determinants

Ex 4.2 Class 12 Maths Question 15.
If A be a square matrix of order 3×3, then | kA | is equal to
(a) k|A|
(b) k² |A|
(c) k³ |A|
(d) 3k|A|
Solution:
Option (c) is correct.

Ex 4.2 Class 12 Maths Question 16.
Which of the following is correct:
(a) Determinant is a square matrix
(b) Determinant is a number associated to a matrix
(c) Determinant is a number associated to a square matrix
(d) None of these
Solution:
Option (c) is correct

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Hindi Medium Ex 4.2

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.3

Get Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.3 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Determinants Exercise 4.3 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 4 Class 12 Determinants Ex 4.3 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.3 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Determinants chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.3

Ex 4.3 Class 12 Maths Question 1.
Find the area of the triangle with vertices at the point given in each of the following:
(i) (1,0), (6,0) (4,3)
(ii) (2,7), (1,1), (10,8)
(iii) (-2,-3), (3,2), (-1,-8)
Solution:
(i) Area of triangle =
= [1(0-3)+1(18-0)]
= 7.5 sq units

Ex 4.3 Class 12 Maths Question 2.
Show that the points A (a, b + c), B (b, c + a) C (c, a+b) are collinear.
Solution:
The vertices of ∆ABC are A (a, b + c), B (b, c + a) and C (c, a + b)

Ex 4.3 Class 12 Maths Question 3.
Find the value of k if area of triangle is 4 square units and vertices are
(i) (k, 0), (4,0), (0,2)
(ii) (-2,0), (0,4), (0, k).
Solution:
(i) Area of ∆ = 4 (Given)

= [-2k+8]
= -k+4
Case (a): -k + 4 = 4 ==> k = 0
Case(b): -k + 4 = -4 ==> k = 8
Hence, k = 0,8
(ii) The area of the triangle whose vertices are (-2,0), (0,4), (0, k)

Ex 4.3 Class 12 Maths Question 4.
(i) Find the equation of line joining (1, 2) and (3,6) using determinants.
(ii) Find the equation of line joining (3,1), (9,3) using determinants.
Solution:
(i) Given: Points (1,2), (3,6)
Equation of the line is

Ex 4.3 Class 12 Maths Question 5.
If area of triangle is 35 sq. units with vertices (2, – 6), (5,4) and (k, 4). Then k is
(a) 12
(b) – 2
(c) -12,-2
(d) 12,-2
Solution:
(d) Area of ∆ =
= [50 – 10k] = 25 – 5k
∴ 25-5k = 35 or 25-5k = -35
-5k = 10 or 5k = 60
=> k = -2 or k = 12

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Hindi Medium Ex 4.3

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.4

Get Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.4 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Determinants Exercise 4.4 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 4 Class 12 Determinants Ex 4.4 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.4 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Determinants chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.4

Ex 4.4 Class 12 Maths Question 1.
Write the minors and cofactors of the elements of following determinants:
(i)
(ii)
Solution:
(i) Let A =
M₁₁ = 3, M₁₂ = 0, M₂₁ = – 4, M₂₂ = 2
For cofactors

Ex 4.4 Class 12 Maths Question 2.
Write Minors and Cofactor of elements of following determinant
(i)
(ii)
Solution:
(i) Minors M₁₁ = = 1

Ex 4.4 Class 12 Maths Question 3.
Using cofactors of elements of second row, evaluate

Solution:
Given

Ex 4.4 Class 12 Maths Question 4.
Using Cofactors of elements of third column, evaluate

Solution:
Elements of third column are yz, zx, xy

Ex 4.4 Class 12 Maths Question 5.
If and Aij is the cofactors of aij? then value of ∆ is given by
(a) a₁₁A₃₁+a₁₂A₃₂+a₁₃A₃₃
(b) a₁₁A₁₁+a₁₂A₂₁+a₁₃A₃₁
(c) a₂₁A₁₁+a₂₂A₁₂+a₂₃A₁₃
(d) a₁₁A₁₁+a₂₁A₂₁+a₃₁A₃₁
Solution:
Option (d) is correct.

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Hindi Medium Ex 4.4

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

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NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5

Get Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Determinants Exercise 4.5 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 4 Class 12 Determinants Ex 4.5 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Determinants chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5

Find the adjoint of each of the matrices in Questions 1 and 2.

Ex 4.5 Class 12 Maths Question 1.

Solution:
Let C_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by
C₁₁ = (-1)¹⁺¹ (4) = 4; C₁₂ = (-1)¹⁺² (3) = -3
C₂₁ = (-1)²⁺¹ (2)= – 2; C₂₂ = (-1)²⁺² (1) = -1
Adj A =
=

Ex 4.5 Class 12 Maths Question 2.

Solution:

Similarly,

Verify A (adjA) = (adjA) •A = |A| I in Qs. 3 and 4.
Ex 4.5 Class 12 Maths Question 3.

Solution:
|A| = 24

Ex 4.5 Class 12 Maths Question 4.

Solution:
A₁₁ = 0, A₁₂ = – 11, A₁₃ = 0,
A₂₁ = – 3, A₂₂ = 1, A₂₃ = 1, A₃₁ = – 2
A₃₂ = 8, A₃₃ = – 3

Find the inverse of each of the matrices (if it exists) given in Questions 5 to 11:

Ex 4.5 Class 12 Maths Question 5.

Solution:

So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by

Ex 4.5 Class 12 Maths Question 6.

Solution:

So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by

Ex 4.5 Class 12 Maths Question 7.

Solution:
|A| = 10

Ex 4.5 Class 12 Maths Question 8.

Solution:

So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by

Ex 4.5 Class 12 Maths Question 9.

Solution:
|A| =
= 2(-1-0)-1(4-0)+3(8-3)
So, A is non-singular matrix and therefore, it is invertible.

Ex 4.5 Class 12 Maths Question 10.

Solution:
|A| =
= 1(8-6)+1(0+9)+2(0-6)
= 2+9-12
= -1≠0
∴A is invertible and

Ex 4.5 Class 12 Maths Question 11.

Solution:
A =
adj A =
First find |A| = -cos²α-sin²α
=-1≠0

Ex 4.5 Class 12 Maths Question 12.
Let , verify that (AB)^-1 = B^-1A^-1
Solution:
Here |A| =
= 15-14
= 1≠0

Ex 4.5 Class 12 Maths Question 13.
If show that A² – 5A + 7I = 0,hence find A^-1
Solution:
A =
A² =

Ex 4.5 Class 12 Maths Question 14.
For the matrix A = find file numbers a and b such that A²+aA+bI²=0. Hence, find A^-1.
Solution:
A =
A²+aA+bI²=0

Ex 4.5 Class 12 Maths Question 15.
For the matrix  Show that A³-6A²+5A+11I₃=0.Hence find A^-1
Solution:
A² =

Ex 4.5 Class 12 Maths Question 16.
If show that A³-6A²+9A-4I=0 and hence, find A^-1
Solution:
We have

Ex 4.5 Class 12 Maths Question 17.
Let A be a non-singular square matrix of order 3×3. Then | Adj A | is equal to:
(a) | A |
(b) | A |²
(c) | A |³
(d) 3 | A |
Solution:
Let A =


Dividing by | A |, |Adj. A| = | A |²
Hence, Part (b) is the correct answer.

Ex 4.5 Class 12 Maths Question 18.
If A is an invertible matrix of order 2, then det. (A^-1) is equal to:
(a) det. (A)
(b)
(c) 1
(d) 0
Solution:
|A|≠0
=> A^-1 exists => AA^-1 = I
|AA^-1| = |I| = I
=> |A||A^-1| = I

Hence option (b) is correct.

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Hindi Medium Ex 4.5

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

The post NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5 appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.6

Get Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.6 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Determinants Exercise 4.6 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 4 Class 12 Determinants Ex 4.6 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.6 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Determinants chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.6

Examine the consistency of the system of equations in Questions 1 to 6:

Ex 4.6 Class 12 Maths Question 1.
x + 2y = 2
2x + 3y = 3
Solution:
x + 2y = 2,
2x + 3y = 3
=>
=> AX = B
Now |A| =
= 3 – 4
= – 1 ≠ 0.
Hence, equations are consistent.

Ex 4.6 Class 12 Maths Question 2.
2x – y = 5
x + y = 4
Solution:
2x – y = 5,
x + y = 4
=>
=> AX = B
Now |A| =
= 2 + 1
= 3 ≠ 0.
Hence, equations are consistent.

Ex 4.6 Class 12 Maths Question 3.
x + 3y = 5,
2x + 6y = 8
Solution:
x + 3y = 5,
2x + 6y = 8
=>
=> AX = B
Now |A| =
= 6 – 6
= 0.

Hence, equations are consistent with no solution

Ex 4.6 Class 12 Maths Question 4.
x + y + z = 1
2x + 3y + 2z = 2
ax + ay + 2az = 4
Solution:
x + y + z = 1
2x + 3y + 2z = 2
x + y + z =

Ex 4.6 Class 12 Maths Question 5.
3x – y – 2z = 2
2y – z = – 1
3x – 5y = 3
Solution:

=> AX = B

Ex 4.6 Class 12 Maths Question 6.
5x – y + 4z = 5
2x + 3y + 5z = 2
5x – 2y + 6z = -1
Solution:
Given
5x – y + 4z = 5
2x + 3y + 5z = 2
5x – 2y + 6z = -1


= 5(18 + 10)+1(12 – 25)+4(-4-15)
= 140-13-76
= 51 ≠ 0
Hence equations are consistent with a unique
solution.

Solve system of linear equations using matrix method in Questions 7 to 14:

Ex 4.6 Class 12 Maths Question 7.
5x + 2y = 4
7x + 3y = 5
Solution:
The given system of equations can be written as

Ex 4.6 Class 12 Maths Question 8.
2x – y = – 2
3x + 3y = 3
Solution:
The given system of equations can be written

Ex 4.6 Class 12 Maths Question 9.
4x – 3y = 3
3x – 5y = 7
Solution:
The given system of equations can be written as

where

Ex 4.6 Class 12 Maths Question 10.
5x + 2y = 3
3x + 2y = 5
Solution:
The given system of equations can be written as

where

Ex 4.6 Class 12 Maths Question 11.
2x + y + z = 1,
x – 2y – z = 3/2
3y – 5z = 9
Solution:
The given system of equations are
2x + y + z = 1,
x – 2y – z = 3/2,
3y – 5z = 9
We know AX = B => X = A^-1B

Ex 4.6 Class 12 Maths Question 12.
x – y + z = 4
2x + y – 3z = 0
x + y + z = 2.
Solution:
The given system of equations can be written

Ex 4.6 Class 12 Maths Question 13.
2x + 3y + 3z = 5
x – 2y + z = – 4
3x – y – 2z = 3
Solution:
The given system of equations can be written as:

Ex 4.6 Class 12 Maths Question 14.
x – y + 2z = 7
3x + 4y – 5z = – 5
2x – y + 3z = 12.
Solution:
The given system of equations can be written

Ex 4.6 Class 12 Maths Question 15.
If A = Find A^-1. Using A^-1. Solve the following system of linear equations 2x – 3y + 5z = 11,3x + 2y – 4z = – 5, x + y – 2z = – 3
Solution:
We have AX = B
where

Ex 4.6 Class 12 Maths Question 16.
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs. 69. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs. 90. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 70. Find the cost of each item per kg by matrix method
Solution:
Let cost of 1 kg onion = Rs x
and cost of 1 kg wheat = Rs y
and cost of 1 kg rice = Rs z
4x+3y+2z=60
2x+4y+6z=90
6x+2y+3z=70

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Hindi Medium Ex 4.6

More Resources for NCERT Solutions Class 12:

• Maths NCERT Solutions
• RD Sharma Class 12 Solutions
• NCERT Solutions Class 12 Physics
• NCERT Solutions Class 12 Chemistry
• NCERT Solutions Class 12 Biology
• NCERT Solutions Class 12 Hindi
• NCERT Solutions Class 12 Vistas English
• NCERT Solutions Class 12th Flamingo English

The post NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.6 appeared first on Learn CBSE.

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The post PACE Scholarship Entrance Exam for IIT JEE, AIMS, NEET – Super 50 Programme appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.2 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2

Differentiate the functions with respect to x in Questions 1 to 8.

Ex 5.2 Class 12 Maths Question 1.
sin(x² + 5)
Solution:
Let y = sin(x2 + 5),
put x² + 5 = t
y = sint
t = x²+5


= cos (x² + 5) × 2x
= 2x cos (x² + 5)

Ex 5.2 Class 12 Maths Question 2.
cos (sin x)
Solution:
let y = cos (sin x)
put sinx = t
∴ y = cost,
t = sinx
∴

Putting the value of t,

Ex 5.2 Class 12 Maths Question 3.
sin(ax+b)
Solution:
let = sin(ax+b)
put ax+bx = t
∴ y = sint
t = ax+b

Ex 5.2 Class 12 Maths Question 4.
sec(tan(√x))
Solution:
let y = sec(tan(√x))
by chain rule

Ex 5.2 Class 12 Maths Question 5.

Solution:
y = =
u = sin(ax+b)

Ex 5.2 Class 12 Maths Question 6.
cos x³ . sin²(x⁵) = y(say)
Solution:
Let u = cos x³ and v = sin² x⁵,
put x³ = t

Ex 5.2 Class 12 Maths Question 7.

Solution:

Ex 5.2 Class 12 Maths Question 8.
cos(√x) = y(say)
Solution:
cos(√x) = y(say)

Ex 5.2 Class 12 Maths Question 9.
Prove that the function f given by f (x) = |x – 1|,x ∈ R is not differential at x = 1.
Solution:
The given function may be written as

Ex 5.2 Class 12 Maths Question 10.
Prove that the greatest integer function defined by f (x)=[x], 0 < x < 3 is not differential at x = 1 and x = 2.
Solution:
(i) At x = 1

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.2

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.3

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.3 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.3 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.3 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.3 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.3

Find in the following

Ex 5.3 Class 12 Maths Question 1.
2x + 3y = sinx
Solution:
2x + 3y = sinx
Differentiating w.r.t x,

=>

Ex 5.3 Class 12 Maths Question 2.
2x + 3y = siny
Solution:
2x + 3y = siny
Differentiating w.r.t x,

=>

Ex 5.3 Class 12 Maths Question 3.
ax + by² = cosy
Solution:
ax + by² = cosy
Differentiate w.r.t. x,

=>

Ex 5.3 Class 12 Maths Question 4.
xy + y² = tan x + y
Solution:
xy + y² = tanx + y
Differentiating w.r.t. x,

Ex 5.3 Class 12 Maths Question 5.
x² + xy + y² = 100
Solution:
x² + xy + xy = 100
Differentiating w.r.t. x,

Ex 5.3 Class 12 Maths Question 6.
x³ + x²y + xy² + y³ = 81
Solution:
Given that
x³ + x²y + xy² + y³ = 81
Differentiating both sides we get

Ex 5.3 Class 12 Maths Question 7.
sin² y + cos xy = π
Solution:
Given that
sin² y + cos xy = π
Differentiating both sides we get

Ex 5.3 Class 12 Maths Question 8.
sin²x + cos²y = 1
Solution:
Given that
sin²x + cos²y = 1
Differentiating both sides, we get

Ex 5.3 Class 12 Maths Question 9.

Solution:

put x = tanθ

Ex 5.3 Class 12 Maths Question 10.

Solution:

put x = tanθ

Ex 5.3 Class 12 Maths Question 11.

Solution:

put x = tanθ

Ex 5.3 Class 12 Maths Question 12.

Solution:

put x = tanθ
we get

Ex 5.3 Class 12 Maths Question 13.

Solution:

put x = tanθ
we get

Ex 5.3 Class 12 Maths Question 14.

Solution:

put x = tanθ
we get

Ex 5.3 Class 12 Maths Question 15.

Solution:

put x = tanθ
we get

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.3

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.3 appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.4 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.4 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Differentiate the following w.r.t.x:

Ex 5.4 Class 12 Maths Question 1.

Solution:

Ex 5.4 Class 12 Maths Question 2.

Solution:


x=sint

Ex 5.4 Class 12 Maths Question 3.

Solution:

Ex 5.4 Class 12 Maths Question 4.

Solution:

Ex 5.4 Class 12 Maths Question 5.

Solution:

Ex 5.4 Class 12 Maths Question 6.
…
Solution:

…

Ex 5.4 Class 12 Maths Question 7.

Solution:
y =

Ex 5.4 Class 12 Maths Question 8.
log(log x),x>1
Solution:
y = log(log x),
put y = log t, t = log x,
differentiating

Ex 5.4 Class 12 Maths Question 9.

Solution:
let

Ex 5.4 Class 12 Maths Question 10.
cos(log x+e^x),x>0
Solution:
y = cos(log x+e^x),x>0
put y = cos t,t = log x+e^x

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.4

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.5 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.5 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.5 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.5 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.5

Differentiate the functions given in Questions 1 to 11 w.r.to x

Ex 5.5 Class 12 Maths Question 1.
cos x. cos 2x. cos 3x
Solution:
Let y = cos x. cos 2x . cos 3x,
Taking log on both sides,
log y = log (cos x. cos 2x. cos 3x)
log y = log cos x + log cos 2x + log cos 3x,
Differentiating w.r.t. x, we get

Ex 5.5 Class 12 Maths Question 2.

Solution:

taking log on both sides
log y = log

Ex 5.5 Class 12 Maths Question 3.
(log x)^cosx
Solution:
let y = (log x)^cosx
Taking log on both sides,
log y = log (log x)^cosx
log y = cos x log (log x),
Differentiating w.r.t. x,

Ex 5.5 Class 12 Maths Question 4.
x – 2^sinx
Solution:
let y = x – 2^sinx,
y = u – v

Ex 5.5 Class 12 Maths Question 5.
(x+3)².(x + 4)³.(x + 5)⁴
Solution:
let y = (x + 3)².(x + 4)³.(x + 5)⁴
Taking log on both side,
logy = log [(x + 3)² • (x + 4)³ • (x + 5)⁴]
= log (x + 3)² + log (x + 4)³ + log (x + 5)⁴
log y = 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5)
Differentiating w.r.t. x, we get

Ex 5.5 Class 12 Maths Question 6.

Solution:
let
let

Ex 5.5 Class 12 Maths Question 7.
(log x)^x + x^logx
Solution:
let y = (log x)^x + x^logx = u+v
where u = (log x)^x
∴ log u = x log(log x)

Ex 5.5 Class 12 Maths Question 8.
(sin x)^x+sin^-1 √x
Solution:
Let y = (sin x)^x + sin^-1 √x
let u = (sin x)x, v = sin^-1 √x

Ex 5.5 Class 12 Maths Question 9.
x^sinx + (sin x)^cosx
Solution:
let y = x^sinx + (sin x)^cosx = u+v
where u = x^sinx
log u = sin x log x

Ex 5.5 Class 12 Maths Question 10.

Solution:

y = u + v

Ex 5.5 Class 12 Maths Question 11.

Solution:

Let u = (x cosx)^x
logu = x log(x cosx)

Find of the functions given in Questions 12 to 15.

Ex 5.5 Class 12 Maths Question 12.
x^y + y^x = 1
Solution:
x^y + y^x = 1
let u = x^y and v = y^x
∴ u + v = 1,

Now u = x

Ex 5.5 Class 12 Maths Question 13.
y^x = x^y
Solution:
y = x
x logy = y logx

Ex 5.5 Class 12 Maths Question 14.
(cos x)^y = (cos y)^x
Solution:
We have
(cos x)^y = (cos y)^x
=> y log (cosx) = x log (cosy)

Ex 5.5 Class 12 Maths Question 15.
xy = e^(x-y)
Solution:
log(xy) = log e^(x-y)
=> log(xy) = x – y
=> logx + logy = x – y

Ex 5.5 Class 12 Maths Question 16.
Find the derivative of the function given by f (x) = (1 + x) (1 + x²) (1 + x⁴) (1 + x⁸) and hence find f'(1).
Solution:
Let f(x) = y = (1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)
Taking log both sides, we get
logy = log [(1 + x)(1 + x²)(1 + x⁴)(1 + x⁸)]
logy = log(1 + x) + log (1 + x²) + log(1 + x⁴) + log(1 + x⁸)

Ex 5.5 Class 12 Maths Question 17.
Differentiate (x² – 5x + 8) (x³ + 7x + 9) in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Do they all give the same answer?
Solution:
(i) By using product rule
f’ = (x² – 5x + 8) (3x² + 7) + (x³ + 7x + 9) (2x – 5)
f = 5x⁴ – 20x³ + 45x² – 52x + 11.
(ii) By expanding the product to obtain a single polynomial, we get

Ex 5.5 Class 12 Maths Question 18.
If u, v and w are functions of w then show that

in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Solution:
Let y = u.v.w
=> y = u. (vw)

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.5

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.5 appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.6

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.6 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.6 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.6 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.6 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.6

If x and y are connected parametrically by the equations given in Questions 1 to 10, without eliminating the parameter. Find .

Ex 5.6 Class 12 Maths Question 1.
x = 2at², y = at⁴
Solution:

Ex 5.6 Class 12 Maths Question 2.
x = a cosθ,y = b cosθ
Solution:

Ex 5.6 Class 12 Maths Question 3.
x = sin t, y = cos 2t
Solution:

Ex 5.6 Class 12 Maths Question 4.

Solution:

Ex 5.6 Class 12 Maths Question 5.
x = cos θ – cos 2θ, y = sin θ – sin 2θ
Solution:

Ex 5.6 Class 12 Maths Question 6.
x = a(θ – sinθ), y = a(1 + cosθ)
Solution:

Ex 5.6 Class 12 Maths Question 7.

Solution:

Ex 5.6 Class 12 Maths Question 8.

Solution:

Ex 5.6 Class 12 Maths Question 9.
x = a sec θ,y = b tan θ
Solution:
x = a sec θ,y = b tan θ

Ex 5.6 Class 12 Maths Question 10.
x = a(cosθ+θsinθ), y = a(sinθ-θcosθ)
Solution:
x = a(cosθ+θsinθ), y = a(sinθ-θcosθ)

Ex 5.6 Class 12 Maths Question 11.
If show that
Solution:
Given that

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.6

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.6 appeared first on Learn CBSE.

ONGC Scholarship 2019: ONGC Scholarship 2019 is a CSR initiative run by the Oil and Natural Gas Company (ONGC) Limited. ONGC Scholarship offers scholarships to SC/ST meritorious students for pursuing professional courses. These are Engineering, Medical and Master degrees in Business Administrative, Geology, and Geophysics. Candidates who are Indian nationals and studying in the specified courses can apply for this scholarship scheme. Candidates can fill the application form and apply for this scholarship on or before the 3rd week of January 2020. 

ONGC scholarship rewards 1000 meritorious candidates every year. 50% of this scholarship is reserved for girl candidates. However, five zones in the states of India are eligible to apply for this scholarship. The article below provides the information that is eligibility, application, and so forth about ONGC Scholarship 2019.

ONGC Scholarship 2019 for SC/ST meritorious students is one of its initiatives to encourage them to pursue higher education. This scholarship scheme helps SC/ST meritorious students to complete professional courses without financial hurdle. ONGC is a multinational oil and gas public enterprise company in India. It is the largest producer of natural gas and oil across the nation. As a part of Corporate Social Responsibility (CSR), the company has introduced various schemes. These schemes focus on areas like education, healthcare, entrepreneurship, women empowerment, water management and many more. 

ONGC Scholarship Overview

ONGC Scholarship 2019 – Important Dates

Scholarships for Students

ONGC Scholarship 2019 – Eligibility Criteria

Applicants must fulfill the following eligibility criteria to apply for ONGC Scholarship:

• Students studying in India and must belong to Indian citizenship.
• The age limit of the students must not be more than 30 years as on 1st November of the current academic session.
• SC/ ST category students can only apply for this scholarship.
• Candidate must be pursuing the following full-time courses in an AICTE/UGC/State or Central Govt approved institutes.
• 1st year of Engineering/ MBBS courses
• 1st year of MBA or Master in Geology/ Geophysics courses
• Candidates pursuing the following courses are also eligible
  • 4 years of Engineering or MBBS courses
  • 2 years Masters course in Business Administration
  • 2 years Masters course in Geology/ Geophysics
• The family income of candidates including all sources should not be more than 4.5 Lakh per annum.
• Candidates must have passed with minimum 60% marks or equivalent CGPA in class 12 for Engineering/ MBBS degrees.
• Candidates must have passed graduation with minimum 60% marks or 6.0 CGPA for MBA or Masters in Geology/ Geophysics.

ONGC Scholarship 2019 – Selection Process

The selection process for ONGC Scholarship is given below

• It is a merit-based scholarship which considers academic merit prescribed for each eligible course for final selection.
• The company prepares the list of final selected scholars and invite them for an interview.
• If candidates having an equal percentage in the exam then the one with lower family income will be selected for this scholarship.
• First preference will be given to candidates from Below Poverty Line (BPL) families.
• If sufficient candidates from BPL families satisfying the eligibility criteria are not available then only other candidates will be considered.
• If the candidate pursuing an Engineering or MBBS degree then the selection is based on the performance in the 12th class examination.
• Whereas the selection of candidates will be based on the performance at the graduation level who are pursuing an MBA or Masters degree in Geology or Geophysics.

ONGC Scholarship 2019 – Application Process

The application for ONGC Scholarship will be available online in the third week of November 2019. Candidates can fill and submit the application in the 3rd week of January 2020. Refer to the below points to apply for ONGC Scholarship 2019.

• Candidates should visit the official website www.ongcindia.com.
• Click on the “Advertisement of ONGC Scholarship to SC/ST” appear under the “News and Update” section of the homepage.
• Download the application format available in pdf form.
• Read ONGC SC/ST Scholarship details carefully. Fill all the essential details in the application form in capital letters.
• Recheck the application thoroughly to avoid rejection by the authorities during verification.
• Attach xerox copies of all required documents along with the application. The documents must be duly certified as well as forwarded by the Head/ Principal of the School/ College/ University.
• Enclose the completed application along with the documents in an envelope. Then, send it to the designated ONGC office of your respective zone.
• The zone from where you can apply is based on the location of your college/university of the qualifying exam.
• The zone is not decided on the basis of your domicile.
• Send the completed application to the below-mentioned address of designated zone offices.

ONGC Scholarship – Zone-wise Address for Application Submission

ONGC Scholarship 2019 – Checklist of Documents

Candidates must include the below-mentioned documents along with the application to apply for ONGC Scholarship.

• Passport size photograph of the candidate
• Caste certificate in English/Hindi language
• Age proof (birth certificate or 10th class mark sheets)
• Marksheets of the qualifying examination
• Annual family income certificate in English/Hindi language
• Bank details of the candidate in the prescribed format
• Xerox copy of PAN Card

ONGC Scholarship 2019 – Terms and Conditions

Candidates must refer to the below terms and conditions to apply for ONGC Scholarship:

• Candidates who are applying for the ONGC scholarship should not be availing any other scholarship or financial assistance.
• The scholars need to show satisfactory performance in each year’s annual exam in order to continue the scholarship every year.
• The conduct of the scholar is also considered for scholarship continuation.
• Candidates have to secure minimum 50% marks or a scale of 5 grade points out of 10.
• If the candidate fails to maintain the academic requirements or conduct then the scholarship will not be paid to them for the respective year.
• If the scholarship is discontinued, candidates can apply for a renewal by filling the scholarship renewal application form.
• The renewal application form should be filled in the specified format.
• If candidates are able to maintain the said percentage or grades again then only they can apply for scholarship renewal.

ONGC Scholarship 2019 – Result

The result for ONGC Scholarship will be released on the official website in the last week of March 2020. Candidates can go to the official website www.ongcindia.com to check the final list of the scholarship beneficiaries. The shortlisted candidate’s documents will be verified after declaring the result. The Scholarships will be awarded only to the shortlisted candidates by ONGC.

ONGC Scholarship 2019 – Contact Details

In case of any queries related to ONGC Scholarship or any assistance contact on the below-mentioned details.

FAQ’s on ONGC Scholarship 2019

Question 1.
How is ONGC Scholarship helpful to SC/ST students?

Answer:
ONGC Scholarship offers scholarships to SC/ST meritorious students for pursuing professional courses. These professional courses include Engineering, Medical and Master degrees in Business Administrative, Geology, and Geophysics.

Question 2.
How many scholarships are awarded to meritorious candidates every year?

Answer:
ONGC scholarship rewards 1000 meritorious candidates every year.

Question 3.
How many scholarships are reserved for girl candidates?

Answer:
50% of ONGC Scholarship is reserved only for girl candidates.

Question 4.
What are the ONGC Scholarship application starting and closing date?

Answer:
The ONGC Scholarship application starting date is the third week of November 2019. The application closing date is the third week of January 2020.

Question 5.
What is the age limit to apply for ONGC Scholarship 2019?

Answer:
The age limit of the students must not be more than 30 years as on 1st November of the current academic session.

Hope this article will help you to get more information about ONGC Scholarship 2019. If you have queries related to ONGC Scholarship, then leave it in the comment box to get in touch with us.

The post ONGC Scholarship 2019 | Important Dates, Eligibility, Rewards, Application appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.7 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.7 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Find the second order derivatives of the functions given in Questions 1 to 10.

Ex 5.7 Class 12 Maths Question 1.
x² + 3x + 2 = y(say)
Solution:

Ex 5.7 Class 12 Maths Question 2.
x²⁰ = y(say)
Solution:

Ex 5.7 Class 12 Maths Question 3.
x.cos x = y(say)
Solution:

Ex 5.7 Class 12 Maths Question 4.
log x = y (say)
Solution:

Ex 5.7 Class 12 Maths Question 5.
x³ log x = y (say)
Solution:
x³ log x = y

Ex 5.7 Class 12 Maths Question 6.
e^x sin5x = y
Solution:
e^x sin5x = y

Ex 5.7 Class 12 Maths Question 7.
e^6x cos3x = y
Solution:
e^6x cos3x = y

Ex 5.7 Class 12 Maths Question 8.
tan^-1 x = y
Solution:

Ex 5.7 Class 12 Maths Question 9.
log(logx) = y
Solution:
log(logx) = y

Ex 5.7 Class 12 Maths Question 10.
sin(log x) = y
Solution:
sin(log x) = y

Ex 5.7 Class 12 Maths Question 11.
If y = 5 cosx – 3 sin x, prove that
Solution:



Hence proved

Ex 5.7 Class 12 Maths Question 12.
If y = cos^-1 x, Find in terms of y alone.
Solution:

Ex 5.7 Class 12 Maths Question 13.
If y = 3 cos (log x) + 4 sin (log x), show that

Solution:
Given that
y = 3 cos (log x) + 4 sin (log x)

Ex 5.7 Class 12 Maths Question 14.

Solution:
Given that

Ex 5.7 Class 12 Maths Question 15.
If y = 500e^7x + 600e^-7x, show that .
Solution:
we have
y = 500e^7x + 600e^-7x

Ex 5.7 Class 12 Maths Question 16.
If e^y(x+1) = 1,show that
Solution:

Ex 5.7 Class 12 Maths Question 17.
If y=(tan^-1 x)² show that (x²+1)²y₂+2x(x²+1)y₁=2
Solution:
we have
y=(tan^-1 x)²

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.7

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.8

Get Free NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.8 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Continuity and Differentiability Exercise 5.8 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Continuity and Differentiability Ex 5.8 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.8 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Continuity and Differentiability chapter are the following:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second Order Derivative
• Mean Value Theorem
• Summary

There are total eight exercises and one misc exercise(144 Questions fully solved) in the class 12th maths chapter 5 Continuity and Differentiability.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.8

Ex 5.8 Class 12 Maths Question 1.
Verify Rolle’s theorem for the function
f(x) = x² + 2x – 8,x∈ [-4,2]
Solution:
Now f(x) = x² + 2x – 8 is a polynomial
∴ it is continuous and derivable in its domain x∈R.
Hence it is continuous in the interval [-4,2] and derivable in the interval (- 4,2)
f(-4) = (-4)² + 2(-4) – 8 = 16 – 8 – 8 = 0,
f(2) = 2² + 4 – 8 = 8 – 8 = 0
Conditions of Rolle’s theorem are satisfied.
f'(x) = 2x + 2
∴ f’ (c) = 2c + 2 = 0
or c = – 1, c = – 1 ∈ [-4,2]
Thus f’ (c) = 0 at c = – 1.

Ex 5.8 Class 12 Maths Question 2.
Examine if Rolle’s theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s theorem from these example?
(i) f(x) = [x] for x ∈ [5,9]
(ii) f (x) = [x] for x ∈ [-2,2]
(iii) f (x) = x² – 1 for x ∈ [1,2]
Solution:
(i) In the interval [5, 9], f (x) = [x] is neither continuous nor derivable at x = 6,7,8 Hence Rolle’s theorem is not applicable
(ii) f (x) = [x] is not continuous and derivable at -1, 0, 1. Hence Rolle’s theorem is not applicable.
(iii) f(x) = (x² – 1),f(1) = 1 – 1 = 0,
f(2) = 22 – 1 = 3
f(a)≠f(b)
Though it is continous and derivable in the interval [1,2].
Rolle’s theorem is not applicable.
In case of converse if f (c)=0, c ∈ [a, b] then conditions of rolle’s theorem are not true.
(i) f (x) = [x] is the greatest integer less than or equal to x.
∴f(x) = 0, But fis neither continuous nor differentiable in the interval [5,9].
(ii) Here also, theough f (x) = 0, but f is neither continuous nor differentiable in the interval [-2,2].
(iii) f (x)=x² – 1, f'(x)=2x. Here f'(x) is not zero in the [1,2], So f (2) ≠ f’ (2).

Ex 5.8 Class 12 Maths Question 3.
If f: [-5,5] –>R is a differentiable function and if f (x) does not vanish anywhere then prove that f (- 5) ≠ f (5).
Solution:
For Rolle’s theorem
If (i) f is continuous in [a, b]
(ii) f is derivable in [a, b]
(iii) f (a) = f (b)
then f’ (c)=0, c e (a, b)
∴ f is continuous and derivable
but f (c) ≠ 0 =>f(a) ≠ f(b) i.e., f(-5)≠f(5)

Ex 5.8 Class 12 Maths Question 4.
Verify Mean Value Theorem, if
f (x) = x² – 4x – 3 in the interval [a, b], where a = 1 and b = 4.
Solution:
f (x) = x² – 4x – 3. It being a polynomial it is continuous in the interval [1,4] and derivable in (1,4), So all the condition of mean value theorem hold.
then f’ (x) = 2x – 4,
f’ (c) = 2c – 4
f(4)= 16 – 16 – 3 = – 3,
f(1)= 1 – 4 – 3 = – 6
Then there exist a value c such that

Ex 5.8 Class 12 Maths Question 5.
Verify Mean Value Theorem, if f (x)=x³ – 5x² – 3x in the interval [a, b], where a = 1 and b = 3. Find all c ∈ (1,3) for which f’ (c) = 0.
Solution:
f (x)=x³ – 5x² – 3x,
It is a polynomial. Therefore it is continuous in the interval [1,3] and derivable in the interval (1,3)
Also, f'(x)=3x²-10x-3

Ex 5.8 Class 12 Maths Question 6.
Examine the applicability of Mean Value theroem for all three functions given in the above Question 2.
Solution:
(i) F (x)= [x] for x ∈ [5,9], f (x) = [x] in the interval [5, 9] is neither continuous, nor differentiable.
(ii) f (x) = [x], for x ∈ [-2,2],
Again f (x) = [x] in the interval [-2,2] is neither continous, nor differentiable.
(iii) f(x) = x²-1 for x ∈ [1,2], It is a polynomial. Therefore it is continuous in the interval [1,2] and differentiable in the interval (1,2)
f (x) = 2x, f(1) = 1 – 1 = 0 ,
f(2) = 4 – 1 = 3, f'(c) = 2c

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Hindi Medium Ex 5.8

NCERT Class 12 Maths Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.8 appeared first on Learn CBSE.

District Merit Scholarship (DMS) 2019: The Directorate of Collegiate Education (DCE), Government of Kerala has released the notification for the District Merit Scholarship (DMS) 2019. The District Merit Scholarship has initiated by the Directorate of Collegiate Education, Govt. of Kerala. This scheme is available only for the students of Kerala State. The government of Kerala grants District Merit Scholarship (DMS) 2019 to students who have passed in all subjects in the SSLC exam with A+ grade. This SSLC exam must be conducted by the Board of Public Examination, Kerala.

The application form for District Merit Scholarship will be available online on August 1, 2019. Candidates can fill the application and apply for this scholarship till 30th September 2019. The motive behind the DMS is to notify candidates regarding engineering courses after class 10th or 12th every year. Read the article below to get more information about District Merit Scholarship 2019.

District Merit Scholarship (DMS) 2019

DMS is available for students of Kerala who are currently studying in Higher Secondary, VHSC, ITI or polytechnic Courses. This Scholarship is one of the initiatives of Govt. of Kerala to encourage Kerala students for pursuing engineering courses. DMS provides financial support to the underprivileged students who cannot afford their education. It also helps the meritorious students to complete engineering courses without financial hassle.

Govt. of Kerala has introduced DMS to award the outstanding meritorious students in the education sector as well as in the economically and socially backward classes. This project envisages the development and implementation of a web-based scholarship management system. It is an efficient and foolproof system which facilitates DCE for selection of students and disbursements of scholarship.

Govt. of Kerala releases both Fresh and Renewal application form to apply for DMS. It is mandatory that candidates applying for DMS Fresh Application should not applied for it earlier. On the other hand, a Renewal Application is for those candidates who have applied for it earlier.

District Merit Scholarship (DMS) Overview

Scholarships for Students

District Merit Scholarship 2019 – Important Dates

District Merit Scholarship 2019 – Eligibility Criteria

Applicants must fulfill the following eligibility criteria to apply for DMS 2019:

• Students must have passed A+ grade in all subjects in the SSLC examination.
• Students must have passed the SSLC examination conducted by the Board of Public Examinations, Kerala State.
• Students studying in Higher Secondary / VHSC / ITI / Polytechnic courses can apply for DMS.

District Merit Scholarship 2019 – Selection Process

The Higher Education Department of Govt. of Kerala is responsible for the selection of the meritorious students. The final selection for DMS depends on the academic record and income of parents.

District Merit Scholarship 2019 – Application Process

The application for DMS will be available online on August 1, 2019. Candidates can fill and submit the application on or before September 30, 2019. Refer to the below points to apply for District Merit Scholarship 2019.

• Candidates should visit the official website of DCE, Govt.of Kerala.
• Select the “District Merit Scholarship (DMS)” from the types of scholarship list. Refer to the picture below:

• Then, click on “Apply Online” to redirect to the Candidate Login page.
• Already registered candidates can enter the login details as shown in the below picture

• Otherwise, click on the “New Registration” to generate an account.
• Fill the correct and valid details in the application form in capital letters.
• Recheck the application thoroughly to avoid rejection by DCE, Govt of Kerala during verification.
• Upload scanned copies of all required documents along with the application. The documents must be attested by the Head of the School/ College/ University.
• Submit the completed application along with the required documents.
• Then, you will receive an automated email regarding login Id and password.
• Keep the credentials safe for your future usage.
• Finally, take a print out of the DMS application form for future purposes.

District Merit Scholarship 2019 – Checklist of Documents

The below-mentioned documents must be uploaded along with the application to apply for DMS 2019.

• Recent passport size photograph
• Attested copies of school/college certificates and mark Sheets
• Candidates annual family income certificate and affidavit for any additional income
• Attested copies of caste certificate
• Age proof (birth certificate or class 10th board certificate)
• Residential proof (electricity bill, ration card, voter Id, aadhar card, and so forth)
• Attested copies of disability certificates, if any
• Candidates Bank Account details in the prescribed format

District Merit Scholarship 2019 – Renewal

The DMS Renewal application form will be released online on August 1, 2019. Candidates can fill the application and apply for renewal on or before September 30, 2019. Candidates who have already registered for DMS must renew it every year. Candidates should have satisfactory performance in each year’s annual exam to renew this scholarship. Candidates should maintain the academic requirements and conduct to apply for DMS Renewal. Refer the below points to Renew the District Merit Scholarship application form.

• Candidates should visit the official website of DCE, Govt. of Kerala to renew the application.
• Login by entering registration Id, date of birth, password and selecting Renewal. Edit the old application form which exists in the DCE portal.
• Make sure that the modified details are accurate and filled in the specified format. For any corrections modify the form accordingly.
• Then, submit the DMS Renewal application form and check the status online.

District Merit Scholarship 2019 – Result

The result for DMS will be released on the official website in October 2019. DCE prepares the final selection list of candidates to award the scholarship. Candidates can go to the official website of DCE, Govt. of Kerala to check the selection list of the scholarship beneficiaries. Candidates can check the selection list by entering the Scholarship Type, State, District, and College/ Institution Name. The list of selected candidates name and institution name will appear on the screen. Candidates can search their name in the selection list to confirm their chances of receiving the scholarship. The selected candidate’s documents will be verified by DCE after declaring the result. The Scholarships will be disbursed only to the selected candidates by DCE, Govt.of Kerala on October 2019.

District Merit Scholarship 2019 – Contact Details

In case of any queries related to District Merit Scholarship or any assistance contact on the below-mentioned details.

FAQ’s on District Merit Scholarship 2019

Question 1.
How is District Merit Scholarship helpful to Kerala students?

Answer:
The government of Kerala grants District Merit Scholarship (DMS) 2019 to students who have passed in all subjects in the SSLC exam with A+ grade. This SSLC exam must be conducted by the Board of Public Examination, Kerala.

Question 2.
Is District Merit Scholarship available to all state students?

Answer:
No, this scholarship is available only for the students of Kerala State. The District Merit Scholarship has initiated by the Directorate of Collegiate Education, Govt. of Kerala.

Question 3.
How much scholarship amount will be disbursed to the meritorious students?

Answer:
The scholarship amount of Rs. 1,250/- per annum will be disbursed to each scholar.

Question 4.
What are the DMS application starting and closing date?

Answer:
The DMS application starting date is August 1, 2019. Whereas the closing date is September 30, 2019.

Question 5.
The District Merit Scholarship is available for which courses?

Answer:
District Merit Scholarship is available for students of Kerala who are currently studying in Higher Secondary, VHSC, ITI or polytechnic Courses.

Hope this article will help you to get more information about District Merit Scholarship (DMS) 2019. If you have queries related to DMS, then leave it in the comment box to get in touch with us.

The post District Merit Scholarship (DMS) 2019 | Important Dates, Eligibility, Application appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives Ex 6.2

Get Free NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives Ex 6.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Application of Derivatives Exercise 6.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 5 Class 12 Application of Derivatives Ex 6.2 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives Ex 6.2 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Applications of Derivatives chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives Ex 6.2

Ex 6.2 Class 12 Maths Question 1.
Show that the function given by f (x) = 3x+17 is strictly increasing on R.
Solution:
f(x) = 3x + 17
∴ f’ (x) = 3>0 ∀ x∈R
⇒ f is strictly increasing on R.

Ex 6.2 Class 12 Maths Question 2.
Show that the function given by f (x) = e^2x is strictly increasing on R.
Solution:
We have f (x) = e^2x
⇒ f’ (x) = 2e^2x
Case I When x > 0, then f’ (x) = 2e^2x

Ex 6.2 Class 12 Maths Question 3.
Show that the function given by f (x) = sin x is
(a) strictly increasing in
(b) strictly decreasing in
(c) neither increasing nor decreasing in (0, π)
Solution:
We have f(x) = sinx
∴ f’ (x) = cosx
(a) f’ (x) = cos x is + ve in the interval
⇒ f(x) is strictly increasing on
(b) f’ (x) = cos x is a -ve in the interval
⇒ f (x) is strictly decreasing in
(c) f’ (x) = cos x is +ve in the interval
while f’ (x) is -ve in the interval
∴ f(x) is neither increasing nor decreasing in (0,π)

Ex 6.2 Class 12 Maths Question 4.
Find the intervals in which the function f given by f(x) = 2x² – 3x is
(a) strictly increasing
(b) strictly decreasing
Solution:
f(x) = 2x² – 3x
⇒ f’ (x) = 4x – 3
⇒ f’ (x) = 0 at x =
The point divides the real

Ex 6.2 Class 12 Maths Question 5.
Find the intervals in which the function f given by f (x) = 2x³ – 3x² – 36x + 7 is
(a) strictly increasing
(b) strictly decreasing
Solution:
f(x) = 2x³ – 3x² – 36x + 7;
f (x) = 6 (x – 3) (x + 2)
⇒ f’ (x) = 0 at x = 3 and x = – 2
The points x = 3, x = – 2, divide the real line into three disjoint intervals viz. (-∞,-2), (-2,3), (3,∞)
Now f’ (x) is +ve in the intervals (-∞, -2) and (3,∞). Since in the interval (-∞, -2) each factor x – 3, x + 2 is -ve.
⇒ f’ (x) = + ve.
(a) f is strictly increasing in (-∞, -2)∪(3,∞)
(b) In the interval (-2,3), x+2 is +ve and x-3 is -ve.
f (x) = 6(x – 3)(x + 2) = + x – = -ve
∴ f is strictly decreasing in the interval (-2,3).

Ex 6.2 Class 12 Maths Question 6.
Find the intervals in which the following functions are strictly increasing or decreasing:
(a) x² + 2x – 5
(b) 10 – 6x – 2x²
(c) – 2x³ – 9x² – 12x + 1
(d) 6 – 9x – x²
(e) (x + 1)³(x – 3)³
Solution:
(c) Let f(x) = – 2x³ – 9x² – 12x + 1
∴ f’ (x) = – 6x² – 18x – 12
= – 6(x² + 3x + 2)
f'(x) = – 6(x + 1)(x + 2), f’ (x) = 0 gives x = -1 or x = -2
The points x = – 2 and x = – 1 divide the real line into three disjoint intervals namely ( – ∞, – 2) ( – 2, – 1) and( – 1 ∞).
In the interval (-∞,-2) i.e.,-∞<x<-2 (x+ 1) (x+2) are -ve.
∴f’ (x) = (-) (-) (- ) = – ve.
⇒ f (x) is decreasing in (-∞,-2)
In the interval (-2, -1) i.e., – 2 < x < -1,
(x + 1) is -ve and (x + 2) is + ve.
∴ f'(x) = (-)(-) (+) = + ve.
⇒ f (x) is increasing in (-2, -1)
In the interval (-1,∞) i.e.,-1 <x<∞,(x + 1) and (x + 2) are both positive. f’ (x) = (-) (+) (+) = -ve.
⇒ f (x) is decreasing in (-1, ∞)
Hence, f (x) is increasing for – 2 < x < – 1 and decreasing for x<-2 and x>-1.

Ex 6.2 Class 12 Maths Question 7.
Show that , is an increasing function of x throughout its domain.
Solution:
let
f’ (x) =
For f (x) to be increasing f’ (x) > 0

Hence, is an increasing function of x for all values of x > – 1.

Ex 6.2 Class 12 Maths Question 8.
Find the values of x for which y = [x (x – 2)]² is an increasing function.
Solution:
y = x⁴ – 4x³ + 4x²
∴ = 4x³ – 12x² + 8x
For the function to be increasing >0
4x³ – 12x² + 8x>0
⇒ 4x(x – 1)(x – 2)>0
For 0 < x < 1, = (+)(-)(-) = +ve and for x > 2, = (+) (+) (+) = +ve
Thus, the function is increasing for 0 < x < 1 and x > 2.

Ex 6.2 Class 12 Maths Question 9.
Prove that is an increasing function of θ in
Solution:

For the function to be increasing > 0
⇒ cosθ(4-cos²θ)>0
⇒ cosθ>0
⇒ θ∈

Ex 6.2 Class 12 Maths Question 10.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Solution:
Let f (x) = log x
Now, f’ (x) = ; When takes the
values x > 0, > 0, when x > 0,
∵ f’ (x) > 0
Hence, f (x) is an increasing function for x > 0 i.e

Ex 6.2 Class 12 Maths Question 11.
Prove that the function f given by f (x) = x² – x + 1 is neither strictly increasing nor strictly decreasing on (-1,1).
Solution:
Given
f (x) = x² – x + 1

∴ f (x) is neither increasing nor decreasing on (-1,1).

Ex 6.2 Class 12 Maths Question 12.
Which of the following functions are strictly decreasing on
(a) cos x
(b) cos 2x
(c) cos 3x
(d) tan x
Solution:
(a) We have f (x) = cos x
∴ f’ (x) = – sin x < 0 in
∴ f’ (x) is a decreasing function.

Ex 6.2 Class 12 Maths Question 13.
On which of the following intervals is the function f given by f (x )= x¹⁰⁰ + sin x – 1 strictly decreasing ?
(a) (0,1)
(b)
(c)
(d) none of these
Solution:
(d) f(x) = x¹⁰⁰ + sin x – 1
∴ f’ (x)= 100x⁹⁹+ cos x
(a) for(-1, 1)i.e.,- 1 <x< 1,-1 <x⁹⁹< 1
⇒ -100<100x⁹⁹<100;
Also 0 ⇒ f’ (x) can either be +ve or -ve on(-1, 1)
∴ f (x) is neither increasing nor decreasing on (-1,1).
(b) for (0,1) i.e. 0<x< 1 x⁹⁹ and cos x are both +ve ∴ f’ (x) > 0
⇒ f (x) is increasing on(0,1)

Ex 6.2 Class 12 Maths Question 14.
Find the least value of a such that the function f given by f (x) = x² + ax + 1 is strictly increasing on (1,2).
Solution:
We have f (x) = x² + ax + 1
∴ f’ (x) = 2x + a.
Since f (x) is an increasing function on (1,2)
f’ (x) > 0 for all 1 < x < 2 Now, f” (x) = 2 for all x ∈ (1,2) ⇒ f” (x) > 0 for all x ∈ (1,2)
⇒ f’ (x) is an increasing function on (1,2)
⇒ f’ (x) is the least value of f’ (x) on (1,2)
But f’ (x)>0 ∀ x∈ (1,2)
∴ f’ (1)>0 =>2 + a>0
⇒ a > – 2 : Thus, the least value of a is – 2.

Ex 6.2 Class 12 Maths Question 15.
Let I be any interval disjoint from (-1,1). Prove that the function f given by is strictly increasing on I.
Solution:
Given


Hence, f’ (x) is strictly increasing on I.

Ex 6.2 Class 12 Maths Question 16.
Prove that the function f given by f (x) = log sin x is strictly increasing on and strictly decreasing on

Solution:
f’ (x) =
when 0 < x < , f’ (x) is +ve; i.e., increasing
When < x < π, f’ (x) is – ve; i.e., decreasing,
∴ f (x) is decreasing. Hence, f is increasing on (0, π/2) and strictly decreasing on (π/2, π).

Ex 6.2 Class 12 Maths Question 17.
Prove that the function f given by f(x) = log cos x is strictly decreasing on and strictly increasing on
Solution:

f’ (x) =
In the interval ,f’ (x) = -ve
∴ f is strictly decreasing.
In the interval , f’ (x) is + ve.
∴ f is strictly increasing in the interval.

Ex 6.2 Class 12 Maths Question 18.
Prove that the function given by
f (x) = x³ – 3x² + 3x -100 is increasing in R.
Solution:
f’ (x) = 3x² – 6x + 3
= 3 (x² – 2x + 1)
= 3 (x -1 )²
Now x ∈ R, f'(x) = (x – 1)²≥0
i.e. f'(x)≥0 ∀ x∈R; hence, f(x) is increasing on R.

Ex 6.2 Class 12 Maths Question 19.
The interval in which y = x² e^-x is increasing is
(a) (-∞,∞)
(b) (-2,0)
(c) (2,∞)
(d) (0,2)
Solution:
(d) f’ (x) = 2xe^-x + x²( – e^-x) = xe^-x(2-x) = e^-xx(2-x)
Now e^-x is positive for all x ∈ R f’ (x) = 0 at x = 0,2
x = 0, x = 2 divide the number line into three disjoint intervals, viz. (-∞, 0), (0,2), (2, ∞)
(a) Interval (-∞,0) x is +ve and (2-x) is +ve
∴ f’ (x) = e^-xx (2- x)=(+)(-) (+) = -ve
⇒ f is decreasing in (-∞,0)
(b) Interval (0,2) f’ (x) = e^-x x (2 – x)
= (+)(+)(+) = +ve
⇒ f is increasing in (0,2)
(c) Interval (2, ∞) f’ (x) = e^-x x (2 – x) = (+) (+) (-)
= – ve
⇒ f is decreasing in the interval (2, ∞)

NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives Hindi Medium Ex 6.2

Class 12 Maths NCERT Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals Ex 7.1
• Chapter 8 Application of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability Ex 13.1

The post NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives Ex 6.2 appeared first on Learn CBSE.