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:

Section Name	Topic Name
4	Determinants
4.1	Introduction
4.2	Determinant
4.3	Properties of Determinants
4.4	Area of a Triangle
4.5	Adjoint and Inverse of a Matrix
4.6	Applications of Determinants and Matrices
4.7	Summary

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.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x & a & x+a \\ y & b & y+b \\ z & c & z+c \end{matrix} \right| =0$
Solution:
L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x & a & x \\ y & b & y \\ z & c & z \end{matrix} \right| +\left| \begin{matrix} x & a & a \\ y & b & b \\ z & c & c \end{matrix} \right|$
(C1 = C3 and C2 = C3)
= 0 + 0
= 0
= R.H.S

Ex 4.2 Class 12 Maths Question 2.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{matrix} \right| =0$
Solution:
L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{matrix} \right| =0$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 2

Ex 4.2 Class 12 Maths Question 3.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| =0$
Solution:
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| =\left| \begin{matrix} 2 & 7 & 0 \\ 3 & 8 & 0 \\ 5 & 9 & 0 \end{matrix} \right|$
Image may be NSFW.
Clik here to view. ${ C }_{ 3 }\rightarrow { C }_{ 3 }-{ C }_{ 1 }-{ 9C }_{ 2 }=0$

Ex 4.2 Class 12 Maths Question 4.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b) \end{matrix} \right| =0$
Solution:
L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b) \end{matrix} \right|$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 4
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Ex 4.2 Class 12 Maths Question 5.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} b+c & q+r & y+z \\ c+a & r+p & z+x \\ a+b & p+q & x+y \end{matrix} \right| =2\left| \begin{matrix} a & p & x \\ b & q & y \\ c & r & z \end{matrix} \right|$
Solution:
L.H.S = ∆ = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} b+c & q+r & y+z \\ c+a & r+p & z+x \\ a+b & p+q & x+y \end{matrix} \right|$
Image may be NSFW.
Clik here to view. vedantu class 12 maths Chapter 4 Determinants 5

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

Ex 4.2 Class 12 Maths Question 6.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 0 & a & -b \\ -a & 0 & -c \\ b & c & 0 \end{matrix} \right| =0$
Solution:
L.H.S = ∆ = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 0 & a & -b \\ -a & 0 & -c \\ b & c & 0 \end{matrix} \right|$ …(i)
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 6

Ex 4.2 Class 12 Maths Question 7.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} { -a }^{ 2 } & ab & ac \\ ba & { -b }^{ 2 } & bc \\ ac & cb & { -c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }$
Solution:
L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} { -a }^{ 2 } & ab & ac \\ ba & { -b }^{ 2 } & bc \\ ac & cb & { -c }^{ 2 } \end{matrix} \right|$
Image may be NSFW.
Clik here to view. vedantu class 12 maths Chapter 4 Determinants 7

Ex 4.2 Class 12 Maths Question 8.
(a) Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| =(a-b)(b-c)(c-a)$
(b) Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ { a }^{ 3 } & { b }^{ 3 } & { c }^{ 3 } \end{matrix} \right| =(a-b)(b-c)(c-a)(a+b+c)$
Solution:
(a) L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right|$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 8
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Clik here to view.
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Ex 4.2 Class 12 Maths Question 9.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x & x^{ 2 } & yx \\ y & { y }^{ 2 } & zx \\ z & { z }^{ 2 } & xy \end{matrix} \right| =(x-y)(y-z)(z-x)(xy+yz+zx)$
Solution:
Let ∆ = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x & x^{ 2 } & yx \\ y & { y }^{ 2 } & zx \\ z & { z }^{ 2 } & xy \end{matrix} \right|$
Applying R1–>R1 – R2, R2–>R2 – R3
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 9

Ex 4.2 Class 12 Maths Question 10.
(a) Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{matrix} \right| =(5x+4){ (4-x) }^{ 2 }$
(b) Image may be NSFW.
Clik here to view. $\left| \begin{matrix} y+x & y & y \\ y & y+k & y \\ y & y & y+k \end{matrix} \right| ={ k }^{ 2 }(3y+k)$
Solution:
(a) L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{matrix} \right|<br />$
Image may be NSFW.
Clik here to view. vedantu class 12 maths Chapter 4 Determinants 10

Ex 4.2 Class 12 Maths Question 11.
(a) Image may be NSFW.
Clik here to view. $\left| \begin{matrix} a-b-c & \quad 2a & \quad 2a \\ 2b & \quad b-c-a & \quad 2b \\ 2c & 2c & \quad c-a-b \end{matrix} \right| ={ (a+b+c) }^{ 3 }$
(b) Image may be NSFW.
Clik here to view. $\left| \begin{matrix} x+y+2z & \quad z & \quad z \\ x & \quad y+z+2x & \quad x \\ y & y & \quad z+x+2y \end{matrix} \right| ={ 2(x+y+z) }^{ 3 }$
Solution:
(a) L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} a-b-c & \quad 2a & \quad 2a \\ 2b & \quad b-c-a & \quad 2b \\ 2c & 2c & \quad c-a-b \end{matrix} \right|$
= Image may be NSFW.
Clik here to view. $\left( a+b+c \right) \left| \begin{matrix} 1 & \quad 1 & \quad 1 \\ 2b & \quad b-c-a & \quad 2b \\ 2c & \quad 2c & \quad c-a-b \end{matrix} \right|$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 11

Ex 4.2 Class 12 Maths Question 12.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & \quad x & { \quad x }^{ 2 } \\ { x }^{ 2 } & \quad 1 & x \\ x & { \quad x }^{ 2 } & 1 \end{matrix} \right| ={ { (1-x }^{ 3 }) }^{ 2 }$
Solution:
L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1 & \quad x & { \quad x }^{ 2 } \\ { x }^{ 2 } & \quad 1 & x \\ x & { \quad x }^{ 2 } & 1 \end{matrix} \right|$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 12
Image may be NSFW.
Clik here to view.

Ex 4.2 Class 12 Maths Question 13.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1+{ a }^{ 2 }-{ b }^{ 2 } & \quad 2ab & \quad -2b \\ 2ab & \quad 1-{ a }^{ 2 }+{ b }^{ 2 } & \quad 2a \\ 2b & \quad -2a & \quad 1-{ a }^{ 2 }+{ b }^{ 2 } \end{matrix} \right| ={ (1+{ a }^{ 2 }+{ b }^{ 2 }) }^{ 3 }$
Solution:
L.H.S = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} 1+{ a }^{ 2 }-{ b }^{ 2 } & \quad 2ab & \quad -2b \\ 2ab & \quad 1-{ a }^{ 2 }+{ b }^{ 2 } & \quad 2a \\ 2b & \quad -2a & \quad 1-{ a }^{ 2 }+{ b }^{ 2 } \end{matrix} \right|$
Image may be NSFW.
Clik here to view. vedantu class 12 maths Chapter 4 Determinants 13

Ex 4.2 Class 12 Maths Question 14.
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} { a }^{ 2 }+1 & \quad ab & \quad ac \\ ab\quad & \quad b^{ 2 }+1 & \quad bc \\ ca\quad & \quad cb & \quad { c }^{ 2 }+1 \end{matrix} \right| =1+{ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }$
Solution:
Let ∆ = Image may be NSFW.
Clik here to view. $\left| \begin{matrix} { a }^{ 2 }+1 & \quad ab & \quad ac \\ ab\quad & \quad b^{ 2 }+1 & \quad bc \\ ca\quad & \quad cb & \quad { c }^{ 2 }+1 \end{matrix} \right|$
Image may be NSFW.
Clik here to view. $\left| \begin{matrix} { a }^{ 2 }+1 & \quad ab+0 & \quad ac+0 \\ ab+0\quad & \quad b^{ 2 }+1 & \quad bc+0 \\ ca+0\quad & \quad cb+0 & \quad { c }^{ 2 }+1 \end{matrix} \right|$
This may be expressed as the sum of 8 determinants
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 14

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

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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 Determinants
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Clik here to view. Class 12 Maths Exercise 4.2 Solutions
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Clik here to view. Determinants Exercise 4.2 properties solutions
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.2 सरणिक
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Clik here to view. 12 Maths Chapter 4 Exercise 4.2 Question 11, 12, 13, 14, 15, 16