RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 Textbook Exercise Questions and Answers.

Rajasthan Board RBSE Solutions for Class 12 Maths in Hindi Medium & English Medium are part of RBSE Solutions for Class 12. Students can also read RBSE Class 12 Maths Important Questions for exam preparation. Students can also go through RBSE Class 12 Maths Notes to understand and remember the concepts easily.

RBSE Class 12 Maths Solutions Chapter 4 Determinants Ex 4.2

Question 1.
\(\left|\begin{array}{lll} x & a & x+a \\ y & b & y+b \\ z & c & z+c \end{array}\right|\) = 0
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 1

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 2.
\(\left|\begin{array}{lll} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{array}\right|\) = 0
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 2

Question 3.
\(\left|\begin{array}{lll} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{array}\right|\) = 0
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 3

Question 4.
\(\left|\begin{array}{lll} 1 & b c & a(b+c) \\ 1 & c a & b(c+a) \\ 1 & a b & c(a+b) \end{array}\right|\) = 0
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 4

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 5.
\(\left|\begin{array}{lll} b+c & q+r & y+z \\ c+a & r+p & z+x \\ a+b & p+q & x+y \end{array}\right|=2\left|\begin{array}{lll} a & p & x \\ b & q & y \\ c & r & z \end{array}\right|\)
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 5

Question 6.
\(\left|\begin{array}{rrr} 0 & a & -b \\ -a & 0 & -c \\ b & c & 0 \end{array}\right|\) = 0
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 6
= - Δ
⇒ Δ + Δ = 0 ⇒ 2Δ = 0
⇒ Δ = 0
Hence proved.

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 7.
\(\left|\begin{array}{rrr} -a^{2} & a b & a c \\ b a & -b^{2} & b c \\ c a & c b & -c^{2} \end{array}\right|\) = 4a2b2c2
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 7

Question 8.
(i) \(\left|\begin{array}{ccc} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}\right|\) = (a - b) (b - c) (c - a)
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 8
Taking (b - a) and (c - a) common from R2 and R3 respectively
Δ = (b - a) (c - a) \(\left|\begin{array}{ccc} 1 & a & a^{2} \\ 0 & 1 & b+a \\ 0 & 1 & c+a \end{array}\right|\)
Expanding along C1
Δ = (b - a) (c - a) \(\left\{1\left|\begin{array}{cc} 1 & b+a \\ 1 & c+a \end{array}\right|\right\}\)
= (b - a) (c - a) {c + a - b - a}
= (b - a) (c - a) (c - b)
= (a - b) (b - c) (c - a)
Hence proved

(ii) \(\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3} \end{array}\right|\) = (a - b) (b - c) (c - a) (a + b + c)
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 9
= (a - b) (b - c) × {b2 + bc + c2 - a2 - ab - b2}
= (a - b) (b - c) {c2 - a2 + bc - ab}
= (a - b) (b - c) × {(c - a) (c + a) + b(c - a)}
= (a - b) (b - c) (c - a) (a + b + c)
Hence Proved.

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 9.
\(\left|\begin{array}{ccc} x & x^{2} & y z \\ y & y^{2} & z x \\ z & z^{2} & x y \end{array}\right|\) = (x - y) (y - z) (z - x) (xy + yz + zx)
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 10
Taking (x - y) and (y - z) common from R1 and R2 respectively
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 11
= (x - y) (y - z) (z - x) × {xy - z2 + z(x + y + z)}
= (x - y) (y - z) (z - x) × (xy - z2 + xz + yz + z2)
= (x - y) (y - z) (z - x) (xy + yz + zx)
Hence Proved.

Question 10.
(i) \(\left|\begin{array}{ccc} x+4 & 2 x & 2 x \\ 2 x & x+4 & 2 x \\ 2 x & 2 x & x+4 \end{array}\right|\) = (5x + 4) (4 - x)2
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 12
= (5x - 4) {(x - 4) (x - 4) - 0 × (4 - x)}
= (5x + 4) (x - 4)2
= (5x + 4) (4 - x)2
Hence Proved.

(ii) \(\left|\begin{array}{ccc} y+k & y & y \\ y & y+k & y \\ y & y & y+k \end{array}\right|\) = k2(3y + k)
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 13

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 11.
(i) \(\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|\) = (a + b + c)3
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 14

(ii) \(\left|\begin{array}{ccc} x+y+2 z & x & y \\ z & y+z+2 x & y \\ z & x & z+x+2 y \end{array}\right|\) = 2(x + y + z)3
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 15

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 12.
\(\left|\begin{array}{ccc} 1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{array}\right|\) = (1 - x3)2
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 16
= (1 + x + x2) (1 - x2) (1 + x + x2)
= (1 + x + x2)2 (1 - x)2 = {(1 + x + x2) (1 - x)}2
= (1 - x3)2
Hence proved.

Question 13.
\(\left|\begin{array}{ccc} 1+a^{2}-b^{2} & 2 a b & -2 b \\ 2 a b & 1-a^{2}+b^{2} & 2 a \\ 2 b & -2 a & 1-a^{2}-b^{2} \end{array}\right|\) = (1 + a2 + b2)3
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 17
= (1 + a2 + b2)2 {1 - a2 - b2 + 2a2 + 2b2}
= (1 + a2 + b2)2 {1 + a2 + b2}
= (1 + a2 + b2)3
Hence Proved.

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 14.
\(\left|\begin{array}{ccc} a^{2}+1 & a b & a c \\ a b & b^{2}+1 & b c \\ c a & c b & c^{2}+1 \end{array}\right|\) = 1 + a2 + b2 + c2
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 18
= (1 + a2 + b2 + c2) {1 (1 × 1 - 0)}
= (1 + a2 + b2 + c2).1
= 1 + a2 + b2 + c2
Hence Proved.

Question 15.
Let 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|
Answer:
RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2 19
Thus, option (C) is correct.

RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.2

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
Answer:
Option (C) is correct.

Bhagya
Last Updated on Nov. 2, 2023, 9:24 a.m.
Published Nov. 1, 2023