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.

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:

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:

Question 3.
\(\left|\begin{array}{lll} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{array}\right|\) = 0
Answer:

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:

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:

Question 6.
\(\left|\begin{array}{rrr} 0 & a & -b \\ -a & 0 & -c \\ b & c & 0 \end{array}\right|\) = 0
Answer:

= - Δ
⇒ Δ + Δ = 0 ⇒ 2Δ = 0
⇒ Δ = 0
Hence proved.

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:

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:

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:

= (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.

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:

Taking (x - y) and (y - z) common from R₁ and R₂ respectively

= (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:

= (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:

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:

(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:

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:

= (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:

= (1 + a² + b²)² {1 - a² - b² + 2a² + 2b²}
= (1 + a² + b²)² {1 + a² + b²}
= (1 + a² + b²)³
Hence Proved.

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:

= (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:

Thus, option (C) is correct.

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.