# RBSE Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.1

## Rajasthan Board RBSE Class 12 Maths Chapter 4 Determinants Ex 4.1

Question 1.
For which value of k, det
will be zero ?
Solution:

Question 2.
, find x : y.
Solution:

Question 3.
, then find the value of x and y.
Solution :

Question 4.
Find x, if

Solution :

⇒ (x – 1) (x – 3) – x(x – 2) = 0
⇒ x2 – 3x – x + 3 – x2 + 2x = 0
⇒ -2x + 3 = 0
⇒ – 2x = -3
So, x = 3/2

Question 5.
Write minors and co-factors of following determinants corresponding to first column, also find the value of determinants:

Solution:

= – 2 – 10 = -12
Cofactor of a
F11 = (- 1)2 M11
= 1 x (- 12) = -12
Minor of a21

= -6 – 10 = -16
Cofactor of a21
F21 = (- 1)3 M21
= -1 x (-16) = 16
Minor of a31

= – 6 – (- 2)
= – 6 + 2 = -4
Cofactor of a31
F31 = (- 1))4M31
= 1 x (-4) = -4
So, Determinant M = a11F11 + a21 F21 + a31 F31
= 1 .(- 12) +4·(16) + 3.(-4)
= – 12 + 64 – 12 = 40
So, M11 = -12, M21 = – 16, M31 =-4
F11 = – 12, F21 = 16, F31 = -4
|A| = 40

(ii)
Solution:
Minor of a11

= a . (bc – f2) + h. (fg – hc) + g. (hf – bg)
= abc – af2 +fgh – h2c + fgh – bg2
= abc + 2fgh – af2 – bg2 – ch2
So M11 = bc – f2, M21= hc – fg, M31 = hf – bg
F11 = bc – f2, F21 = fg – hc, F31 = hf – bg
|A| = abc + 2fgh – af2 – bg2 – hc2

Question 6.
Find the value of determinant

Solution:

= -5 (0 – 3)
= -5 x (-3) = 15

Question 7.
Prove that :

Solution:

= 1 + c2 + a2 – abc + abc + b2
= 1 + a2 + b2 + c2 = R.H.S. Proved.