## 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

⇒ x^{2} – 3x – x + 3 – x^{2} + 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

F_{11} = (- 1)^{2} M_{11}

= 1 x (- 12) = -12

Minor of a^{21}

= -6 – 10 = -16

Cofactor of a_{21}

F_{21} = (- 1)^{3} M_{21}

= -1 x (-16) = 16

Minor of a_{31}

= – 6 – (- 2)

= – 6 + 2 = -4

Cofactor of a_{31}

F_{31} = (- 1))^{4}M_{31}

= 1 x (-4) = -4

So, Determinant M = a_{11}F_{11} + a_{21} F_{21} + a_{31} F_{31}

= 1 .(- 12) +4·(16) + 3.(-4)

= – 12 + 64 – 12 = 40

So, M_{11} = -12, M_{21} = – 16, M_{31} =-4

F_{11} = – 12, F_{21} = 16, F_{31} = -4

|A| = 40

(ii)

Solution:

Minor of a_{11}

= a . (bc – f^{2}) + h. (fg – hc) + g. (hf – bg)

= abc – af^{2} +fgh – h^{2}c + fgh – bg^{2}

= abc + 2fgh – af^{2} – bg^{2} – ch^{2}

So M_{11} = bc – f^{2}, M_{21}= hc – fg, M_{31} = hf – bg

F_{11} = bc – f^{2}, F_{21} = fg – hc, F_{31} = hf – bg

|A| = abc + 2fgh – af^{2} – bg^{2} – hc^{2}

Question 6.

Find the value of determinant

Solution:

= -5 (0 – 3)

= -5 x (-3) = 15

Question 7.

Prove that :

Solution:

= 1 + c^{2} + a^{2} – abc + abc + b^{2}

= 1 + a^{2} + b^{2} + c^{2} = R.H.S. Proved.