# RBSE Solutions for Class 12 Maths Chapter 4 Determinants Miscellaneous Exercise

## Rajasthan Board RBSE Class 12 Maths Chapter 4 Determinants Miscellaneous Exercise

RBSE Solutions For Class 12 Maths Chapter 4 Miscellaneous Question 1.
Value of determinant is:
(a) 0
(b) 1
(c) -1
(d) none of these
Solution: = cos 80° sin 10° – (- cos 109) sin 80°
= cos 80° sin 10° + cos 10° sin 80°
= sin (10° + 80°)
= sin 90° = 1
So, option (b) is correct.

RBSE Class 12 Maths Chapter 4 Miscellaneous Question 2.
Co-factors of first column of determinant (a) -1, 3
(b) -1,- 3
(c)- 1, 20
(d) – 1,- 20
Solution: Co-factor of a11
F11 = (- 1)2 M11
= 1 x (- 1) = – 1
Co-factor of a12
F21 = (-1)11 M21
= (- 1) × 20 = – 20
So, option (d) is correct.

RBSE Solutions For Class 12 Maths Chapter 4 Question 3.
If , then the value of will be :
(a) – 2∆
(b) 8∆
(c) -8∆
(d) -6∆
Solution: So, option (c) is correct.

Ch 4 Miscellaneous Class 12 Maths Question 4.
Which of the following determinant is identical to determinant : Solution:  So, option (c) is correct.

RBSE Solution Class 12 Maths Question 5.
Value of (a) 0
(b) 1
(c) 1/2
(d) – 1/2
Solution: = cos 50° cos 10° – sin 50° sin 10°
= cos (50° + 10°)
= cos 60° = $$\frac { 1 }{ 2 }$$
So, option (c) is correct.

Class 12th Maths RBSE Solution Question 6.
Value of (a) ab + bc + ca
(b) 0
(c) 1
(d) abc
Solution: RBSE Class 10 Maths Chapter 4 Miscellaneous Solutions Question 7.
If ω is a cube root of unity, then value of (a) ω2
(b) ω
(c) 1
(d) o
11 04 081
Solution: RBSE Solution Class 12th Maths Question 8.
If , then x will be:
(a) 6
(b) 7
(c) 8
(d) 0
Solution: ⇒ (4 – 2)2 = (3x – 2) – (x + 6)
⇒ (2)2 = 3x – 2 – x – 6
⇒ 4 = 2x – 8
⇒ 4 + 8 = 2x
x = 6
So, option (a) is correct.

RBSE Solution Maths Class 12 Question 9.
If and F11, F12, F13, …, are the corresponding cofactors of a11, a12, a13, …, then which of the following is true :
(a) a12F12+ a22F22 + a32F32 = 0
(b) a12F12 + a22F22 + a32F32 ≠ ∆
(c) a12F12 + a22F22 + a32F32 = ∆
(d) a12F12 + a22F22 + a32F32 = – ∆.
Solution:
a12F12 + a22F22 + a32F32 = ∆
So, option (c) is correct.

RBSE Solution Class 12th Math Question 10.
Value of determinant (a) x + y + z
(b) 2(x + y + z)
(c) 1
(d) 0
Solution: So, option (d) is correct.

RBSE Solution Class 12 Math Question 11.
Solve the following equation : Solution: ⇒ 1(9x – 48) – 2(36 – 42) + 3(32 – 7x) = 0
⇒ 9x – 48 + 12 + 96 – 21x =0
⇒ – 12x + 60 = 0
⇒ – 12x = -60
⇒ x = $$\frac { 60 }{ 12 }$$ = -5
So, x = 5.

RBSE Solutions Maths Class 12 Question 12.
Find the value of determinant : Solution: = 1(27 – 1) – 3(9 – 9) + 9(3 – 81)
= 26 – 0 – 702
= -676.

Class 12 RBSE Solutions Maths Question 13.
Find the value of determinant Solution: Miscellaneous Exercise 4 Class 12 Question 14.
Prove that Solution: = a2b2c2 [- 1(1 – 1) – 1(-1 – 1) + 1(1 + 1)]
= a2b2c2 (0 + 2 + 2)
= 4a2b2c2 = R.H.S. Hence Proved.

RBSE Solution For Class 12 Maths Question 15.
Prove that x = 2 is a root of following equation. Also find its remaining roots : Solution: ∵ R1 = R2
∴ Determinant will be zero.
∴ Root of equation is 2. RBSE Solutions For Class 12 Maths Chapter 5 Miscellaneous Question 16.
Prove that: Solution: RBSE Solutions Class 12th Maths Question 17.
Prove that .
Solution:  = (a + b + c)[(- b – c – a) (- c – a – b) – 0]
= (a + b + c)(b + c + a)(c + a + b)
= (a + b + c)3 = R.H.S.
Proved.

Class 12 Maths Chapter 4 Miscellaneous Solution Question 18.
Prove that : Solution: Miscellaneous Exercise Chapter 4 Class 12 Question 19.
Prove that: Solution:  = (a – b) (b – c) [(b2 + c2 + bc) – (a2 + b2 + ab)]
= (a – b) (b – c) (b2 + c2 + bc – a2 – b2 – ab)
= (a – b)(b – c) [bc + c2 – a2 – ab]
= (a – b)(b – c) [bc – ab + c2 – a2 ]
= (a – b) (b – c) [b(c – a) + (c2 – a)]
= (a – b)(b – c)(c – a)(b + c + a)
= R.H.S. Proved.

Miscellaneous Exercise On Chapter 4 Class 12 Question 20.
Prove that: Solution:  Maths RBSE Solutions Class 12 Question 21.
If a + b + c = 0, then solve: Solution: ⇒ (-x) [(c – a) (a – c + x) – (b – c + x) (b – x – a)] = 0
⇒ (-x) [(ac – c2 + cx – a2 + ac – ax)
⇒ (b2 – bx – ab – bc + cx + ac – xb – x2 – ax)] = 0
= (-x) [x2 – (a2 + b2 – ab – bc – bc – ca)]= 0
If – x = 0, then x = 0
Now, if x2 = (a2 + b2 + c2 + bc – ab – ca) = 0 .  Class 12 Maths Ch 4 Miscellaneous Question 22.
Prove that Solution: Chapter 4 Miscellaneous Class 12 Question 23.
If p + q + r = 0, then prove that Solution: Miscellaneous Chapter 4 Class 12 Question 24.
Prove that Solution: 