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

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.

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 a_{11}

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

= 1 x (- 1) = – 1

Co-factor of a_{12}

F_{21} = (-1)^{11} M_{21}

= (- 1) × 20 = – 20

So, option (d) is correct.

Question 3.

If , then the value of will be :

(a) – 2∆

(b) 8∆

(c) -8∆

(d) -6∆

Solution:

So, option (c) is correct.

Question 4.

Which of the following determinant is identical to determinant

:

Solution:

So, option (c) is correct.

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.

Question 6.

Value of

(a) ab + bc + ca

(b) 0

(c) 1

(d) abc

Solution:

Question 7.

If ω is a cube root of unity, then value of

(a) ω^{2}

(b) ω

(c) 1

(d) o

11 04 081

Solution:

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.

Question 9.

If and F_{11}, F_{12}, F_{13}, …, are the corresponding cofactors of a_{11}, a_{12}, a_{13}, …, then which of the following is true :

(a) a_{12}F_{12}+ a_{22}F_{22} + a_{32}F_{32} = 0

(b) a_{12}F_{12} + a_{22}F_{22} + a_{32}F_{32} ≠ ∆

(c) a_{12}F_{12} + a_{22}F_{22} + a_{32}F_{32} = ∆

(d) a_{12}F_{12} + a_{22}F_{22} + a_{32}F_{32} = – ∆.

Solution:

a_{12}F_{12} + a_{22}F_{22} + a_{32}F_{32} = ∆

So, option (c) is correct.

Question 10.

Value of determinant

(a) x + y + z

(b) 2(x + y + z)

(c) 1

(d) 0

Solution:

So, option (d) is correct.

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.

Question 12.

Find the value of determinant :

Solution:

= 1(27 – 1) – 3(9 – 9) + 9(3 – 81)

= 26 – 0 – 702

= -676.

Question 13.

Find the value of determinant

Solution:

Question 14.

Prove that

Solution:

= a^{2}b^{2}c^{2} [- 1(1 – 1) – 1(-1 – 1) + 1(1 + 1)]

= a^{2}b^{2}c^{2} (0 + 2 + 2)

= 4a^{2}b^{2}c^{2} = R.H.S. Hence Proved.

Question 15.

Prove that x = 2 is a root of following equation. Also find its remaining roots :

Solution:

∵ R_{1} = R_{2}

∴ Determinant will be zero.

∴ Root of equation is 2.

Question 16.

Prove that:

Solution:

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.

Question 18.

Prove that :

Solution:

Question 19.

Prove that:

Solution:

= (a – b) (b – c) [(b^{2} + c^{2} + bc) – (a^{2} + b^{2} + ab)]

= (a – b) (b – c) (b^{2} + c^{2} + bc – a^{2} – b^{2} – ab)

= (a – b)(b – c) [bc + c^{2} – a^{2} – ab]

= (a – b)(b – c) [bc – ab + c^{2} – a^{2} ]

= (a – b) (b – c) [b(c – a) + (c^{2} – a)]

= (a – b)(b – c)(c – a)(b + c + a)

= R.H.S. Proved.

Question 20.

Prove that:

Solution:

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 – c^{2} + cx – a^{2} + ac – ax)

⇒ (b^{2} – bx – ab – bc + cx + ac – xb – x^{2} – ax)] = 0

= (-x) [x^{2} – (a^{2} + b^{2} – ab – bc – bc – ca)]= 0

If – x = 0, then x = 0

Now, if x^{2} = (a^{2} + b^{2} + c^{2} + bc – ab – ca) = 0 .

Question 22.

Prove that

Solution:

Question 23.

If p + q + r = 0, then prove that

Solution:

Question 24.

Prove that

Solution:

##### RBSE Solutions for Class 12 Maths

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