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## Determinants Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ Questions of Determinants Class 12 with answers so you can apply your skills during the exam.

Question 1.

If \(\left|\begin{array}{rr}

x & 2 \\

18 & x

\end{array}\right|\) = \(\left|\begin{array}{rr}

6 & 2 \\

18 & 6

\end{array}\right|\), then x is equal to

(a) 6

(b) ±6

(c) -6

(d) 6, 6

## Answer

Answer: (a) 6

Question 2.

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

Answer: (c) k³|A|

Question 3.

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

Answer: (c) Determinant is a number associated to a square matrix

Question 4.

If area of triangle is 35 sq. units with vertices (2, -6), (5, 4) and (k, 4). Then k is

(a) 12

(b) -2

(c) -12, -2

(d) 12, -2.

## Answer

Answer: (d) 12, -2.

Question 5.

If A = \(\left[\begin{array}{lll}

a_{11} & a_{12} & a_{13} \\

a_{21} & a_{22} & a_{23} \\

a_{31} & a_{32} & a_{33}

\end{array}\right]\) and A_{ij} is co-factors of a_{ij}, then A is given by

(a) a_{11}A_{31} +a_{12}A_{32} + a_{13}A_{33}

(b) a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{33}

(c) a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13}

(d) a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}

## Answer

Answer: (d) a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}

Question 6.

Let A be a non-singular matrix of order 3 × 3. Then |adj. A| is equal to

(a) |A|

(b) |A|²

(c) |A|³

(d) 3|A|

## Answer

Answer: (b) |A|²

Question 7.

If A is any square matrix of order 3 x 3 such that |a| = 3, then the value of |adj. A| is?

(a) 3

(b) \(\frac { 1 }{3}\)

(c) 9

(d) 27

## Answer

Answer: (c) 9

Question 8.

If A is an invertible matrix of order 2, then det (A^{-1}) is equal to

(a) det (A)

(b) \(\frac { 1 }{det(A)}\)

(c) 1

(d) 0

## Answer

Answer: (b) \(\frac { 1 }{det(A)}\)

Question 9.

If a, b, c are in A.P., then determinant

(a) 0

(b) 1

(c) x

(d) 2x

## Answer

Answer: (a) 0

Question 10.

If x, y, z are non-zero real numbers, then the inverse of matrix A = \(\left[\begin{array}{lll}

x & 0 & 0 \\

0 & y & 0 \\

0 & 0 & z

\end{array}\right]\) is

## Answer

Answer:

Question 11.

Let A =

where 0 ≤ θ ≤ 2π then

(a) Det (A) = 0

(b) Det (A) ∈ (2, ∞)

(c) Det (A) ∈ (2, 4)

(d) Det (A) ∈ [2, 4]

## Answer

Answer: (d) Det (A) ∈ [2, 4]

Question 12.

If \(\left|\begin{array}{rr}

2x & 5 \\

8 & x

\end{array}\right|\) = \(\left|\begin{array}{rr}

6 & -2 \\

7 & 3

\end{array}\right|\), then value of ‘x’ is

(a) 3

(b) ±3

(c) ±6

(d) 6

## Answer

Answer: (c) ±6

Question 13.

Let Δ = \(\left|\begin{array}{lll}

\mathbf{A} x & x^{2} & 1 \\

B y & y^{2} & 1 \\

C z & z^{2} & 1

\end{array}\right|\) and Δ_{1} = \(\left|\begin{array}{rrr}

\mathbf{A} & \mathbf{B} & \mathbf{C} \\

\boldsymbol{x} & \boldsymbol{y} & z \\

z \boldsymbol{y} & z x & x y

\end{array}\right|\), then

(a) Δ_{1} = -Δ

(b) Δ ≠ Δ_{1}

(c) Δ – Δ_{1} = 0

(d) None of these

## Answer

Answer: (c) Δ – Δ_{1} = 0

Question 14.

If x, y ∈R, then the determinant:

lies in the interval

(a) [-√2, √2]

(b) [-1, 1]

(c) [√2, 1]

(d) [-1, √2]

## Answer

Answer: (a) [-√2, √2]

Question 15.

The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value of ‘k’ will be:

(a) 9

(b) 3

(c) -9

(d) 6.

## Answer

Answer: (b) 3

Question 16.

If A, B and C are angles of a triangle, then the determinant:

\(\left|\begin{array}{ccc}

-1 & \cos C & \cos B \\

\cos C & -1 & \cos A \\

\cos B & \cos A & -1

\end{array}\right|\) is equal to

(a) 0

(b) -1

(c) 1

(d) None of these.

## Answer

Answer: (a) 0

Question 17.

Let A be a square matrix all of whose entries are integers. Then which of the following is true?

(a) If det A = ± 1, then A^{-1} need not exist

(b) If det A = ± 1, then A^{-1} exists but all entries are not necessarily integers.

(c) If det A ≠ ± 1, then A^{-1} exists and all its entries are non-integers

(d) If det A = ± 1, then A^{-1} exists and all its entries are integers.

## Answer

Answer: (d) If det A = ± 1, then A^{-1} exists and all its entries are integers.

Hint:

Since each entry of A is an integer,

∴ co-factor of each entry is also an integer.

Hence, each entry of the adjoint is an integer.

Also det A = ± 1 and A^{-1} = \(\frac { 1 }{det(A)}\) (adj A).

Hence, all entries of A^{-1} are integers.

Question 18.

The number of values of ‘k’ for which the linear equations:

4x + ky + 2z = 0

kx + 4y + z = 0

2x + 2y + z = 0

possesses a non-zero solution is

(a) 3

(b) 2

(c) 1

(d) zero.

## Answer

Answer: (b) 2

Hint:

The system possesses non-zero solution

If \(\left|\begin{array}{lll}

4 & k & 2 \\

k & 4 & 1 \\

2 & 2 & 1

\end{array}\right|\) = 0

If 4(4 – 2) + k (k – 2) + 2(2k – 8) = 0

if = 8 – k² + 2k + 4k – 16 = 0

if k² – 6k + 8 = 0

if (k – 2)(k – 4) = 0

if k = 2 or 4

k = 2.

Question 19.

If A = \(\left[\begin{array}{lll}

1 & \alpha & 3 \\

1 & 3 & 3 \\

2 & 4 & 4

\end{array}\right]\) is the adjoint of a 3 × 3 matrix A and |A| = 4 then α is equal to

(a) 11

(b) 5

(c) 0

(d) 4

## Answer

Answer: (a) 11

Hint:

Here |adj A| = |A|^{3-1}

= |A|² = 4²

= 16

⇒ 1. (12 – 12) -α (4 – 6)+ 3(4 – 6) = 16

⇒ 2α – 6 = 16

⇒ 2α = 22.

Hence, α = 11.

Question 20.

If α, ß ≠ 0 and f(x) = α” + ß” and

= k (1 – α)²(1 – ß)²(α – ß)², then ‘4k’ is equal to:

(a) \(\frac { 1 }{αß}\)

(b) 1

(c) -1

(d) αß

## Answer

Answer: (b) 1

Hint:

= [(α – 1) (ß² – 1) – (α² – 1) (ß – 1)]²

= (α – 1)²(ß – 1)²(α – ß)2.

Hence, k = 1

Question 21.

The system of linear equations:

x + λy – z = 0

λr – y – z = 0

x + y – λz = 0

has a non-trivial solution for

(a) Exactly one value of λ

(b) Exactly two values of λ

(c) Exactly three values of λ

(d) Infinitely many values of λ.

## Answer

Answer: (c) Exactly three values of λ

Hint:

The system AX = O has non-trivial solution if det A = 0

i.,e if \(\left|\begin{array}{rrr}

1 & \lambda & -1 \\

\lambda & -1 & -1 \\

1 & 1 & -\lambda

\end{array}\right|\) = 0

⇒ (1)(λ + 1) -λ(-λ² + 1) + (-1)(λ + 1) = 0

⇒ λ + 1 + λ³ – λ – λ – 1 = 0

⇒ λ³ – λ = 0

⇒ λ(λ² – 1) = 0

⇒ λ = 0, 1, -1.

Hence, λ = -1, 0, 1.

Question 22.

Let on be a complex number such that 2ω + 1 = z, where z = √-3

If \(\left|\begin{array}{ccc}

1 & 1 & 1 \\

1 & -\omega^{2}-1 & \omega^{2} \\

1 & \omega^{2} & \omega

\end{array}\right|\) = 3k the k is equal to

(a) -1

(b) 1

(c) z

(d) -z

## Answer

Answer: (c) z

Hint:

[Operating R_{1} → R_{1} + R_{2} + R_{3}]

= 3[- ω (ω² + 1) – ω^{4}]

= 3[- ω³ – ω – ω] = 3[- 1 – 2ω]

= – 3(1 + 2ω) = – 3z.

Thus 3k = – 3z.

Hence, k = -z.

Question 23.

If Sis the set of distinct values of ‘h’ for which the following system of linear equations:

x + y + z = 1,

x + ay + z = 1,

ax + by + z = 1

has no solution, then S is

(a) a finite set containing two or more elements

(b) a singleton

(c) an empty set

(d) an infinite set.

## Answer

Answer: (b) a singleton

Hint:

D = \(\left|\begin{array}{lll}

1 & 1 & 1 \\

1 & a & 1 \\

a & b & 1

\end{array}\right|\) = 0

⇒ a – 1

⇒ x + y + z = 1

and x + by + z = 0.

The planes are parallel

⇒ b = 1.

Hence, S is a singleton.

Question 24.

If A = \(\left[\begin{array}{rr}

2 & -3 \\

-4 & 1

\end{array}\right]\), then adj. (3A² + 12A) is equal to

## Answer

Answer:

Hint:

Question 25.

If the system of linear equations:

x + ky + 3z = 0

3x + ky – 2z = 0

2x + 4y – 3z = 0

has a non zero solution (x, y, z), then \(\frac {xz}{y^2}\) is equal to:

(a) -10

(b) 10

(c) -30

(d) 30

## Answer

Answer: (b) 10

Hint:

The given system has non-zero solution

⇒ \(\left|\begin{array}{ccc}

1 & k & 3 \\

3 & k & -2 \\

2 & 4 & -3

\end{array}\right|\)

⇒ 1(-3k + 8)-k (-9 + 4) + 3 (12 – 2k) = 0

⇒ 44 – 4k = 0

⇒ k = 11

Let z = λ

Thus x + 11 y = -3λ

and 3x + 11 y = 2λ

Fill in the blanks

Question 1.

If \(\left|\begin{array}{ll}

x & 2 \\

8 & x

\end{array}\right|\) = \(\left|\begin{array}{ll}

3 & 2 \\

9 & 6

\end{array}\right|\), then the value of x is ……………..

## Answer

Answer: ±4

Hint:

\(\left|\begin{array}{ll}

x & 2 \\

8 & x

\end{array}\right|\) = \(\left|\begin{array}{ll}

3 & 2 \\

9 & 6

\end{array}\right|\)

⇒ x² – 16 = 18 – 18

⇒ x² = 16

⇒ x = ±4.

Question 2.

Let A be a 3 x 3 determinant and |A| = 7. Then the value of |2A| is ……………….

## Answer

Answer: 56

Hint:

|2A| = 2³ |A| = 8 x 7 = 56.

Question 3.

If A = \(\left[\begin{array}{ll}

1 & 2 \\

4 & 2

\end{array}\right]\) Then the value of k = ……………

if |2A| = k|A|

## Answer

Answer: 4

Hint:

if |2A| = 2²|A| = 4|A|

k = 4

Question 4.

If A is a skew-symmetric matrix of order 3, then det A = ……………..

## Answer

Answer: 0.

Question 5.

The value of \(\left[\begin{array}{ccc}

102 & 18 & 36 \\

1 & 3 & 4 \\

17 & 3 & 6

\end{array}\right]\) is ……………..

## Answer

Answer: 0

Hint:

Δ = 6\(\left|\begin{array}{ccc}

17 & 3 & 6 \\

1 & 3 & 4 \\

17 & 3 & 6

\end{array}\right|\) = 6(0) = 0

Question 6.

If Δ = \(\left|\begin{array}{ll}

1 & a \\

1 & b

\end{array}\right|\), then minor of ‘b’ is ………………

## Answer

Answer: 1

Question 7.

Minor of ‘d’ is = \(\left|\begin{array}{ll}

a & c \\

b & d

\end{array}\right|\), is ………………

## Answer

Answer: a

Question 8.

A square matrix A has inverse if and only if A is ………………

## Answer

Answer: Invertible.

Question 9.

Co-factor of ‘b’ in \(\left|\begin{array}{ll}

a & c \\

b & d

\end{array}\right|\) is ……………

## Answer

Answer: -c

Question 10.

If Δ = \(\left|\begin{array}{lll}

1 & 2 & 3 \\

2 & 0 & 1 \\

5 & 3 & 8

\end{array}\right|\) then minor of a_{22} is …………….

## Answer

Answer: -7

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