Determinants

Previous Years' CBSE Board Questions

4.2 Determinant

MCQ

The value of the determinant 2 7 1 1 is 10 8 1 (a) 47 (b) -79 (c) 49 (d) -51 (2023)

1. If <math>\begin{vmatrix} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{vmatrix} = 0</math>, then the value of <math>\alpha</math> is
2. (b) 2 (c) 3 (d) 4

(2023)

1. If <math>A = \begin{bmatrix} \alpha & 2 \\ 2 & \alpha \end{bmatrix}</math> and <math>|A^3|</math> = 27, then the value of <math>\alpha</math> is
2. <math>\pm 1</math> (b) <math>\pm 2</math> (c) <math>\pm \sqrt{5}</math> (d) ±√7 (Term I, 2021-22) (II)

1. If <math>\begin{vmatrix} 5 & 3 & -1 \\ -7 & x & -3 \\ 9 & 6 & -2 \end{vmatrix} = 0</math>, then the value of x is
2. 3 (b) 5 (c) 7

1. The determinant <math>\begin{vmatrix} y+k & y & y \\ y & y+k & y \end{vmatrix}</math> is equal to<br> (a) <math>k(3y+k^2)</math> (b) <math>3y+k^3</math> (c) <math>3y + k^2</math> (d) <math>k^2(3y + k)</math> (Term I, 2021-22)

| 1 | 2 | 3 | The value of | 2 | 3 | 3 | 4 | 4 | is 3 4 5

1. 12 (b) -12 (c) 24 (d) -24 (Term I, 2021-22)

If A is a non-singular square matrix of order 3 such that <math>A^2 = 3A</math>, then value of <math>|A|</math> is

1. - 3 (b) 3
2. 9 (d) 27

X 0 8

1. The roots of the equation <math>|4 \ 1 \ 3|=0</math> are 2 0 x
2. (b) 2, -4 (c) 2, 4 (d) 2,8

(2020 C)

If A is a square matrix of order 3 and |A| = 5, then the value of |2A'| is

1. -10 (b) 10
2. -40 (d) 40 (2020) U

If A is a skew-symmetric matrix of order 3, then the value of |A| is

1. 3<br>(c) 9 27 (2020)

is a 3 × 3 matrix such that |A| = 8, then |3A| equals

1. 8 (b) 24

72 (d) 216 (2020)

(1 mark)

1. If <math>A = \begin{bmatrix} 3 & -5 \\ 2 & 0 \end{bmatrix}</math> and <math>B = \begin{bmatrix} 1 & 17 \\ 0 & -10 \end{bmatrix}</math>, then <math>|AB| = </math>_____. (2020 C)
2. If A and B are square matrices each of order 3 and |A| = 5, |B| = 3, then the value of |3AB| is _____. (2020)

If A and B are square matrices of the same order 3,

such that <math>|A| = 2</math> and <math>AB = 2I</math>, write the value of <math>|B|</math>.

(Delhi 2019) An

1. Find the maximum value of <math display="block"> \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+\sin\theta & 1 \\ 1 & 1 & 1+\cos\theta \end{vmatrix} . </math> (Delhi 2016)
2. 9<br>(Term I, 2021-22) (R) 16. If <math>x \in N</math> and <math>\begin{vmatrix} x+3 & -2 \\ -3x & 2x \end{vmatrix} = 8</math>, then find the value of x. (Al 2016)

<math>x \sin\theta \cos\theta</math><br> 17. If <math>-\sin\theta - x</math> 1 = 8, write the value of x.<br> <math>\cos\theta</math> 1 x (Foreign 20) (Foreign 2016) (An)

1. Write the value of <math>\Delta = \begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ -3 & -3 & -3 \end{vmatrix}</math>. (Al 2015)
2. If <math>A = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}</math> and <math>B = \begin{bmatrix} 1 & 3 \\ -1 & 1 \end{bmatrix}</math>, write the value of <math>|AB|</math>. (Delhi 2015C)
3. If <math>\begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 6 & -2 \\ 7 & 3 \end{vmatrix}</math>, write the value of x. (Delhi 2014)
4. If <math>\begin{vmatrix} 3x & 7 \\ -2 & 4 \end{vmatrix} = \begin{vmatrix} 8 & 7 \\ 6 & 4 \end{vmatrix}</math>, find the value of x. (Al 2014)

If A is a 3 × 3 matrix, |A| ≠ 0 and |3A| = k|A|, then write

the value of k. (Foreign 2014)

23. Write the value of the determinant <math>\begin{bmatrix} p & p+1 \\ p-1 & p \end{bmatrix}</math>.

(Delhi 2014C) 65

1. Write the value of 3 8 75 (AI 2014C) Ap 5 9 86

SAII (3 marks)

X <math>sin\theta</math> cos0l Show that the determinant -sin0 -x 1 is cos0 1 independent of <math>\theta</math>. (2023)

44 Minors and Cofactors

VSA. (1 mark)

1. Find the cofactors of all the elements of (2020) Ap

Find the cofactor of the element a₂₃ of the 5 3 8 determinant 2 0 1. (2019C) 1 2 3

1. If <math>A = \begin{bmatrix} 5 & 6 & -3 \\ -4 & 3 & 2 \\ -4 & -7 & 3 \end{bmatrix}</math>, then write the cofactor of the element a21 of its 2nd row. (Foreign 2015)

4.5 Adjoint and Inverse of a Matrix

MCQ

1. The inverse of <math>\begin{bmatrix} -4 & 3 \\ 7 & -5 \end{bmatrix}</math> is
2. | -5 3 | (b) | 5 3 | 7 4
3. [-5 7] (d) [-5 -3]<br>-7 -4 (Term I, 2021-22) (📭

1. If <math>A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 59 & 69 & -1 \end{bmatrix}</math>, then <math>A^{-1}</math>
2. is A (b) is (-A)
3. is A² (d) does not exist (Term I, 2021-22)

1. If A = <math display="block">\begin{bmatrix} 1 & -2 & 4 \\ 2 & -1 & 3 \\ 4 & 2 & 0 \end{bmatrix}</math> is the adjoint of a square matrix B, then B-1 is equal to
2. <math>\pm A</math> (b) <math>\pm \sqrt{2}A</math> (c) <math>\pm \frac{1}{\sqrt{2}}B</math> (d) <math>\pm \frac{1}{\sqrt{2}}A</math> (Term I, 2021-22) [7]

1. If <math>\begin{bmatrix} 1 & -\tan\theta \\ \tan\theta & 1 \end{bmatrix} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}</math>, then
2. <math>a = 1 = b</math> (b) a = cos 2θ, b = sin 2θ
3. <math>a = \sin 2\theta</math>, <math>b = \cos \theta</math> (d) <math>a = \cos \theta</math>, <math>b = \sin \theta</math> (Term I, 2021-22)

1. If <math>A = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{bmatrix}</math>, then the value of |adj A| is (a) 64 (b) 16 (c) 0 (d) (2020)

34. If A is a square matrix of order 3, such that A (adj A) =

10 I, then |adj A| is equal to

1. 1 (b) 10 (c) 100 (d) 101 (2020)

(1 mark)

1. If A is a square matrix of order 3 such that A(adj A) = 0 -2 0 , then find |A|.<br>0 0 -2 (2021 C)
2. If <math>A = \begin{bmatrix} 2 & 0 & 0 \\ -1 & 2 & 3 \\ 3 & 3 & 5 \end{bmatrix}</math>, then find A (adj A). (2020) Ap
3. If A is a square matrix of order 3 with |A| = 9, then write the value of |2 · adj A|. (Al 2019)

If A is a 3 × 3 invertible matrix, then what will be the

value of k if <math>det(A^{-1}) = (det A)^k</math>? (Delhi 2017) R

If for any 2 × 2 square matrix A, <math>A(\text{adj }A) = \begin{bmatrix} 8 & 0 \\ 0 & 8 \end{bmatrix}</math>, then write the value of <math>|A|</math>. (AI 2017)

1. In the interval <math>\pi/2 < x < \pi</math>, find the value of x for which the matrix <math>\begin{bmatrix} 2\sin x & 3 \\ 1 & 2\sin x \end{bmatrix}</math> is singular. (AI 2015C)
2. Find (adj A), if <math>A = \begin{bmatrix} 5 & 2 \\ 7 & 3 \end{bmatrix}</math>. (Delhi 2014C) (Ev)

(2 marks)

1. For the matrix <math>A = \begin{bmatrix} 2 & 3 \\ -4 & -6 \end{bmatrix}</math>, verify the following: <math>A (adi A) = (adi A) A = |A|I</math> (2020 C)
2. Find <math>(AB)^{-1}</math> if <math>A = \begin{bmatrix} 1 & 0 \\ -4 & 2 \end{bmatrix}</math> and <math>B^{-1} = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}</math>. (2020) Ap
3. Given <math>A = \begin{bmatrix} 2 & -3 \\ -4 & 7 \end{bmatrix}</math>, compute <math>I^{-1}</math> and show that <math>2A^{-1} = 9I - A</math>. (2018) Cr

(4 marks)

45. Gautam buys 5 pens, 3 bags and 1 instrument box

and pays a sum of ₹160. From the same shop Vikram

buys 2 pens, 1 bag and 3 instrument boxes and pays

a sum of ₹190. Also Ankur buys 1 pen, 2 bags and 4

instrument boxes and pays a sum of ₹250.