Matrices and Determinants jee mains questions | JEE Main Maths Matrices and Determinants Previous Year Question

Q 11 :

Let $A = [\begin{matrix} 200 & 50 \\ 10 & 2 \end{matrix}]$ and $B = [\begin{matrix} 50 & 40 \\ 2 & 3 \end{matrix}],$ then $| A B |$ is equal to:

460
2000
3000
- 7000

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(4)

Ans. -7000

Explanation:

$| A B |$ $=$ $| A |$ $| B |$ $= (400 - 500) (150 - 80)$

$= (- 100) (70) = - 7000$

Q 12 :

If A is a square matrix of order 3, such that $A (adj A) = 10 I,$ then $| adj A |$ is equal to:

1
10
100
101

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(3)

Ans. 100

Explanation:

Given: $A (adj A) = 10 I$

We know that, $A . adj A = | A | . I$

$∴$ $| A | = 10$

$∵$ $| adj A | = | A |^{n - 1}$

$∴$ $| adj A |$ $= | A |^{3 - 1} = 10^{2} = 100$

Q 13 :

If A is a $3 \times 3$ matrix such that $| A |$ $= 8$ , then $| 3 A |$ equals:

8
24
72
216

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(4)

Ans. 216

Explanation:

Given: $| A |$ $= 8$

We know that, $| k A |$ $= k^{n}$ $| A |$ $,$ where $k$ is a constant

$∴$ $| 3 A |$ $= 3^{3}$ $| A |$ $= 27 \times 8 = 216$

Q 14 :

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

3
0
9
27

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(2)

Ans. 0

Explanation:

Since, A is a skew-symmetric matrix,

$∴$ $A^{T} = - A$

$\Rightarrow$ $| A^{T} |$ $=$ $| - A |$

Since $| A^{T} |$ $=$ $| A |$ and for a $3 \times 3$ matrix,

$| - A |$ $= {(- 1)}^{3}$ $| A |$ $,$

we get $| A |$ $= {(- 1)}^{3}$ $| A |$

Hence, $| A |$ $= -$ $| A |$ $,$

$\Rightarrow$ $2$ $| A |$ $= 0$

$\Rightarrow$ $| A |$ $= 0$

Q 15 :

If $| \begin{matrix} x & 2 \\ 18 & x \end{matrix} |$ $=$ $| \begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix} |$ $,$ then the value of $x$ is:

$\pm 2$
$0$
$\pm 3$
$\pm 6$

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(4)

Ans. $\pm 6$

Explanation:

$∵$ $| \begin{matrix} x & 2 \\ 18 & x \end{matrix} |$ $=$ $| \begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix} |$

$\Rightarrow$ $x^{2} - 36 = 36 - 36$

$\Rightarrow$ $x^{2} - 36 = 0$

$\Rightarrow$ $x = \pm 6$

Q 16 :

The determinant $| \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} |$ is:

Independent of $θ$ only
Independent of $x$ only
Independent of both $θ$ and $x$
None of the above

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(1)

Ans. Independent of $θ$ only

Explanation:

Let $Δ =$ $| \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} |$

$= x (- x^{2} - 1) - \sin θ (- x \sin θ - \cos θ) + \cos θ (- \sin θ + x \cos θ)$

$= - x^{3} - x + x \sin^{2} θ + \sin θ \cos θ - \sin θ \cos θ + x \cos^{2} θ$

$= - x^{3} - x + x (\sin^{2} θ + \cos^{2} θ)$

$= - x^{3} - x + x$ $[∵ \sin^{2} θ + \cos^{2} θ = 1]$

$= - x^{3},$ which is independent of $θ$ .

Q 17 :

If $M_{11} = - 40$ , $M_{12} = - 10$ and $M_{13} = 35$ of the determinant $Δ =$ $| \begin{matrix} 1 & 3 & - 2 \\ 4 & - 5 & 6 \\ 3 & 5 & 2 \end{matrix} |$ $,$ then the value of $Δ$ is:

$- 80$
$60$
$70$
$100$

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(1)

Ans. $- 80$

Explanation:

$Δ = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13}$

$= a_{11} M_{11} - a_{12} M_{12} + a_{13} M_{13}$

$= 1 (- 40) - 3 (- 10) + (- 2) (35)$

$= - 40 + 30 - 70$ $= - 80$

Q 18 :

Let $A$ be a square matrix of order $3 \times 3$ , then $| k A |$ is equal to:

$k$ $| A |$
$k^{2}$ $| A |$
$k^{3}$ $| A |$
$3 k$ $| A |$

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(3)

Ans. $k^{3}$ $| A |$

Explanation:

We know that for $n \times n$ matrix,

$| (λ A) |$ $= λ^{n} | A |$

Hence, $| k A |$ $= k^{3}$ $| A |$

Q 19 :

Which of the following is correct?

Determinant is a square matrix
Determinant is a number associated to a matrix
Determinant is a number associated to a square matrix
None of the above

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(3)

Ans. Determinant is a number associated to a square matrix.

Explanation:

We know that to every square matrix, $A = [a_{i j}]$ of order $n$ , we can associate a number called the determinant of the square matrix A, where $a_{i j}$ $= {(i, j)}^{th}$ element of A. Thus, the determinant is a number associated to a square matrix.

Q 20 :

Value of $| \begin{matrix} \cos 15 ° & \sin 15 ° \\ \sin 15 ° & \cos 15 ° \end{matrix} |$ is:

$1$
$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
None of these

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(3)

Ans. $\frac{\sqrt{3}}{2}$

Explanation:

$Δ =$ $| \begin{matrix} \cos 15 ° & \sin 15 ° \\ \sin 15 ° & \cos 15 ° \end{matrix} |$

$= \cos^{2} 15 ° - \sin^{2} 15 °$

$= \cos (2 \times 15 °) = \cos 30 °$

$= \frac{\sqrt{3}}{2}$

Q 21 :

If A is a square matrix $x,$ such that $A^{2} = I,$ then $A^{- 1}$ is equal to:

2A
0
A
A + 1

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(3)

Ans. A

Explanation:

$A^{2} = I$

then $A \cdot A = I$

or $A^{- 1} (A \cdot A) = A^{- 1} I$

$\Rightarrow$ $(A^{- 1} A) A = A^{- 1}$

$\Rightarrow$ $I A = A^{- 1}$

$\Rightarrow$ $A^{- 1} = A$

Q 22 :

If $a$ , $b$ and $c$ are all different from zero and $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0,$ then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is:

$0$
$a b c$
$- 1$
$\frac{1}{a b c}$

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Ans. $- 1$

Explanation:

Given, $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0$

Expanding the determinant,

$(1 + a) [(1 + b) (1 + c) - 1] - 1 (1 + c - b) + 1 (1 - 1 - b) = 0$

$(1 + a) (b + c + b c) - c - b = 0$

$b + c + b c + a b + a c + a b c - b - c = 0$

$a b + b c + c a = - a b c$

Dividing both sides by $a b c$ ,

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = - 1$

Q 23 :

Let $A$ be a $3 \times 3$ matrix of non-negative real numbers such that $A$ $[\begin{matrix} 2 \\ 2 \\ 2 \end{matrix}]$ $= 4$ $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ then ${(\det A)}_{\max}$ is

12
8
16
32

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(2)

$a_{1} + a_{2} + a_{3} = 2,$ $b_{1} + b_{2} + b_{3} = 2,$ $c_{1} + c_{2} + c_{3} = 2$

$(\det A) is maximum when a_{1} = 2,$ $b_{2} = 2,$ $c_{3} = 2 and all others are 0 .$

Q 24 :

$If a, b, c are real numbers satisfying$ $| \begin{matrix} {(a^{2} + 1)}^{2} & {(a b + 1)}^{2} & {(a c + 1)}^{2} \\ {(a b + 1)}^{2} & {(b^{2} + 1)}^{2} & {(b c + 1)}^{2} \\ {(a c + 1)}^{2} & {(b c + 1)}^{2} & {(c^{2} + 1)}^{2} \end{matrix} |$ $= k {(a - b)}^{2} {(b - c)}^{2} {(c - a)}^{2} then k =$

0
1
2
4

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(3)

$| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} |$ $| \begin{matrix} 1 & 1 & 1 \\ 2 a & 2 b & 2 c \\ a^{2} & b^{2} & c^{2} \end{matrix} |$

Q 25 :

Let $S = {A = [\begin{matrix} a & b \\ c & d \end{matrix}] : a, b, c, d \in {0, 1, 2, 3, 4} and A^{2} - 4 A + 3 I = 0}$ be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

20
17
21
19

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(4)

We have, Tr = (A) = 4

$\Rightarrow$ a + d = 4 ... (i)

and |A| = 3

[ $∵ A^{2} - 4 A + 3 I = 0$ is characteristic equation of matrix A]

$\Rightarrow$ ad – bc = 3

Case 1: If a = 0, d = 4, then bc = –3 (Reject)

Case 2: If a = 1, d = 3, then 3 – bc = 3

$\Rightarrow$ bc = 0

If b = 0, then 'c' has 5 choices and if c = 0, 'b' has 4 choices, so total 9 cases.

Case 3: If a = 2, d = 2, then 4 – bc = 3

$\Rightarrow$ bc = 1 $\Rightarrow$ b = 1, c = 1 (only 1 case)

Case 4: If a = 3, d = 1, then 3 – bc = 3

$\Rightarrow$ bc = 0 $\Rightarrow$ 9 cases

Case 5: If a = 4, d = 0, then bc = -3 (Reject)

$∴$ Total 19 cases.

Q 26 :

Let $A = [\begin{matrix} 1 & 2 & 7 \\ 4 & - 2 & 8 \\ 3 & 8 & - 7 \end{matrix}]$ and $\det (A - α I) = 0$ , where $α$ is a real number. If the largest possible value of $α$ is p, then the circle ${(x - p)}^{2} + {(y - 2 p)}^{2} = 320$ , intersects the co-ordinate axes at [2026]

1 point
2 points
3 points
4 points

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4

(3)

$A = [\begin{matrix} 1 & 2 & 7 \\ 4 & - 2 & 8 \\ 3 & 8 & - 7 \end{matrix}]$

Now, $| A - α I | = 0$

$\Rightarrow [\begin{matrix} 1 - α & 2 & 7 \\ 4 & - 2 - α & 8 \\ 3 & 8 & - 7 - α \end{matrix}] = 0$

$\Rightarrow (1 - α) {α^{2} + 9 α + 14 - 64} - 2 {- 4 α - 28 - 24} + 7 {32 + 6 + 3 α} = 0$

$\Rightarrow α^{2} + 9 α - 50 - α^{3} - 9 α^{2} + 50 α + 8 α + 104 + 266 + 21 α = 0$

$\Rightarrow α^{3} + 8 α^{2} - 88 α - 320 = 0$

$\Rightarrow α^{2} (α - 8) + 16 α (α - 8) + 40 (α - 8) = 0$

$\Rightarrow (α - 8) + (α^{2} + 16 α + 40) = 0$

$\Rightarrow α = 8, α = \frac{- 16 \pm \sqrt{96}}{2}$

$\Rightarrow α = - 8 \pm \sqrt{24}$

Largest possible value of $α$ is 8.

$∴ p = 8$

$\Rightarrow$ ${(x - 8)}^{2} = {(y - 16)}^{2} = 320$

For y = 0

${(x - 8)}^{2} = 320 - 256 = 64$

$\Rightarrow x - 8 = \pm 8 \Rightarrow x = 16, 0$

For x = 0

${(y - 16)}^{2} = 320 - 64$

$\Rightarrow {(y - 16)}^{2} = 256 \Rightarrow y - 16 = \pm 16$

y = 32, 0

So, points are (0, 32), (0, 0), (16, 0).

Q 27 :

If $f : N \to Z$ is defined by $f (n)$ $=$ $| \begin{matrix} n & - 1 & - 5 \\ - 2 n^{2} & 3 (2 k + 1) & 2 k + 1 \\ - 3 n^{3} & 3 k (2 k + 1) & 3 k (k + 2) + 1 \end{matrix} |$ $, k \in N$ and $\sum_{n = 1}^{k} f (n) = 98$ , then k is equal to : [2026]

3
4
5
6

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2

3

4

(1)

We have,

$\sum_{} f (n) =$ $| \begin{matrix} \sum_{} n & - 1 & - 5 \\ - 2 \sum_{} n^{2} & 3 (2 k + 1) & 2 k + 1 \\ - 3 \sum_{} n^{3} & 3 k (2 k + 1) & 3 k (k + 2) + 1 \end{matrix} |$ $, k \in N$

Now, $\sum_{} n = \frac{k (k + 1)}{2}, \sum_{} n^{2} = \frac{k (k + 1) (2 k + 1)}{6}$

and $\sum_{} n^{3} = \frac{k^{2} {(k + 1)}^{2}}{4}$

$\sum_{n = 1}^{k} f (n) =$ $| \begin{matrix} \frac{k (k + 1)}{2} & - 1 & - 5 \\ - 2 \cdot \frac{k \cdot (k + 1) (2 k + 1)}{6} & 3 (2 k + 1) & 2 k + 1 \\ - 3 \cdot \frac{k^{2} {(k + 1)}^{2}}{4} & 3 k (2 k + 1) & 3 k (k + 2) + 1 \end{matrix} |$

$\Rightarrow \frac{7 k (k + 1) (2 k + 1)}{6} = 98$

$\Rightarrow k (k + 1) (2 k + 1) = 84$

$∴ k = 3$

Q 28 :

Match the following columns:

Column I (A,B,C are matrices) Column-II

a. If $| A_{3 \times 3} |$ $= 2$ , then $| 2 A^{- 1} |$ $=$ p. 1

b. If $| A_{3 \times 3} |$ $= \frac{1}{8}$ , then $| a d j (a d j (2 A)) |$ $=$ q. 4

c. If ${(A + B)}^{2} = A^{2} + B^{2}$ , and $| A_{2 \times 2} |$ $= 2$ , then $| B_{2 \times 2} |$ $=$ r. 8

d. $| A_{2 \times 2} |$ $= 2,$ $| B_{3 \times 3} |$ $= 3$ and $| C_{4 \times 4} |$ $= 4$ , then $| A B C |$ is equal to s. 0

t. Does not exist

a-q; b-p; c-s; d-t
a-p; b-r; c-q; d-s
a-p; b-q; c-r; d-s
a-s; b-q; c-r; d-p

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(1)

a. $| A |$ $= 2 \Rightarrow$ $| 2 A^{- 1} |$ $= 2^{3} /$ $| A |$ $= 4$

b. $| a d j (a d j (2 A)) |$ $=$ $| 2 A |^{4}$ $= 2^{12}$ $| A |^{4}$ $= 2^{12} / 2^{12} = 1$

c. ${(A + B)}^{2} = A^{2} + B^{2} \Rightarrow A B + B A = O$

$\Rightarrow$ $| A B |$ $=$ $| - B A |$ $= -$ $| B A |$ $= -$ $| A B |$ $\Rightarrow$ $| A B |$ $= 0 \Rightarrow$ $| B |$ $= 0$

d. Product ABC is not defined.

Topic Question Set

Q 11 :

Let $A = [\begin{matrix} 200 & 50 \\ 10 & 2 \end{matrix}]$ and $B = [\begin{matrix} 50 & 40 \\ 2 & 3 \end{matrix}],$ then $| A B |$ is equal to:

Q 12 :

If A is a square matrix of order 3, such that $A (adj A) = 10 I,$ then $| adj A |$ is equal to:

Q 13 :

If A is a $3 \times 3$ matrix such that $| A |$ $= 8$ , then $| 3 A |$ equals:

Q 14 :

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

Q 15 :

If $| \begin{matrix} x & 2 \\ 18 & x \end{matrix} |$ $=$ $| \begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix} |$ $,$ then the value of $x$ is:

Q 16 :

The determinant $| \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} |$ is:

Q 17 :

If $M_{11} = - 40$ , $M_{12} = - 10$ and $M_{13} = 35$ of the determinant $Δ =$ $| \begin{matrix} 1 & 3 & - 2 \\ 4 & - 5 & 6 \\ 3 & 5 & 2 \end{matrix} |$ $,$ then the value of $Δ$ is:

Q 18 :

Let $A$ be a square matrix of order $3 \times 3$ , then $| k A |$ is equal to:

Q 19 :

Which of the following is correct?

Q 20 :

Value of $| \begin{matrix} \cos 15 ° & \sin 15 ° \\ \sin 15 ° & \cos 15 ° \end{matrix} |$ is:

Q 21 :

If A is a square matrix $x,$ such that $A^{2} = I,$ then $A^{- 1}$ is equal to:

Q 22 :

If $a$ , $b$ and $c$ are all different from zero and $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0,$ then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is:

Q 23 :

Let $A$ be a $3 \times 3$ matrix of non-negative real numbers such that $A$ $[\begin{matrix} 2 \\ 2 \\ 2 \end{matrix}]$ $= 4$ $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ then ${(\det A)}_{\max}$ is

Q 25 :

Let $S = {A = [\begin{matrix} a & b \\ c & d \end{matrix}] : a, b, c, d \in {0, 1, 2, 3, 4} and A^{2} - 4 A + 3 I = 0}$ be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

Q 26 :

Let $A = [\begin{matrix} 1 & 2 & 7 \\ 4 & - 2 & 8 \\ 3 & 8 & - 7 \end{matrix}]$ and $\det (A - α I) = 0$ , where $α$ is a real number. If the largest possible value of $α$ is p, then the circle ${(x - p)}^{2} + {(y - 2 p)}^{2} = 320$ , intersects the co-ordinate axes at [2026]

Q 27 :

If $f : N \to Z$ is defined by $f (n)$ $=$ $| \begin{matrix} n & - 1 & - 5 \\ - 2 n^{2} & 3 (2 k + 1) & 2 k + 1 \\ - 3 n^{3} & 3 k (2 k + 1) & 3 k (k + 2) + 1 \end{matrix} |$ $, k \in N$ and $\sum_{n = 1}^{k} f (n) = 98$ , then k is equal to : [2026]

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	Column I (A,B,C are matrices)		Column-II
a.	If $\| A_{3 \times 3} \|$ $= 2$ , then $\| 2 A^{- 1} \|$ $=$	p.	1
b.	If $\| A_{3 \times 3} \|$ $= \frac{1}{8}$ , then $\| a d j (a d j (2 A)) \|$ $=$	q.	4
c.	If ${(A + B)}^{2} = A^{2} + B^{2}$ , and $\| A_{2 \times 2} \|$ $= 2$ , then $\| B_{2 \times 2} \|$ $=$	r.	8
d.	$\| A_{2 \times 2} \|$ $= 2,$ $\| B_{3 \times 3} \|$ $= 3$ and $\| C_{4 \times 4} \|$ $= 4$ , then $\| A B C \|$ is equal to	s.	0
		t.	Does not exist

Topic Question Set

Q 11 : Let A=[20050102] and B=[504023], then |AB| is equal to:

Q 12 : If A is a square matrix of order 3, such that A(adjA)=10I, then |adjA| is equal to:

Q 13 : If A is a 3×3 matrix such that |A|=8, then |3A| equals:

Q 14 : If A is a skew-symmetric matrix of order 3, then the value of |A| is:

Q 15 : If |x218x|=|62186|, then the value of x is:

Q 16 : The determinant |xsinθcosθ-sinθ-x1cosθ1x| is:

Q 17 : If M11=-40, M12=-10 and M13=35 of the determinant Δ=|13-24-56352|, then the value of Δ is:

Q 18 : Let A be a square matrix of order 3×3, then |kA| is equal to:

Q 19 : Which of the following is correct?

Q 20 : Value of |cos15°sin15°sin15°cos15°| is:

Q 21 : If A is a square matrix x, such that A2=I, then A-1 is equal to:

Q 22 : If a, b and c are all different from zero and |1+a1111+b1111+c|=0, then the value of 1a+1b+1c is:

Q 23 : Let A be a 3×3 matrix of non-negative real numbers such that A[222]=4[111] then (detA)max is

Q 24 : If a,b,c are real numbers satisfying |(a2+1)2(ab+1)2(ac+1)2(ab+1)2(b2+1)2(bc+1)2(ac+1)2(bc+1)2(c2+1)2|=k(a-b)2(b-c)2(c-a)2 then k=

Q 25 : Let S={A=[abcd]:a,b,c,d∈{0,1,2,3,4} and A2–4A+3I=0} be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

Q 26 : Let A=[1274–2838–7] and det(A–αI)=0, where α is a real number. If the largest possible value of α is p, then the circle (x–p)2+(y–2p)2=320, intersects the co-ordinate axes at [2026]

Q 27 : If f:N→Z is defined by f(n)=|n–1–5–2n23(2k+1)2k+1–3n33k(2k+1)3k(k+2)+1|, k∈N and ∑n=1kf(n)=98, then k is equal to : [2026]

Q 28 : Match the following columns: Column I (A,B,C are matrices) Column-II a. If |A3×3|=2, then |2A-1|= p. 1 b. If |A3×3|=18, then |adj(adj(2A))|= q. 4 c. If (A+B)2=A2+B2, and |A2×2|=2, then |B2×2|= r. 8 d. |A2×2|=2,|B3×3|=3 and |C4×4|=4, then |ABC| is equal to s. 0 t. Does not exist

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Q 11 :

Let $A = [\begin{matrix} 200 & 50 \\ 10 & 2 \end{matrix}]$ and $B = [\begin{matrix} 50 & 40 \\ 2 & 3 \end{matrix}],$ then $| A B |$ is equal to:

Q 12 :

If A is a square matrix of order 3, such that $A (adj A) = 10 I,$ then $| adj A |$ is equal to:

Q 13 :

If A is a $3 \times 3$ matrix such that $| A |$ $= 8$ , then $| 3 A |$ equals:

Q 14 :

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

Q 15 :

If $| \begin{matrix} x & 2 \\ 18 & x \end{matrix} |$ $=$ $| \begin{matrix} 6 & 2 \\ 18 & 6 \end{matrix} |$ $,$ then the value of $x$ is:

Q 16 :

The determinant $| \begin{matrix} x & \sin θ & \cos θ \\ - \sin θ & - x & 1 \\ \cos θ & 1 & x \end{matrix} |$ is:

Q 17 :

If $M_{11} = - 40$ , $M_{12} = - 10$ and $M_{13} = 35$ of the determinant $Δ =$ $| \begin{matrix} 1 & 3 & - 2 \\ 4 & - 5 & 6 \\ 3 & 5 & 2 \end{matrix} |$ $,$ then the value of $Δ$ is:

Q 18 :

Let $A$ be a square matrix of order $3 \times 3$ , then $| k A |$ is equal to:

Q 19 :

Which of the following is correct?

Q 20 :

Value of $| \begin{matrix} \cos 15 ° & \sin 15 ° \\ \sin 15 ° & \cos 15 ° \end{matrix} |$ is:

Q 21 :

If A is a square matrix $x,$ such that $A^{2} = I,$ then $A^{- 1}$ is equal to:

Q 22 :

If $a$ , $b$ and $c$ are all different from zero and $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0,$ then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is:

Q 23 :

Let $A$ be a $3 \times 3$ matrix of non-negative real numbers such that $A$ $[\begin{matrix} 2 \\ 2 \\ 2 \end{matrix}]$ $= 4$ $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ then ${(\det A)}_{\max}$ is

Q 25 :

Let $S = {A = [\begin{matrix} a & b \\ c & d \end{matrix}] : a, b, c, d \in {0, 1, 2, 3, 4} and A^{2} - 4 A + 3 I = 0}$ be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

Q 26 :

Let $A = [\begin{matrix} 1 & 2 & 7 \\ 4 & - 2 & 8 \\ 3 & 8 & - 7 \end{matrix}]$ and $\det (A - α I) = 0$ , where $α$ is a real number. If the largest possible value of $α$ is p, then the circle ${(x - p)}^{2} + {(y - 2 p)}^{2} = 320$ , intersects the co-ordinate axes at [2026]

Q 27 :

If $f : N \to Z$ is defined by $f (n)$ $=$ $| \begin{matrix} n & - 1 & - 5 \\ - 2 n^{2} & 3 (2 k + 1) & 2 k + 1 \\ - 3 n^{3} & 3 k (2 k + 1) & 3 k (k + 2) + 1 \end{matrix} |$ $, k \in N$ and $\sum_{n = 1}^{k} f (n) = 98$ , then k is equal to : [2026]