Let $A = \left\{ X = ( x , y , z ) ^ { T } : P X = 0 \text{ and } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 \right\}$ where $P = \left[ \begin{array} { c c c } 1 & 2 & 1 \\ - 2 & 3 & - 4 \\ 1 & 9 & - 1 \end{array} \right]$ then the set $A$
(1) Is a singleton.
(2) Is an empty set.
(3) Contains more than two elements
(4) Contains exactly two elements

Suppose the vectors $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ are the solutions of the system of linear equations, $A x = b$ when the vector $b$ on the right side is equal to $b _ { 1 } , b _ { 2 }$ and $b _ { 3 }$ respectively. If $x _ { 1 } = \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] , x _ { 2 } = \left[ \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right] , x _ { 3 } = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] ; b _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] , b _ { 2 } = \left[ \begin{array} { l } 0 \\ 2 \\ 0 \end{array} \right] , b _ { 3 } = \left[ \begin{array} { l } 0 \\ 0 \\ 2 \end{array} \right]$, then the determinant of $A$ is equal to
(1) 4
(2) 2
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$

Let $a , b , c \in R$ be all non-zero and satisfies $a ^ { 3 } + b ^ { 3 } + c ^ { 3 } = 2$. If the matrix $A = \left[ \begin{array} { c c c } a & b & c \\ b & c & a \\ c & a & b \end{array} \right]$ satisfies $A ^ { T } A = I$, then a value of $a b c$ can be
(1) $- \frac { 1 } { 3 }$
(2) $\frac { 1 } { 3 }$
(3) 3
(4) $\frac { 2 } { 3 }$

If $a + x = b + y = c + z + 1$, where $a, b, c, x, y, z$ are non-zero distinct real numbers, then $\left|\begin{array}{lll} x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c \end{array}\right|$ is equal to:
(1) $y(b-a)$
(2) $y(a-b)$
(3) $0$
(4) $y(a-c)$

The number of distinct real roots of $\left| \begin{array} { c c c } \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{array} \right| = 0$ in the interval $- \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 4 }$ is:
(1) 4
(2) 1
(3) 2
(4) 3

The maximum value of $f ( x ) = \left| \begin{array} { c c c } \sin ^ { 2 } x & 1 + \cos ^ { 2 } x & \cos 2x \\ 1 + \sin ^ { 2 } x & \cos ^ { 2 } x & \cos 2x \\ \sin ^ { 2 } x & \cos ^ { 2 } x & \sin 2x \end{array} \right| , x \in R$ is
(1) $\sqrt { 7 }$
(2) $\frac { 3 } { 4 }$
(3) $\sqrt { 5 }$
(4) 5

If for the matrix, $A = \left[ \begin{array} { c c } 1 & - \alpha \\ \alpha & \beta \end{array} \right] , A A ^ { T } = I _ { 2 }$, then the value of $\alpha ^ { 4 } + \beta ^ { 4 }$ is:
(1) 3
(2) 1
(3) 2
(4) 4

Let $A = \left[ \begin{array} { l l } 2 & 3 \\ a & 0 \end{array} \right] , a \in R$ be written as $P + Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname { det } ( Q ) = 9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:
(1) 36
(2) 24
(3) 45
(4) 18

Let $A$ be a $3 \times 3$ matrix with $\operatorname { det } ( A ) = 4$. Let $R _ { i }$ denote the $i ^ { \text {th} }$ row of $A$. If a matrix $B$ is obtained by performing the operation $R _ { 2 } \rightarrow 2 R _ { 2 } + 5 R _ { 3 }$ on $2 A$, then $\operatorname { det } ( B )$ is equal to:
(1) 64
(2) 16
(3) 128
(4) 80

The value of $k \in R$, for which the following system of linear equations $3 x - y + 4 z = 3$ $x + 2 y - 3 z = - 2$ $6 x + 5 y + k z = - 3$ has infinitely many solutions, is:
(1) 3
(2) - 5
(3) 5
(4) - 3

Let $A = \left[ \begin{array} { c c } i & - i \\ - i & i \end{array} \right] , i = \sqrt { - 1 }$. Then, the system of linear equations $A ^ { 8 } \left[ \begin{array} { l } x \\ y \end{array} \right] = \left[ \begin{array} { c } 8 \\ 64 \end{array} \right]$ has :
(1) A unique solution
(2) Infinitely many solutions
(3) No solution
(4) Exactly two solutions

If $A = \left[ \begin{array} { c c } 0 & \sin \alpha \\ \sin \alpha & 0 \end{array} \right]$ and $\operatorname { det } \left( A ^ { 2 } - \frac { 1 } { 2 } \mathrm { I } \right) = 0$, then a possible value of $\alpha$ is
(1) $\frac { \pi } { 2 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 6 }$

If $P = \left[ \begin{array} { c c } 1 & 0 \\ \frac { 1 } { 2 } & 1 \end{array} \right]$, then $P ^ { 50 }$ is:
(1) $\left[ \begin{array} { l l } 1 & 0 \\ 25 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { l l } 1 & 50 \\ 0 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { l l } 1 & 25 \\ 0 & 1 \end{array} \right]$
(4) $\left[ \begin{array} { l l } 1 & 0 \\ 50 & 1 \end{array} \right]$

If $A = \left[ \begin{array} { c c } 0 & - \tan \left( \frac { \theta } { 2 } \right) \\ \tan \left( \frac { \theta } { 2 } \right) & 0 \end{array} \right]$ and $\left( I _ { 2 } + A \right) \left( I _ { 2 } - A \right) ^ { - 1 } = \left[ \begin{array} { c c } a & - b \\ b & a \end{array} \right]$, then $13 \left( a ^ { 2 } + b ^ { 2 } \right)$ is equal to

Let $P = \left[ \begin{array} { c c c } - 30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{array} \right]$ and $A = \left[ \begin{array} { c c c } 2 & 7 & \omega ^ { 2 } \\ - 1 & - \omega & 1 \\ 0 & - \omega & - \omega + 1 \end{array} \right]$ where $\omega = \frac { - 1 + i \sqrt { 3 } } { 2 }$, and $I _ { 3 }$ be the identity matrix of order 3 . If the determinant of the matrix $\left( P ^ { - 1 } A P - I _ { 3 } \right) ^ { 2 }$ is $\alpha \omega ^ { 2 }$, then the value of $\alpha$ is equal to $\_\_\_\_$.

Let $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ and $B = \left[ \begin{array} { l } \alpha \\ \beta \end{array} \right] \neq \left[ \begin{array} { l } 0 \\ 0 \end{array} \right]$ such that $A B = B$ and $a + d = 2021$, then the value of $a d - b c$ is equal to $\_\_\_\_$.

The total number of $3 \times 3$ matrices $A$ having entries from the set $\{ 0,1,2,3 \}$ such that the sum of all the diagonal entries of $A A ^ { T }$ is 9 , is equal to $\_\_\_\_$.

Let $\theta = \frac { \pi } { 5 }$ and $A = \begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{bmatrix}$. If $B = A + A ^ { 4 }$, then $\det ( B )$

Let $A$ be a $2 \times 2$ matrix with $\operatorname { det } ( A ) = - 1$ and $\operatorname { det } ( ( A + I ) ( \operatorname { Adj } ( A ) + I ) ) = 4$. Then the sum of the diagonal elements of $A$ can be:
(1) $- 1$
(2) 2
(3) 1
(4) $- \sqrt { 2 }$

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?
(1) $A ^ { 4 } - B ^ { 4 }$ is a symmetric matrix
(2) $A B - B A$ is a symmetric matrix
(3) $B ^ { 5 } - A ^ { 5 }$ is a skew-symmetric matrix
(4) $A B + B A$ is a skew-symmetric matrix

Let $A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$. If $M$ and $N$ are two matrices given by $M = \sum _ { k = 1 } ^ { 10 } A ^ { 2k }$ and $N = \sum _ { k = 1 } ^ { 10 } A ^ { 2k - 1 }$ then $MN^{2}$ is
(1) a non-identity symmetric matrix
(2) a skew-symmetric matrix
(3) neither symmetric nor skew-symmetric matrix
(4) an identity matrix

Which of the following matrices can NOT be obtained from the matrix $\begin{pmatrix} -1 & 2 \\ 1 & -1 \end{pmatrix}$ by a single elementary row operation?
(1) $\begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}$
(2) $\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$
(3) $\begin{pmatrix} -1 & 2 \\ -2 & 7 \end{pmatrix}$
(4) $\begin{pmatrix} -1 & 2 \\ -1 & 3 \end{pmatrix}$

Let $A$ be a $3 \times 3$ invertible matrix. If $| \operatorname { adj } ( 24 A ) | = | \operatorname { adj } ( 3 \operatorname { adj } ( 2A ) ) |$, then $| A | ^ { 2 }$ is equal to
(1) $2 ^ { 6 }$
(2) $2 ^ { 12 }$
(3) 512
(4) $6 ^ { 6 }$

Let $A$ be a $3 \times 3$ real matrix such that $A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$; $A \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ and $A \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. If $X = \begin{pmatrix} x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{pmatrix}$ and $I$ is an identity matrix of order 3, then the system $( A - 2I ) X = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$ has
(1) no solution
(2) infinitely many solutions
(3) unique solution
(4) exactly two solutions

The probability that a randomly chosen $2 \times 2$ matrix with all the entries from the set of first 10 primes, is singular, is equal to
(1) $\frac { 133 } { 10 ^ { 4 } }$
(2) $\frac { 19 } { 10 ^ { 3 } }$
(3) $\frac { 18 } { 10 ^ { 3 } }$
(4) $\frac { 271 } { 10 ^ { 4 } }$