Let $m$ and $M$ be respectively the minimum and maximum values of $\left| \begin{array} { c c c } \cos ^ { 2 } x & 1 + \sin ^ { 2 } x & \sin 2 x \\ 1 + \cos ^ { 2 } x & \sin ^ { 2 } x & \sin 2 x \\ \cos ^ { 2 } x & \sin ^ { 2 } x & 1 + \sin 2 x \end{array} \right|$. Then the ordered pair $( \mathrm { m } , \mathrm { M } )$ is equal to:
(1) $( 3,3 )$
(2) $( - 3 , - 1 )$
(3) $( 4,1 )$
(4) $( 1,3 )$

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) $\frac { 3 } { 2 }$

If the minimum and the maximum values of the function $f : \left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right] \rightarrow R$, defined by $f ( \theta ) = \left| \begin{array} { c c c } - \sin ^ { 2 } \theta & - 1 - \sin ^ { 2 } \theta & 1 \\ - \cos ^ { 2 } \theta & - 1 - \cos ^ { 2 } \theta & 1 \\ 12 & 10 & - 2 \end{array} \right|$ are $m$ and $M$ respectively, then the ordered pair $( \mathrm { m } , \mathrm { M } )$ is equal to :
(1) $( 0,2 \sqrt { 2 } )$
(2) $( - 4,0 )$
(3) $( - 4,4 )$
(4) $( 0,4 )$

If the system of linear equations $$x + y + 3z = 0$$ $$x + 3y + k^2z = 0$$ $$3x + y + 3z = 0$$ has a non-zero solution $(x, y, z)$ for some $k \in \mathrm{R}$, then $x + \left(\frac{y}{z}\right)$ is equal to:
(1) $-3$
(2) $9$
(3) $3$
(4) $-9$

The values of $\lambda$ and $\mu$ for which the system of linear equations $x + y + z = 2 , x + 2 y + 3 z = 5$, $x + 3 y + \lambda z = \mu$ has infinitely many solutions, are respectively
(1) 6 and 8
(2) 5 and 7
(3) 5 and 8
(4) 4 and 9

For the system of linear equations: $$x - 2 y = 1 , x - y + k z = - 2 , k y + 4 z = 6 , k \in R$$ Consider the following statements:
(A) The system has unique solution if $k \neq 2 , k \neq - 2$.
(B) The system has unique solution if $k = - 2$.
(C) The system has unique solution if $k = 2$.
(D) The system has no-solution if $k = 2$.
(E) The system has infinitely many solutions if $k = - 2$.

Consider the following system of equations: $$\begin{aligned} & x + 2 y - 3 z = a \\ & 2 x + 6 y - 11 z = b \\ & x - 2 y + 7 z = c \end{aligned}$$ where $a , b$ and $c$ are real constants. Then the system of equations :
(1) has a unique solution when $5 a = 2 b + c$
(2) has no solution for all $a , b$ and $c$
(3) has infinite number of solutions when $5 a = 2 b + c$
(4) has a unique solution for all $a , b$ and $c$

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

The system of linear equations $3 x - 2 y - k z = 10$ $2 x - 4 y - 2 z = 6$ $x + 2 y - z = 5 m$ is inconsistent if:
(1) $k = 3 , \quad m \neq \frac { 4 } { 5 }$
(2) $k = 3 , \quad m = \frac { 4 } { 5 }$
(3) $k \neq 3 , \quad m \in R$
(4) $k \neq 3 , \quad m \neq \frac { 4 } { 5 }$

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 $A = \left[ \begin{array} { c c } i & - i \\ - i & i \end{array} \right] , i = \sqrt { - 1 }$. 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) $\frac { \pi } { 6 }$

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as ``$ARB$ iff there exists a non-singular matrix $P$ such that $PAP^{-1} = B$''. Then which of the following is true?

The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + zk = k^2$ has no solution if $k$ is equal to:
(1) 0
(2) 1
(3) $-1$
(4) $-2$

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) $\left[ \begin{array} { l l } 1 & 0 \\ 50 & 1 \end{array} \right]$

Let $A = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { c c } 1 , & \text { if } i = j \\ - x , & \text { if } | i - j | = 1 \\ 2 x + 1 , & \text { otherwise } \end{array} \right.$ Let a function $f : R \rightarrow R$ be defined as $f ( x ) = \operatorname { det } ( A )$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:
(1) $- \frac { 20 } { 27 }$
(2) $\frac { 88 } { 27 }$
(3) $\frac { 20 } { 27 }$
(4) $- \frac { 88 } { 27 }$

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 $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ 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 the system of linear equations $x + y + a z = 2$ $3 x + y + z = 4$ $x + 2 z = 1$ have a unique solution $\left( x ^ { * } , y ^ { * } , z ^ { * } \right)$. If $\left( \left( a , x ^ { * } \right) , \left( y ^ { * } , \alpha \right) \right.$ and $\left( x ^ { * } , - y ^ { * } \right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is:
(1) 4
(2) 3
(3) 2
(4) 1