If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$, then adj$(3A^2 + 12A)$ is equal to:
(1) $\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}$
(2) $\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}$
(3) $\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}$
(4) $\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}$

If $A = \begin{pmatrix} 5a & -b \\ 3 & 2 \end{pmatrix}$ and $A$ adj $A = A A^{T}$, then $5a + b$ is equal to: (1) $-1$ (2) $5$ (3) $4$ (4) $13$

If $A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$, then adj$(3A^2 + 12A)$ is equal to: (1) $\begin{pmatrix} 72 & -84 \\ -63 & 51 \end{pmatrix}$ (2) $\begin{pmatrix} 51 & 63 \\ 84 & 72 \end{pmatrix}$ (3) $\begin{pmatrix} 51 & 84 \\ 63 & 72 \end{pmatrix}$ (4) $\begin{pmatrix} 72 & -63 \\ -84 & 51 \end{pmatrix}$

If $\alpha$, $\beta \neq 0$, and $f(n) = \alpha^n + \beta^n$ and $$\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2,$$ then $K$ is equal to: (1) $\alpha\beta$ (2) $\frac{1}{\alpha\beta}$ (3) 1 (4) $-1$

If $S = \left\{ x \in [ 0,2 \pi ] : \left| \begin{array} { c c c } 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{array} \right| = 0 \right\}$, then $\sum _ { x \in S } \tan \left( \frac { \pi } { 3 } + x \right)$ is equal to:
(1) $4 + 2 \sqrt { 3 }$
(2) $- 4 - 2 \sqrt { 3 }$
(3) $- 2 + \sqrt { 3 }$
(4) $- 2 - \sqrt { 3 }$

$\left| \begin{array} { c c c } x - 4 & 2 x & 2 x \\ 2 x & x - 4 & 2 x \\ 2 x & 2 x & x - 4 \end{array} \right| = ( A + B x ) ( x - A ) ^ { 2 }$, then the ordered pair $( A , B )$ is equal to
(1) $( 4,5 )$
(2) $( - 4 , - 5 )$
(3) $( - 4,3 )$
(4) $( - 4,5 )$

If $A = \left[\begin{array}{ccc} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{array}\right]$, then $A$ is:
(1) Invertible only if $t = \pi$
(2) Not invertible for any $t \in R$
(3) Invertible only if $t = \frac{\pi}{2}$
(4) Invertible for all $t \in R$

Let $\mathrm { A } = \left[ a _ { i j } \right]$ and $\mathrm { B } = \left[ b _ { i j } \right]$ be two $3 \times 3$ real matrices such that $b _ { i j } = ( 3 ) ^ { ( i + j - 2 ) } a _ { i j }$, where $i , j = 1,2,3$. If the determinant of B is 81, then determinant of A is
(1) $\frac { 1 } { 3 }$
(2) 3
(3) $\frac { 1 } { 81 }$
(4) $\frac { 1 } { 9 }$

Let $a - 2 b + c = 1$. If $f ( x ) = \left| \begin{array} { l l l } x + a & x + 2 & x + 1 \\ x + b & x + 3 & x + 2 \\ x + c & x + 4 & x + 3 \end{array} \right|$, then:
(1) $f ( - 50 ) = 501$
(2) $f ( - 50 ) = - 1$
(3) $f ( 50 ) = - 501$
(4) $f ( 50 ) = 1$

If $\Delta = \left| \begin{array} { c c c } x - 2 & 2 x - 3 & 3 x - 4 \\ 2 x - 3 & 3 x - 4 & 4 x - 5 \\ 3 x - 5 & 5 x - 8 & 10 x - 17 \end{array} \right| = A x ^ { 3 } + B x ^ { 2 } + C x + D$, then $B + C$ is equal to:
(1) $- 1$
(2) $1$
(3) $- 3$
(4) $9$

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) $( 0,4 )$

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 $\theta=\frac{\pi}{5}$ and $A=\left[\begin{array}{cc}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{array}\right]$. If $B=A+A^{4}$, then $\det(B)$:
(1) is one
(2) lies in $(2,3)$
(3) is zero
(4) lies in $(1,2)$

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 $A = \left[ \begin{array} { c c } 0 & \sin \alpha \\ \sin \alpha & 0 \end{array} \right]$ 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) $\frac { \pi } { 6 }$

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 $A = \left[ a _ { i j } \right]$ 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) $- \frac { 88 } { 27 }$

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 $\det ( B )$

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 $S = \{ \sqrt { n } : 1 \leqslant n \leqslant 50$ and $n$ is odd $\}$. Let $a \in S$ and $A = \left[ \begin{array} { r r r } 1 & 0 & a \\ - 1 & 1 & 0 \\ - a & 0 & 1 \end{array} \right]$. If $\Sigma _ { a \in S } \operatorname { det } ( \operatorname { adj } A ) = 100 \lambda$, then $\lambda$ is equal to
(1) 218
(2) 221
(3) 663
(4) 1717

Let $A$ be a matrix of order $3 \times 3$ and $\operatorname{det}(A) = 2$. Then $\operatorname{det}\left(\operatorname{det}(A)\operatorname{adj}\left(5\operatorname{adj}\left(A^3\right)\right)\right)$ is equal to
(1) $256 \times 10^6$
(2) $1024 \times 10^6$
(3) $512 \times 10^6$
(4) $256 \times 10^{11}$

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 $S$ be the set containing all $3 \times 3$ matrices with entries from $\{ - 1,0,1 \}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A ^ { T } A$ is 6 is $\_\_\_\_$ .

Let $A = \left( \begin{array} { c c } 4 & - 2 \\ \alpha & \beta \end{array} \right)$. If $A ^ { 2 } + \gamma A + 18 I = O$, then $\operatorname { det } ( A )$ is equal to
(1) - 18
(2) 18
(3) - 50
(4) 50