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) $\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}$

Let $A$, be a $3 \times 3$ matrix, such that $A ^ { 2 } - 5 A + 7 I = O$. Statement - I : $A ^ { - 1 } = \frac { 1 } { 7 } ( 5 I - A )$. Statement - II : The polynomial $A ^ { 3 } - 2 A ^ { 2 } - 3 A + I$, can be reduced to $5 ( A - 4 I )$. Then :
(1) Both the statements are true
(2) Both the statements are false
(3) Statement - I is true, but Statement - II is false
(4) Statement - I is false, but Statement - II is true

If $A = \left[ \begin{array} { c c } - 4 & - 1 \\ 3 & 1 \end{array} \right]$, then the determinant of the matrix $\left( A ^ { 2016 } - 2 A ^ { 2015 } - A ^ { 2014 } \right)$ is :
(1) $- 175$
(2) 2014
(3) 2016
(4) $- 25$

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) $-1$

The set of all values of $\lambda$ for which the system of linear equations $$x - 2 y - 2 z = \lambda x$$ $$x + 2 y + z = \lambda y$$ $$- x - y = \lambda z$$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements

If $A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$, then $\text{Adj}(3A^2 + 12A)$ 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) $\begin{pmatrix} 72 & -63 \\ -84 & 51 \end{pmatrix}$

Let $A$ be any $3 \times 3$ invertible matrix. Then which one of the following is not always true?
(1) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | ^ { 2 } \cdot ( \operatorname { adj } ( \mathrm {~A} ) ) ^ { - 1 }$
(2) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot ( \operatorname { adj } ( \mathrm { A } ) ) ^ { - 1 }$
(3) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot A$
(4) $\operatorname { adj } ( \mathrm { A } ) = | A | \cdot A ^ { - 1 }$

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) $- 2 - \sqrt { 3 }$

Let $A$ be a matrix such that $A \cdot \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$ is a scalar matrix and $| 3 A | = 108$. Then, $A ^ { 2 }$ equals :
(1) $\left[ \begin{array} { c c } 4 & 0 \\ - 32 & 36 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 36 & - 32 \\ 0 & 4 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 36 & 0 \\ - 32 & 4 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 4 & - 32 \\ 0 & 36 \end{array} \right]$

Let $A$ be a matrix such that $A$. $\left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$ is a scalar matrix and $| 3 A | = 108$. Then $A ^ { 2 }$ equals
(1) $\left[ \begin{array} { c c } 4 & - 32 \\ 0 & 36 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 4 & 0 \\ - 32 & 36 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 36 & 0 \\ - 32 & 4 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 36 & - 32 \\ 0 & 4 \end{array} \right]$

If the system of linear equations $x + k y + 3 z = 0$ $3 x + k y - 2 z = 0$ $2 x + 4 y - 3 z = 0$ has a non-zero solution $( x , y , z )$, then $\frac { x z } { y ^ { 2 } }$ is equal to:
(1) 30
(2) - 10
(3) 10
(4) - 30

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

If $\left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right] \ldots \left[ \begin{array} { c c } 1 & n - 1 \\ 0 & 1 \end{array} \right] = \left[ \begin{array} { c c } 1 & 78 \\ 0 & 1 \end{array} \right]$, then the inverse of $\left[ \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right]$ is:
(1) $\left[ \begin{array} { c c } 1 & - 12 \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ 12 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 1 & 0 \\ 13 & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & - 13 \\ 0 & 1 \end{array} \right]$

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 $t \in R$

The total number of matrices $A = \left( \begin{array} { c c c } 0 & 2 y & 1 \\ 2 x & y & - 1 \\ 2 x & - y & 1 \end{array} \right) , ( x , y \in R , x \neq y )$ for which $A ^ { T } A = 3 I _ { 3 }$ is:
(1) 6
(2) 3
(3) 4
(4) 2

If the system of equations $2 x + 3 y - z = 0 , x + k y - 2 z = 0$ and $2 x - y + z = 0$ has a non-trivial solution $( x , y , z )$, then $\frac { x } { y } + \frac { y } { z } + \frac { z } { x } + k$ is equal to
(1) $- \frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $- 4$
(4) $\frac { 3 } { 4 }$

Let $\alpha$ be a root of the equation $x ^ { 2 } + x + 1 = 0$ and the matrix $A = \frac { 1 } { \sqrt { 3 } } \left[ \begin{array} { c c c } 1 & 1 & 1 \\ 1 & \alpha & \alpha ^ { 2 } \\ 1 & \alpha ^ { 2 } & \alpha ^ { 4 } \end{array} \right]$, then the matrix $A ^ { 31 }$ is equal to
(1) $A ^ { 3 }$
(2) $I _ { 3 }$
(3) $A ^ { 2 }$
(4) $A$

If $A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix}$ and $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, then $10A^{-1}$ is equal to.
(1) $A - 4I$
(2) $6I - A$
(3) $A - 6I$
(4) $4I - A$

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0, 1\}$ and $|A| \neq 0$. Consider the following two statements: $(P)$ If $A \neq I_{2}$, then $|A| = -1$ $(Q)$ If $|A| = 1$, then $\operatorname{tr}(A) = 2$ Where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then
(1) $(P)$ is false and $(Q)$ is true
(2) Both $(P)$ and $(Q)$ are false
(3) $(P)$ is true and $(Q)$ is false
(4) Both $(P)$ and $(Q)$ are true

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 $(1,2)$

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) $f ( 50 ) = 1$

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations $$2x - y + 2z = 2$$ $$x - 2y + \lambda z = -4$$ $$x + \lambda y + z = 4$$ has no solution. Then the set $S$
(1) Contains more than two elements
(2) Is an empty set
(3) Is a singleton
(4) Contains exactly two elements

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) $9$

If the system of equations $x + y + z = 2$ $2 x + 4 y - z = 6$ $3 x + 2 y + \lambda z = \mu$ has infinitely many solutions, then:
(1) $\lambda + 2 \mu = 14$
(2) $2 \lambda - \mu = 5$
(3) $\lambda - 2 \mu = - 5$
(4) $2 \lambda + \mu = 14$