Let $d \in R$, and $A = \left[ \begin{array} { c c c } - 2 & 4 + d & ( \sin \theta ) - 2 \\ 1 & ( \sin \theta ) + 2 & d \\ 5 & ( 2 \sin \theta ) - d & ( - \sin \theta ) + 2 + 2 d \end{array} \right] , \theta \in [ 0,2 \pi ]$. If the minimum value of $\operatorname { det } ( A )$ is 8, then a value of $d$ is:
(1) $2 ( \sqrt { 2 } + 2 )$
(2) $2 ( \sqrt { 2 } + 1 )$
(3) $- 5$
(4) $- 7$

If $A = \left[ \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right]$, then for all $\theta \in \left( \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } \right) , \operatorname { det } ( A )$ lies in the interval :
(1) $\left( 1 , \frac { 5 } { 2 } \right]$
(2) $\left[ \frac { 5 } { 2 } , 4 \right)$
(3) $\left( \frac { 3 } { 2 } , 3 \right]$
(4) $\left( 0 , \frac { 3 } { 2 } \right]$

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + x + 1 = 0$. Then for $y \neq 0$ in $R , \left| \begin{array} { c c c } y + 1 & \alpha & \beta \\ \alpha & y + \beta & 1 \\ \beta & 1 & y + \alpha \end{array} \right|$ is equal to
(1) $y ^ { 3 }$
(2) $y \left( y ^ { 2 } - 1 \right)$
(3) $y ^ { 3 } - 1$
(4) $y \left( y ^ { 2 } - 3 \right)$

If $A = \left[ \begin{array} { c c c } 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & - 1 & 3 \end{array} \right] , B = \mathrm{adj}\, A$ and $C = 3 A$, then $\frac { | \mathrm{adj}\, B | } { | C | }$ is equal to
(1) 8
(2) 16
(3) 72
(4) 2

If $x , y , z$ are in arithmetic progression with common difference $d , x \neq 3 d$, and the determinant of the matrix $\left[ \begin{array} { c c c } 3 & 4 \sqrt { 2 } & x \\ 4 & 5 \sqrt { 2 } & y \\ 5 & k & z \end{array} \right]$ is zero, then the value of $k ^ { 2 }$ is
(1) 72
(2) 12
(3) 36
(4) 6

Let $A = \left\{ a _ { i j } \right\}$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { l l } ( - 1 ) ^ { j - i } & \text { if } i < j \\ 2 & \text { if } i = j \\ ( - 1 ) ^ { i + j } & \text { if } i > j \end{array} \right.$ then $\det \left( 3 \operatorname{Adj} \left( 2 A ^ { - 1 } \right) \right)$ is equal to $\underline{\hspace{1cm}}$.

If $\mathrm { a } _ { \mathrm { r } } = \cos \frac { 2 \mathrm { r } \pi } { 9 } + i \sin \frac { 2 \mathrm { r } \pi } { 9 } , \mathrm { r } = 1,2,3 , \ldots , i = \sqrt { - 1 }$, then the determinant $\left| \begin{array} { l l l } a _ { 1 } & a _ { 2 } & a _ { 3 } \\ a _ { 4 } & a _ { 5 } & a _ { 6 } \\ a _ { 7 } & a _ { 8 } & a _ { 9 } \end{array} \right|$ is equal to $:$
(1) $\mathrm { a } _ { 9 }$
(2) $a _ { 1 } a _ { 9 } - a _ { 3 } a _ { 7 }$
(3) $a _ { 5 }$
(4) $a _ { 2 } a _ { 6 } - a _ { 4 } a _ { 8 }$

Let $A = \left[ \begin{array} { c c c } { [ x + 1 ] } & { [ x + 2 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 3 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 2 ] } & { [ x + 4 ] } \end{array} \right]$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. If $\operatorname { det } ( A ) = 192$, then the set of values of $x$ is in the interval: (1) $[ 62,63 )$ (2) $[ 65,66 )$ (3) $[ 60,61 )$ (4) $[ 68,69 )$

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 $A$ and $B$ be two $3 \times 3$ matrices such that $A B = I$ and $| A | = \frac { 1 } { 8 }$ then $| \operatorname{adj} ( B \operatorname{adj} ( 2 A ) ) |$ is equal to
(1) 128
(2) 32
(3) 64
(4) 102

Let $A$ be a $n \times n$ matrix such that $|A| = 2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2A^{-1}\right)\right)$ is $2^{84}$, then $n$ is equal to $\_\_\_\_$.

Let $A$ be a $3 \times 3$ matrix and $\det(A) = 2$. If $n = \det(\text{adj}(\text{adj}(\cdots(\text{adj}(A))\cdots)))$ where adj is applied 6 times, then the remainder when $n$ is divided by 9 is $\_\_\_\_$.

Let $\mathrm { A } = \left[ \begin{array} { c c c } 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{array} \right]$ and $\mathrm { P } = \left[ \begin{array} { l l l } 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{array} \right]$. The sum of the prime factors of $\left| \mathrm { P } ^ { - 1 } \mathrm { AP } - 2 \mathrm { I } \right|$ is equal to
(1) 26
(2) 27
(3) 66
(4) 23

For a $3 \times 3$ matrix $M$, let trace ( $M$ ) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $| A | = \frac { 1 } { 2 }$ and trace $( A ) = 3$. If $B = \operatorname { adj } ( \operatorname { adj } ( 2 A ) )$, then the value of $| B | +$ trace $( \mathrm { B } )$ equals :
(1) 56
(2) 132
(3) 174
(4) 280

For some $\mathrm{a}, \mathrm{b}$, let $f(x) = \left| \begin{array}{ccc} \mathrm{a} + \frac{\sin x}{x} & 1 & \mathrm{b} \\ \mathrm{a} & 1 + \frac{\sin x}{x} & \mathrm{b} \\ \mathrm{a} & 1 & \mathrm{b} + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \rightarrow 0} f(x) = \lambda + \mu\mathrm{a} + \nu\mathrm{b}$. Then $(\lambda + \mu + \nu)^{2}$ is equal to:
(1) 16
(2) 25
(3) 9
(4) 36

Q69. Let $\alpha \in ( 0 , \infty )$ and $A = \left[ \begin{array} { c c c } 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{array} \right]$. If $\operatorname { det } \left( \operatorname { adj } \left( 2 A - A ^ { T } \right) \cdot \operatorname { adj } \left( A - 2 A ^ { T } \right) \right) = 2 ^ { 8 }$, then $( \operatorname { det } ( A ) ) ^ { 2 }$ is equal to:
(1) 36
(2) 16
(3) 1
(4) 49

Q68. Let $A$ and $B$ be two square matrices of order 3 such that $| A | = 3$ and $| B | = 2$. Then $\left| \mathrm { A } ^ { \mathrm { T } } \mathrm { A } ( \operatorname { adj } ( 2 \mathrm {~A} ) ) ^ { - 1 } ( \operatorname { adj } ( 4 \mathrm {~B} ) ) ( \operatorname { adj } ( \mathrm { AB } ) ) ^ { - 1 } \mathrm { AA } ^ { \mathrm { T } } \right|$ is equal to :
(1) 108
(2) 32
(3) 81
(4) 64

Q69. Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to :
(1) 64
(2) 216
(3) 343
(4) 125
Q70. $x + y + z = 4$, The values of $m , n$, for which the system of equations $2 x + 5 y + 5 z = 17$, has infinitely many solutions, $x + 2 y + \mathrm { m } z = \mathrm { n }$ satisfy the equation:
(1) $m ^ { 2 } + n ^ { 2 } - m n = 39$
(2) $m ^ { 2 } + n ^ { 2 } - m - n = 46$
(3) $m ^ { 2 } + n ^ { 2 } + m + n = 64$
(4) $m ^ { 2 } + n ^ { 2 } + m n = 68$

Q70. If $A$ is a square matrix of order 3 such that $\operatorname { det } ( A ) = 3$ and $\operatorname { det } \left( \operatorname { adj } \left( - 4 \operatorname { adj } \left( - 3 \operatorname { adj } \left( 3 \operatorname { adj } \left( ( 2 \mathrm {~A} ) ^ { - 1 } \right) \right) \right) \right) \right) = 2 ^ { \mathrm { m } } 3 ^ { \mathrm { n } }$, then $\mathrm { m } + 2 \mathrm { n }$ is equal to :
(1) 2
(2) 3
(3) 6
(4) 4

Q68. Let $A = \left[ \begin{array} { l l l } 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{array} \right]$. If $A ^ { 3 } = 4 A ^ { 2 } - A - 21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a + 3 b$ is equal to
(1) - 9
(2) - 13
(3) - 10
(4) - 12

Q86. Let $A$ be a non-singular matrix of order 3 . If $\operatorname { det } ( 3 \operatorname { adj } ( 2 \operatorname { adj } ( ( \operatorname { det } A ) A ) ) ) = 3 ^ { - 13 } \cdot 2 ^ { - 10 }$ and $\operatorname { det } ( 3 \operatorname { adj } ( 2 \mathrm {~A} ) ) = 2 ^ { \mathrm { m } } \cdot 3 ^ { \mathrm { n } }$, then $| 3 \mathrm {~m} + 2 \mathrm { n } |$ is equal to

If $\mathrm{A} = \left[\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}\right], \mathrm{B} = \left[\begin{array}{ll}1 & 1 \\ \beta & 1\end{array}\right]$ and $\mathrm{A}^{2} - 4\mathrm{A} + 2\mathrm{I} = 0 ; \mathrm{B}^{2} - 2\mathrm{B} + \mathrm{I} = 0$, then $\left|\operatorname{adj}\left(\mathrm{A}^{3} - \mathrm{B}^{3}\right)\right|$ is equal to
(A) 7 (B) 11 (C) -11 (D) 121