Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $\alpha \in R$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then, a value of $\alpha$ is:
(1) 0
(2) $\frac{\pi}{16}$
(3) $\frac{\pi}{64}$
(4) $\frac{\pi}{32}$

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) $y \left( y ^ { 2 } - 3 \right)$

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 the system of linear equations $$\begin{aligned} & 2x + 2ay + az = 0 \\ & 2x + 3by + bz = 0 \\ & 2x + 4cy + cz = 0 \end{aligned}$$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then
(1) $\frac { 1 } { a } , \frac { 1 } { b } , \frac { 1 } { c }$ are in $A.P$.
(2) $a, b, c$ are in $G.P$.
(3) $a + b + c = 0$
(4) $a, b, c$ are in $A.P$.

The system of linear equations $\lambda x + 2y + 2z = 5$ $2\lambda x + 3y + 5z = 8$ $4x + \lambda y + 6z = 10$ has
(1) no solution when $\lambda = 8$
(2) a unique solution when $\lambda = -8$
(3) no solution when $\lambda = 2$
(4) infinitely many solutions when $\lambda = 2$

The following system of linear equations $7 x + 6 y - 2 z = 0$ $3 x + 4 y + 2 z = 0$ $x - 2 y - 6 z = 0$, has
(1) infinitely many solutions, ( $x , y , z$ ) satisfying $y = 2z$
(2) no solution
(3) infinitely many solutions, $( x , y , z )$ satisfying $x = 2z$
(4) only the trivial solution

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

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) $\frac { 2 } { 3 }$

Let $\lambda \in \mathrm { R }$. The system of linear equations $2 x _ { 1 } - 4 x _ { 2 } + \lambda x _ { 3 } = 1$ $x _ { 1 } - 6 x _ { 2 } + x _ { 3 } = 2$ $\lambda x _ { 1 } - 10 x _ { 2 } + 4 x _ { 3 } = 3$ is inconsistent for :
(1) exactly one positive value of $\lambda$
(2) exactly one negative value of $\lambda$
(3) every value of $\lambda$
(4) exactly two values of $\lambda$

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) $y(a-c)$

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

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) $[ 68,69 )$

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

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) $a _ { 2 } a _ { 6 } - a _ { 4 } a _ { 8 }$

If $1 , \log _ { 10 } \left( 4 ^ { x } - 2 \right)$ and $\log _ { 10 } \left( 4 ^ { x } + \frac { 18 } { 5 } \right)$ are in arithmetic progression for a real number $x$ then the value of the determinant $\left| \begin{array} { c c c } 2 \left( x - \frac { 1 } { 2 } \right) & x - 1 & x ^ { 2 } \\ 1 & 0 & x \\ x & 1 & 0 \end{array} \right|$ is equal to:

Let $A = \left[ \begin{array} { l l l } x & y & z \\ y & z & x \\ z & x & y \end{array} \right]$, where $x , y$ and $z$ are real numbers such that $x + y + z > 0$ and $x y z = 2$. If $A ^ { 2 } = I _ { 3 }$, then the value of $x ^ { 3 } + y ^ { 3 } + z ^ { 3 }$ is

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 $A = \left\{ a _ { i j } \right\}$ 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 $\underline{\hspace{1cm}}$.

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) $6 ^ { 6 }$

The number of $\theta \in (0,4\pi)$ for which the system of linear equations $3(\sin 3\theta) x - y + z = 2$ $3(\cos 2\theta) x + 4y + 3z = 3$ $6x + 7y + 7z = 9$ has no solution is
(1) 6
(2) 7
(3) 8
(4) 9

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 $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^ { 2 } = 3 A + \alpha I$. If $A ^ { 4 } = 21 A + \beta I$, then
(1) $\alpha = 1$
(2) $\alpha = 4$
(3) $\beta = 8$
(4) $\beta = - 8$

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 $R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ be a non-zero $3 \times 3$ matrix, where $x\sin\theta = y\sin\left(\theta + \frac{2\pi}{3}\right) = z\sin\left(\theta + \frac{4\pi}{3}\right) \neq 0$, $\theta \in (0, 2\pi)$. For a square matrix $M$, let Trace $M$ denote the sum of all the diagonal entries of $M$. Then, among the statements: I. Trace$(R) = 0$ II. If Trace$(\operatorname{adj}(\operatorname{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.
(1) Both (I) and (II) are true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Only (I) is true