Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x + y + z = 1$, $2 x + N y + 2 z = 2$, $3 x + 3 y + N z = 3$ has unique solution is $\frac { k } { 6 }$, then the sum of value of $k$ and all possible values of $N$ is
(1) 18
(2) 19
(3) 20
(4) 21

Let for $A = \begin{pmatrix} 1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{pmatrix}$, $|A| = 2$. If $| 2 \operatorname { adj } ( 2 \operatorname { adj } ( 2 A ) ) | = 32 ^ { n }$, then $3 n + \alpha$ is equal to
(1) 9
(2) 11
(3) 12
(4) 10

Let the determinant of a square matrix $A$ of order $m$ be $m - n$, where m and $n$ satisfy $4 m + n = 22$ and $17 m + 4 n = 93$. If $\operatorname { det } ( n \operatorname { adj } ( \operatorname { adj } ( m A ) ) ) = 3 ^ { a } 5 ^ { b } 6 ^ { c }$, then $a + b + c$ is equal to
(1) 84
(2) 96
(3) 101
(4) 109

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $$\lambda x + y + z = 1$$ $$x + \lambda y + z = 1$$ $$x + y + \lambda z = 1$$ is inconsistent, then $\sum_{\lambda \in S} (\lambda^2 + \lambda)$ is equal to
(1) 2
(2) 12
(3) 4
(4) 6

If $A = \frac { 1 } { 5! \cdot 6! \cdot 7! } \begin{vmatrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{vmatrix}$, then $|\text{adj}(\text{adj}(2A))|$ is equal to
(1) $2 ^ { 20 }$
(2) $2 ^ { 8 }$
(3) $2 ^ { 12 }$
(4) $2 ^ { 16 }$

Let $f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$, $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then
(1) $\beta^2 - 2\sqrt{\alpha} = \frac{19}{4}$
(2) $\beta^2 + 2\sqrt{\alpha} = \frac{19}{4}$
(3) $\alpha^2 - \beta^2 = 4\sqrt{3}$
(4) $\alpha^2 + \beta^2 = \frac{9}{2}$

Let $x , y , z > 1$ and $A = \left[ \begin{array} { l l l } 1 & \log _ { x } y & \log _ { x } z \\ \log _ { y } x & 2 & \log _ { y } z \\ \log _ { z } x & \log _ { z } y & 3 \end{array} \right]$. Then $\left| \operatorname { adj } \left( \operatorname { adj } \mathrm { A } ^ { 2 } \right) \right|$ is equal to
(1) $6 ^ { 4 }$
(2) $2 ^ { 8 }$
(3) $4 ^ { 8 }$
(4) $2 ^ { 4 }$

Let $A , B , C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements $( S 1 ) A ^ { 13 } B ^ { 26 } - B ^ { 26 } A ^ { 13 }$ is symmetric (S2) $A ^ { 26 } C ^ { 13 } - C ^ { 13 } A ^ { 26 }$ is symmetric Then,
(1) Only $S 2$ is true
(2) Only S1 is true
(3) Both $S 1$ and $S 2$ are false
(4) Both $S 1$ and $S 2$ are true

If $P$ is a $3 \times 3$ real matrix such that $P^{T} = aP + (a-1)I$, where $a > 1$, then
(1) $P$ is a singular matrix
(2) $|\operatorname{Adj} P| > 1$
(3) $|\operatorname{Adj} P| = \frac{1}{2}$
(4) $|\operatorname{Adj} P| = 1$

Let $A = [a_{ij}]$, $a_{ij} \in Z \cap [0,4]$, $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $\_\_\_\_$.

Let $\mathrm { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 10 } } & \frac { 3 } { \sqrt { 10 } } \\ \frac { - 3 } { \sqrt { 10 } } & \frac { 1 } { \sqrt { 10 } } \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { c c } 1 & - \mathrm { i } \\ 0 & 1 \end{array} \right]$, where $\mathrm { i } = \sqrt { - 1 }$. If $\mathrm { M } = \mathrm { A } ^ { \mathrm { T } } \mathrm { BA }$, then the inverse of the matrix $\mathrm { AM } ^ { 2023 } \mathrm {~A} ^ { \mathrm { T } }$ is
(1) $\left[ \begin{array} { c c } 1 & - 2023 i \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { l l } 1 & 0 \\ - 2023 i & 1 \end{array} \right]$
(3) $\left[ \begin{array} { l l } 1 & 0 \\ 2023 i & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & 2023 i \\ 0 & 1 \end{array} \right]$

Let $A = \left[ \begin{array} { c c } 1 & \frac { 1 } { 51 } \\ 0 & 1 \end{array} \right]$. If $B = \left[ \begin{array} { c c } 1 & 2 \\ - 1 & - 1 \end{array} \right] A \left[ \begin{array} { c c } - 1 & - 2 \\ 1 & 1 \end{array} \right]$, then the sum of all the elements of the matrix $\sum _ { n = 1 } ^ { 50 } B ^ { n }$ is equal to
(1) 75
(2) 125
(3) 50
(4) 100

Let $P$ be a square matrix such that $P^{2} = I - P$. For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $P^{\alpha} + P^{\beta} = \gamma I - 29P$ and $P^{\alpha} - P^{\beta} = \delta I - 13P$, then $\alpha + \beta + \gamma - \delta$ is equal to
(1) 18
(2) 40
(3) 22
(4) 24

The set of all values of $t \in \mathbb { R }$, for which the matrix $\left[ \begin{array} { c c c } \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) & \mathrm { e } ^ { - t } ( - 2 \sin t - \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( 2 \sin t + \cos t ) & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } \cos t & \mathrm { e } ^ { - t } \sin t \end{array} \right]$ is invertible, is (1) $\left\{ ( 2 k + 1 ) \frac { \pi } { 2 } , k \in \mathbb { Z } \right\}$ (2) $\left\{ k \pi + \frac { \pi } { 4 } , k \in \mathbb { Z } \right\}$ (3) $\{ k \pi , k \in \mathbb { Z } \}$ (4) $\mathbb { R }$

If a point $P(\alpha, \beta, \gamma)$ satisfying $\begin{pmatrix} \alpha & \beta & \gamma \end{pmatrix} \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ lies on the plane $2x + 4y + 3z = 5$, then $6\alpha + 9\beta + 7\gamma$ is equal to $\_\_\_\_$.

For the system of equations $x + y + z = 6$ $x + 2y + \alpha z = 10$ $x + 3y + 5z = \beta$, which one of the following is NOT true?
(1) System has no solution for $\alpha = 3, \beta = 24$
(2) System has a unique solution for $\alpha = -3, \beta = 14$
(3) System has infinitely many solutions for $\alpha = 3, \beta = 14$
(4) System has a unique solution for $\alpha = 3, \beta = 14$

Let A be a symmetric matrix such that $| A | = 2$ and $\left[ \begin{array} { l l } 2 & 1 \\ 3 & \frac { 3 } { 2 } \end{array} \right] A = \left[ \begin{array} { l l } 1 & 2 \\ \alpha & \beta \end{array} \right]$. If the sum of the diagonal elements of A is $s$, then $\frac { \beta s } { \alpha ^ { 2 } }$ is equal to $\_\_\_\_$.

Let $A = \left[ \begin{array} { c c c } 2 & 1 & 0 \\ 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array} \right]$. If $| \mathrm{adj} ( \mathrm{adj} ( \mathrm{adj}\, 2 A ) ) | = ( 16 ) ^ { n }$, then $n$ is equal to
(1) 8
(2) 10
(3) 9
(4) 12

Let $P = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $Q = P A P ^ { T }$. If $P ^ { T } Q ^ { 2007 } P = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ then $2 a + b - 3 c - 4 d$ is equal to
(1) 2004
(2) 2005
(3) 2007
(4) 2006

If $f(x) = \begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x \in \mathbb{R}$, then $2f(0) + f'(0)$ is equal to
(1) 48
(2) 24
(3) 42
(4) 18

Let $A = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array} \right]$ and $| 2 A | ^ { 3 } = 2 ^ { 21 }$ where $\alpha , \beta \in Z$, Then a value of $\alpha$ is
(1) 3
(2) 5
(3) 17
(4) 9

The values of $\alpha$, for which $$2 \alpha + 3 \quad 3 \alpha + 1 \quad 0$$ (1) ( - 2, 1)
(2) ( - 3, 0)
(3) $- \frac { 3 } { 2 } , \frac { 3 } { 2 }$
(4) $( 0,3 )$

If $\alpha \neq \mathrm { a } , \beta \neq \mathrm { b } , \gamma \neq \mathrm { c }$ and $\left| \begin{array} { c c c } \alpha & \mathrm { b } & \mathrm { c } \\ \mathrm { a } & \beta & \mathrm { c } \\ \mathrm { a } & \mathrm { b } & \gamma \end{array} \right| = 0$, then $\frac { \mathrm { a } } { \alpha - \mathrm { a } } + \frac { \mathrm { b } } { \beta - \mathrm { b } } + \frac { \gamma } { \gamma - \mathrm { c } }$ is equal to: (1) 3 (2) 0 (3) 1 (4) 2

Let A be a square matrix such that $\mathrm { AA } ^ { \mathrm { T } } = \mathrm { I }$. Then $\frac { 1 } { 2 } \mathrm { ~A} \left[ \left( \mathrm { ~A} + \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } + \left( \mathrm { A } - \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } \right]$ is equal to
(1) $A ^ { 2 } + I$
(2) $A ^ { 3 } + I$
(3) $A ^ { 2 } + A ^ { T }$
(4) $\mathrm { A } ^ { 3 } + \mathrm { A } ^ { \mathrm { T } }$

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