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

Consider a matrix $\mathrm { A } = \left[ \begin{array} { c c c } \alpha & \beta & \gamma \\ \alpha ^ { 2 } & \beta ^ { 2 } & \gamma ^ { 2 } \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{array} \right]$, where $\alpha , \beta , \gamma$ are three distinct natural numbers. If $\frac { \operatorname { det } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } A ) ) ) } { ( \alpha - \beta ) ^ { 16 } ( \beta - \gamma ) ^ { 16 } ( \gamma - \alpha ) ^ { 16 } } = 2 ^ { 32 } \times 3 ^ { 16 }$, then the number of such 3-tuples $( \alpha , \beta , \gamma )$ is $\_\_\_\_$ .

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 $\_\_\_\_$.

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 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 $f(x) = \begin{vmatrix} 1 + \sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$, $x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. 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) $2 ^ { 4 }$

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

If $A$ is a $3 \times 3$ matrix and $|A| = 2$, then $|3\, \text{adj}(3A)| \cdot |A^2|$ is equal to
(1) $3^{12} \cdot 6^{11}$
(2) $3^{12} \cdot 6^{10}$
(3) $3^{10} \cdot 6^{11}$
(4) $3^{11} \cdot 6^{10}$

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) $( 0,3 )$

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of the correct observations respectively, then $15 \mu + \mu ^ { 2 } + \sigma ^ { 2 }$ is equal to $\_\_\_\_$ .

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

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 of order 3 such that $\operatorname { det } ( A ) = - 2$ and $\operatorname { det } ( 3 \operatorname { adj } ( - 6 \operatorname { adj } ( 3 A ) ) ) = 2 ^ { \mathrm { m } + \mathrm { n } } \cdot 3 ^ { \mathrm { mn } } , \mathrm { m } > \mathrm { n }$. Then $4 \mathrm {~m} + 2 \mathrm { n }$ is equal to $\_\_\_\_$

Let $A$ be a $3 \times 3$ matrix such that $X^T A X = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $$A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix},\quad A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix},$$ and $\det(\operatorname{adj}(2(A+I))) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is $\underline{\hspace{2cm}}$.

Let $\mathrm{A} = [\mathrm{a}_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}$. If $\mathrm{A}_{ij}$ is the cofactor of $\mathrm{a}_{ij}$, $\mathrm{C}_{ij} = \sum_{\mathrm{k}=1}^{2} \mathrm{a}_{i\mathrm{k}} \mathrm{A}_{j\mathrm{k}}$, $1 \leq i, j \leq 2$, and $\mathrm{C} = [\mathrm{C}_{ij}]$, then $8|\mathrm{C}|$ is equal to:
(1) 288
(2) 222
(3) 242
(4) 262

Let M and m respectively be the maximum and the minimum values of $$f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{vmatrix}, \quad x \in \mathbf{R}.$$ Then $M^4 - m^4$ is equal to:
(1) 1280
(2) 1295
(3) 1215
(4) 1040

Q86. Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = 3 \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]$. Then the maximum value of $\operatorname { det } ( \mathrm { A } )$ is $\_\_\_\_$