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 $\_\_\_\_$ .

Consider the matrices : $A = \left[ \begin{array} { c c } 2 & - 5 \\ 3 & m \end{array} \right] , B = \left[ \begin{array} { l } 20 \\ m \end{array} \right]$ and $X = \left[ \begin{array} { l } x \\ y \end{array} \right]$. Let the set of all $m$, for which the system of equations $A X = B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval ( $a , b$ ). Then $8 \int _ { a } ^ { b } | A | d m$ is equal to $\_\_\_\_$

Let $A = \left[ \begin{array} { c c } 2 & - 1 \\ 1 & 1 \end{array} \right]$. If the sum of the diagonal elements of $A ^ { 13 }$ is $3 ^ { n }$, then $n$ is equal to $\_\_\_\_$

Let $A = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - 2 \\ 0 & 1 \end{array} \right]$ and $P = \left[ \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right] , \theta > 0$. If $\mathrm { B } = \mathrm { PAP } ^ { \mathrm { T } } , \mathrm { C } = \mathrm { P } ^ { \mathrm { T } } \mathrm { B } ^ { 10 } \mathrm { P }$ and the sum of the diagonal elements of $C$ is $\frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is :
(1) 127
(2) 258
(3) 65
(4) 2049

If $\mathrm { A } , \mathrm { B }$, and $\left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right)$ are non-singular matrices of same order, then the inverse of $\mathrm { A } \left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right) ^ { - 1 } \mathrm {~B}$, is equal to
(1) $\mathrm { AB } ^ { - 1 } + \mathrm { A } ^ { - 1 } \mathrm {~B}$
(2) $\operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right)$
(3) $\frac { A B ^ { - 1 } } { | A | } + \frac { B A ^ { - 1 } } { | B | }$
(4) $\frac { 1 } { | A B | } ( \operatorname { adj } ( B ) + \operatorname { adj } ( A ) )$

Let $A = \left[ a _ { i j } \right]$ be a matrix of order $3 \times 3$, with $a _ { i j } = ( \sqrt { 2 } ) ^ { i + j }$. If the sum of all the elements in the third row of $A ^ { 2 }$ is $\alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then $\alpha + \beta$ is equal to :
(1) 280
(2) 224
(3) 210
(4) 168

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

If the system of equations $$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$ $$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$ $$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$ has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to
(1) 6
(2) 10
(3) 20
(4) 12

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

Let $A = \left[ a _ { i j } \right]$ be $3 \times 3$ matrix such that $A \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] , A \left[ \begin{array} { l } 4 \\ 1 \\ 3 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right]$ and $A \left[ \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, then $a _ { 23 }$ equals :
(1) $- 1$
(2) 2
(3) 1
(4) 0

Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm { S } = \{ - 3 , - 2 , - 1,1,2 \}$. Let
$$\begin{aligned} & \mathrm { S } _ { 1 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 2 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = - \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 3 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : a _ { 11 } + a _ { 22 } + a _ { 33 } = 0 \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} . \end{aligned}$$
If $n \left( \mathrm { S } _ { 1 } \cup \mathrm { S } _ { 2 } \cup \mathrm { S } _ { 3 } \right) = 125 \alpha$, then $\alpha$ equals

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 $\mathrm{S} = \{\mathrm{m} \in \mathbf{Z}: \mathrm{A}^{\mathrm{m}^2} + \mathrm{A}^{\mathrm{m}} = 3\mathrm{I} - \mathrm{A}^{-6}\}$, where $\mathrm{A} = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Then $\mathrm{n}(\mathrm{S})$ is equal to \_\_\_\_ .

Q71. Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]$ and $B = I + \operatorname { adj } ( A ) + ( \operatorname { adj } A ) ^ { 2 } + \ldots + ( \operatorname { adj } A ) ^ { 10 }$. Then, the sum of all the elements of the matrix $B$ is:
(1) - 124
(2) 22
(3) - 88
(4) - 110

Q71. The integral $\int _ { 1 / 4 } ^ { 3 / 4 } \cos \left( 2 \cot ^ { - 1 } \sqrt { \frac { 1 - x } { 1 + x } } \right) d x$ is equal to
(1) $1 / 2$
(2) $- 1 / 2$
(3) $- 1 / 4$
(4) $1 / 4$

Q83. Let $A$ be a square matrix of order 2 such that $| A | = 2$ and the sum of its diagonal elements is - 3 . If the points $( x , y )$ satisfying $\mathrm { A } ^ { 2 } + x \mathrm {~A} + y \mathrm { I } = \mathrm { O }$ lie on a hyperbola, whose length of semi major axis is $x$ and semi minor axis is $y$, eccentricity is e and the length of the latus rectum is $l$, then $81 \left( e ^ { 4 } + l ^ { 2 } \right)$ is equal to

Q84. Let $A$ be a $2 \times 2$ symmetric matrix such that $A \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] = \left[ \begin{array} { l } 3 \\ 7 \end{array} \right]$ and the determinant of $A$ be 1 . If $A ^ { - 1 } = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals $\_\_\_\_$

Q85. Consider the matrices : $A = \left[ \begin{array} { c c } 2 & - 5 \\ 3 & m \end{array} \right] , B = \left[ \begin{array} { l } 20 \\ m \end{array} \right]$ and $X = \left[ \begin{array} { l } x \\ y \end{array} \right]$. Let the set of all $m$, for which the system of equations $A X = B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval ( $a , b$ ). Then $8 \int _ { a } ^ { b } | A | d m$ is equal to $\_\_\_\_$

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 $\_\_\_\_$

Q86. Let $\alpha \beta \gamma = 45 ; \alpha , \beta , \gamma \in \mathbb { R }$. If $x ( \alpha , 1,2 ) + y ( 1 , \beta , 2 ) + z ( 2,3 , \gamma ) = ( 0,0,0 )$ for some $x , y , z \in \mathbb { R } , x y z \neq 0$, then $6 \alpha + 4 \beta + \gamma$ is equal to $\_\_\_\_$

Q86. Let $A = \left[ \begin{array} { c c } 2 & - 1 \\ 1 & 1 \end{array} \right]$. If the sum of the diagonal elements of $A ^ { 13 }$ is $3 ^ { n }$, then $n$ is equal to $\_\_\_\_$

If $A = \left[ \begin{array} { l l } 2 & 3 \\ 3 & 5 \end{array} \right]$, then the value of $\left| \vec { A } ^ { 2025 } - 3 A ^ { 2024 } + A ^ { 2023 } \right|$ is

Let $\mathrm { A } = \left[ \begin{array} { l l } 3 & - 4 \\ 1 & - 1 \end{array} \right]$ and B be a $2 \times 2$ matrix such that $\mathrm { A } ^ { 100 } = \underline { 100 \mathrm {~B} } + \mathrm { I }$, then sum of all elements of $B ^ { 100 }$ is

Number of matrices A of order $3 \times 2$ such that all of its elements are from the set $\{ - 2 , - 1,0,1,2 \}$ such that trace of $\mathrm { AA } ^ { \mathrm { T } }$ is 5 , is equal to
(A) 120
(B) 192
(C) 312
(D) 126

Given the matrices $A = \left( \begin{array} { c c c c } 1 & 3 & 4 & 1 \\ 1 & a & 2 & 2 - a \\ - 1 & 2 & a & a - 2 \end{array} \right)$ and $M = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)$; find: a) ( 1.5 points) Study the rank of A as a function of the real parameter a. b) ( 1 point) Calculate, if possible, the inverse of the matrix AM for the case a $= 0$.