The number of distinct real values of $\lambda$ for which the system of linear equations $$x + y + z = 0$$ $$x + \lambda y + z = 0$$ $$x + y + \lambda z = 0$$ has a non-trivial solution is
(A) 0
(B) 1
(C) 2
(D) 3

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) The minimum value of $\frac { x ^ { 2 } + 2 x + 4 } { x + 2 }$ is
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $( A + B ) ( A - B ) = ( A - B ) ( A + B )$. If $( A B ) ^ { t } = ( - 1 ) ^ { k } A B$, where $( A B ) ^ { t }$ is the transpose of the matrix $A B$, then the possible values of $k$ are
(C) Let $a = \log _ { 3 } \log _ { 3 } 2$. An integer $k$ satisfying $1 < 2 ^ { \left( - k + 3 ^ { - a } \right) } < 2$, must be less than
(D) If $\sin \theta = \cos \varphi$, then the possible values of $\frac { 1 } { \pi } \left( \theta \pm \varphi - \frac { \pi } { 2 } \right)$ are
Column II
(p) 0
(q) 1
(r) 2
(s) 3

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations: $$\begin{array}{r} 3x-y-z=0\\ -3x+z=0\\ -3x+2y+z=0 \end{array}$$ Then the number of such points for which $x^{2}+y^{2}+z^{2}\leq100$ is

Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices in $\mathscr { A }$ is
(A) 12
(B) 6
(C) 9
(D) 3

Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices $A$ in $\mathscr { A }$ for which the system of linear equations
$$A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$$
has a unique solution, is
(A) less than 4
(B) at least 4 but less than 7
(C) at least 7 but less than 10
(D) at least 10

Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices $A$ in $\mathscr { A }$ for which the system of linear equations
$$A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$$
is inconsistent, is
(A) 0
(B) more than 2
(C) 2
(D) 1

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $A \left[ \begin{array} { l } x \\ y \\ z \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$ has exactly two distinct solutions, is
A) 0
B) $2 ^ { 9 } - 1$
C) 168
D) 2

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname { det } ( \mathrm { A } )$ divisible by p is
A) $( p - 1 ) ^ { 2 }$
B) $2 ( p - 1 )$
C) $( p - 1 ) ^ { 2 } + 1$
D) $2 p - 1$

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that the trace of $A$ is not divisible by $p$ but $\operatorname { det } ( A )$ is divisible by $p$ is [Note : The trace of a matrix is the sum of its diagonal entries.]
A) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - \mathrm { p } + 1 \right)$
B) $\mathrm { p } ^ { 3 } - ( \mathrm { p } - 1 ) ^ { 2 }$
C) $( p - 1 ) ^ { 2 }$
D) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - 2 \right)$

Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that $\operatorname { det } ( A )$ is not divisible by $p$ is
A) $2 p ^ { 2 }$
B) $p ^ { 3 } - 5 p$
C) $p ^ { 3 } - 3 p$
D) $p ^ { 3 } - p ^ { 2 }$

Let $\omega$ be the complex number $\cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$. Then the number of distinct complex numbers $z$ satisfying $\left| \begin{array} { c c c } z + 1 & \omega & \omega ^ { 2 } \\ \omega & z + \omega ^ { 2 } & 1 \\ \omega ^ { 2 } & 1 & z + \omega \end{array} \right| = 0$ is equal to

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $$\left[ \begin{array} { c c c } 1 & a & b \\ \omega & 1 & c \\ \omega ^ { 2 } & \omega & 1 \end{array} \right]$$ where each of $a , b$, and $c$ is either $\omega$ or $\omega ^ { 2 }$. Then the number of distinct matrices in the set $S$ is
(A) 2
(B) 6
(C) 4
(D) 8

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = \left[ p _ { i j } \right]$ be a $n \times n$ matrix with $p _ { i j } = \omega ^ { i + j }$. Then $P ^ { 2 } \neq 0$, when $n =$
(A) 57
(B) 55
(C) 58
(D) 56

For $3 \times 3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?
(A) $\quad N ^ { T } M N$ is symmetric or skew symmetric, according as $M$ is symmetric or skew symmetric
(B) $M N - N M$ is skew symmetric for all symmetric matrices $M$ and $N$
(C) $M N$ is symmetric for all symmetric matrices $M$ and $N$
(D) $\quad ( \operatorname { adj } M ) ( \operatorname { adj } N ) = \operatorname { adj } ( M N )$ for all invertible matrices $M$ and $N$

Let $M$ and $N$ be two $3 \times 3$ matrices such that $MN = NM$. Further, if $M \neq N^2$ and $M^2 = N^4$, then
(A) determinant of $\left(M^2 + MN^2\right)$ is 0
(B) there is a $3 \times 3$ non-zero matrix $U$ such that $\left(M^2 + MN^2\right)U$ is the zero matrix
(C) determinant of $\left(M^2 + MN^2\right) \geq 1$
(D) for a $3 \times 3$ matrix $U$, if $\left(M^2 + MN^2\right)U$ equals the zero matrix then $U$ is the zero matrix

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if
(A) the first column of $M$ is the transpose of the second row of $M$
(B) the second row of $M$ is the transpose of the first column of $M$
(C) $M$ is a diagonal matrix with nonzero entries in the main diagonal
(D) the product of entries in the main diagonal of $M$ is not the square of an integer

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
(A) $\quad Y ^ { 3 } Z ^ { 4 } - Z ^ { 4 } Y ^ { 3 }$
(B) $X ^ { 44 } + Y ^ { 44 }$
(C) $X ^ { 4 } Z ^ { 3 } - Z ^ { 3 } X ^ { 4 }$
(D) $X ^ { 23 } + Y ^ { 23 }$

Which of the following values of $\alpha$ satisfy the equation $$\left| \begin{array} { c c c } ( 1 + \alpha ) ^ { 2 } & ( 1 + 2 \alpha ) ^ { 2 } & ( 1 + 3 \alpha ) ^ { 2 } \\ ( 2 + \alpha ) ^ { 2 } & ( 2 + 2 \alpha ) ^ { 2 } & ( 2 + 3 \alpha ) ^ { 2 } \\ ( 3 + \alpha ) ^ { 2 } & ( 3 + 2 \alpha ) ^ { 2 } & ( 3 + 3 \alpha ) ^ { 2 } \end{array} \right| = - 648 \alpha$$?
(A) $-4$
(B) $9$
(C) $-9$
(D) $4$

Let $P = \left[\begin{array}{ccc} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q = [q_{ij}]$ is a matrix such that $PQ = kI$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order 3. If $q_{23} = -\frac{k}{8}$ and $\det(Q) = \frac{k^2}{2}$, then
(A) $\alpha = 0, k = 8$
(B) $4\alpha - k + 8 = 0$
(C) $\det(P\operatorname{adj}(Q)) = 2^9$
(D) $\det(Q\operatorname{adj}(P)) = 2^{13}$

Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?
[A] $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
[B] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[C] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[D] $\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$

How many $3 \times 3$ matrices $M$ with entries from $\{ 0,1,2 \}$ are there, for which the sum of the diagonal entries of $M ^ { T } M$ is 5 ?
[A] 126
[B] 198
[C] 162
[D] 135

Let $$P_1 = I = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_2 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right], \quad P_3 = \left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right],$$ $$P_4 = \left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right], \quad P_5 = \left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right], \quad P_6 = \left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$$ and $X = \sum_{k=1}^{6} P_k \left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?
(A) If $X\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] = \alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha = 30$
(B) $X$ is a symmetric matrix
(C) The sum of diagonal entries of $X$ is 18
(D) $X - 30I$ is an invertible matrix

Let $$M = \left[ \begin{array} { c c } \sin ^ { 4 } \theta & - 1 - \sin ^ { 2 } \theta \\ 1 + \cos ^ { 2 } \theta & \cos ^ { 4 } \theta \end{array} \right] = \alpha I + \beta M ^ { - 1 }$$ where $\alpha = \alpha ( \theta )$ and $\beta = \beta ( \theta )$ are real numbers, and $I$ is the $2 \times 2$ identity matrix. If $\alpha ^ { * }$ is the minimum of the set $\{ \alpha ( \theta ) : \theta \in [ 0,2 \pi ) \}$ and $\beta ^ { * }$ is the minimum of the set $\{ \beta ( \theta ) : \theta \in [ 0,2 \pi ) \}$, then the value of $\alpha ^ { * } + \beta ^ { * }$ is
(A) $- \frac { 37 } { 16 }$
(B) $- \frac { 31 } { 16 }$
(C) $- \frac { 29 } { 16 }$
(D) $- \frac { 17 } { 16 }$

Let $x \in \mathbb{R}$ and let $$P = \left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \quad Q = \left[\begin{array}{ccc}2 & x & x \\ 0 & 4 & 0 \\ x & x & 6\end{array}\right] \text{ and } R = PQP^{-1}$$
Then which of the following options is/are correct?
(A) There exists a real number $x$ such that $PQ = QP$
(B) $\det R = \det\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right] + 8$, for all $x \in \mathbb{R}$
(C) For $x = 0$, if $R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right] = 6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]$, then $a + b = 5$
(D) For $x = 1$, there exists a unit vector $\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ for which $R\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right] = \left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$

If $M = \left( \begin{array} { r r } \frac { 5 } { 2 } & \frac { 3 } { 2 } \\ - \frac { 3 } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$, then which of the following matrices is equal to $M ^ { 2022 }$ ?
(A) $\quad \left( \begin{array} { r r } 3034 & 3033 \\ - 3033 & - 3032 \end{array} \right)$
(B) $\quad \left( \begin{array} { l l } 3034 & - 3033 \\ 3033 & - 3032 \end{array} \right)$
(C) $\quad \left( \begin{array} { r r } 3033 & 3032 \\ - 3032 & - 3031 \end{array} \right)$
(D) $\quad \left( \begin{array} { r r } 3032 & 3031 \\ - 3031 & - 3030 \end{array} \right)$