Let $A$ and $B$ be two $3 \times 3$ matrices such that $( A + B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$. Which of the following must be true?
(A) $A$ and $B$ are zero matrices.
(B) $A B$ is the zero matrix.
(C) $( A - B ) ^ { 2 } = A ^ { 2 } - B ^ { 2 }$
(D) $( A - B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$

Consider the system of equations $$\begin{aligned} & x - 2 y + 3 z = - 1 \\ & - x + y - 2 z = k \\ & x - 3 y + 4 z = 1 . \end{aligned}$$ STATEMENT-1 : The system of equations has no solution for $k \neq 3$. and STATEMENT-2 : The determinant $\left| \begin{array} { c c c } 1 & 3 & - 1 \\ - 1 & - 2 & k \\ 1 & 4 & 1 \end{array} \right| \neq 0$, for $k \neq 3$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let k be a positive real number and let
$$A = \left[ \begin{array} { c c c } 2 k - 1 & 2 \sqrt { k } & 2 \sqrt { k } \\ 2 \sqrt { k } & 1 & - 2 k \\ - 2 \sqrt { k } & 2 k & - 1 \end{array} \right] \text { and } B = \left[ \begin{array} { c c c } 0 & 2 k - 1 & \sqrt { k } \\ 1 - 2 k & 0 & 2 \sqrt { k } \\ - \sqrt { k } & - 2 \sqrt { k } & 0 \end{array} \right]$$
If $\operatorname { det } ( \operatorname { adj } \mathrm { A } ) + \operatorname { det } ( \operatorname { adj } \mathrm { B } ) = 10 ^ { 6 }$, then $[ \mathrm { k } ]$ is equal to [Note : adj M denotes the adjoint of a square matrix M and $[ \mathrm { k } ]$ denotes the largest integer less than or equal to k].

Let $P = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array} \right]$ and $I$ be the identity matrix of order 3. If $Q = \left[ q _ { i j } \right]$ is a matrix such that $P ^ { 50 } - Q = I$, then $\frac { q _ { 31 } + q _ { 32 } } { q _ { 21 } }$ equals
(A) 52
(B) 103
(C) 201
(D) 205

The total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{array}\right| = 10$ is

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{ - 1,0,1 \}$. Then, the maximum possible value of the determinant of $P$ is $\_\_\_\_$ .

Let $$M = \left[ \begin{array} { l l l } 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{array} \right] \quad \text { and } \quad \operatorname { adj } M = \left[ \begin{array} { r r r } - 1 & 1 & - 1 \\ 8 & - 6 & 2 \\ - 5 & 3 & - 1 \end{array} \right]$$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?
(A) $a + b = 3$
(B) $\quad ( \operatorname { adj } M ) ^ { - 1 } + \operatorname { adj } M ^ { - 1 } = - M$
(C) $\operatorname { det } \left( \operatorname { adj } M ^ { 2 } \right) = 81$
(D) If $M \left[ \begin{array} { l } \alpha \\ \beta \\ \gamma \end{array} \right] = \left[ \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right]$, then $\alpha - \beta + \gamma = 3$

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is $-18$, then the value of the determinant of $A$ is $\_\_\_\_$

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M ^ { - 1 } = \operatorname { adj } ( \operatorname { adj } M )$, then which of the following statements is/are ALWAYS TRUE?
(A) $M = I$
(B) $\operatorname { det } M = 1$
(C) $M ^ { 2 } = I$
(D) $( \operatorname { adj } M ) ^ { 2 } = I$

Let $M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix}$ and $\text{adj}(M) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
where $a$ and $b$ are real numbers. Which of the following statements is(are) TRUE?
(A) $(a+b)^2 = 9$
(B) $\det(\text{adj}(M^2)) = 81$
(C) $\text{adj}(\text{adj}(M)) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
(D) $\det(\text{adj}(2M)) = 2^8$

Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $|M|$ is ____.

Let $\alpha$, $\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x + 2y + 3z = \alpha$$ $$4x + 5y + 6z = \beta$$ $$7x + 8y + 9z = \gamma - 1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M = \begin{pmatrix} \alpha & 1 & 0 \\ \beta & 2 & 1 \\ \gamma & 1 & 0 \end{pmatrix}.$$ Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0, 1, 0)$ from the plane $P$.
The value of $D$ is ____.

Let $\beta$ be a real number. Consider the matrix
$$A = \left( \begin{array} { c c c } \beta & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{array} \right)$$
If $A ^ { 7 } - ( \beta - 1 ) A ^ { 6 } - \beta A ^ { 5 }$ is a singular matrix, then the value of $9 \beta$ is $\_\_\_\_$ .

Consider the matrix
$$P = \left( \begin{array} { l l l } 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array} \right)$$
Let the transpose of a matrix $X$ be denoted by $X ^ { T }$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that
$$Q ^ { - 1 } = Q ^ { T } \text { and } P Q = Q P$$
is

(A)

32

(B)

8

(C)

16

(D)

24

Let $A = \left[ \begin{array} { c c c } 5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5 \end{array} \right]$. If $\left| A ^ { 2 } \right| = 25$, then $| \alpha |$ equals
(1) $5 ^ { 2 }$
(2) 1
(3) $1 / 5$
(4) 5

If $D = \left| \begin{array} { c c c } 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{array} \right|$ for $x \neq 0 , y \neq 0$ then $D$ is
(1) divisible by neither $x$ nor $y$
(2) divisible by both $x$ and $y$
(3) divisible by $x$ but not $y$
(4) divisible by $y$ but not $x$

If $P = \left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to
(1) 5
(2) 0
(3) 4
(4) 11

If $B$ is a $3 \times 3$ matrix such that $B ^ { 2 } = 0$, then det. $\left[ ( I + B ) ^ { 50 } - 50 B \right]$ is equal to:
(1) 1
(2) 2
(3) 3
(4) 50

The set of all values of $\lambda$ for which the system of linear equations: $2x_1 - 2x_2 + x_3 = \lambda x_1$ $2x_1 - 3x_2 + 2x_3 = \lambda x_2$ $-x_1 + 2x_2 = \lambda x_3$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements

The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:
(1) 0
(2) 2
(3) 1
(4) 3

If $A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to:
(1) $(2, -1)$
(2) $(-2, 1)$
(3) $(2, 1)$
(4) $(-2, -1)$

$A = \left[ \begin{array} { c c c } 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array} \right]$ is a matrix satisfying the equation $A A ^ { T } = 9 I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $( a , b )$ is equal to
(1) $( - 2 , - 1 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( 2,1 )$

The set of all values of $\lambda$ for which the system of linear equations: $2 x _ { 1 } - 2 x _ { 2 } + x _ { 3 } = \lambda x _ { 1 }$ $2 x _ { 1 } - 3 x _ { 2 } + 2 x _ { 3 } = \lambda x _ { 2 }$ $- x _ { 1 } + 2 x _ { 2 } = \lambda x _ { 3 }$ has a non-trivial solution,
(1) Contains more than two elements.
(2) Is an empty set.
(3) Is a singleton.
(4) Contains two elements.

The number of real values of $\lambda$ for which the system of linear equations $$2 x + 4 y - \lambda z = 0$$ $$4 x + \lambda y + 2 z = 0$$ $$\lambda x + 2 y + 2 z = 0$$ has infinitely many solutions, is:
(1) 0
(2) 1
(3) 2
(4) 3

The mean and variance for seven observations are 8 and 16 respectively. If 5 of the observations are $2, 4, 10, 12, 14$, then the product of the remaining two observations is
(1) 48
(2) 45
(3) 49
(4) 40