If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ (where $bc \neq 0$) satisfies the equation $x^2 + k = 0$,then:

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix}$. Let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be the roots of $\det(A - \lambda I_{3}) = 0$, where $I_{3}$ denotes the identity matrix. If $\lambda_{1} + \lambda_{2} + \lambda_{3} = \sqrt{2} + 1$, then the set of possible values of $t$ for $-\pi \leq t < \pi$ is:

If $P$ is a square matrix with $P^2=P$ and if $I$ is the unit matrix of the same order as of $P$,then $(P+I)^4=$

Suppose $\det \begin{bmatrix} \sum_{k=0}^n k & \sum_{k=0}^n {^nC_k} k^2 \\ \sum_{k=0}^n {^nC_k} k & \sum_{k=0}^n {^nC_k} 3^k \end{bmatrix} = 0$ holds for some positive integer $n$. Then $\sum_{k=0}^n \frac{{^nC_k}}{k+1}$ equals

Match the Statements / Expressions in Column $I$ with the Statements / Expressions in Column $II$.

Column $I$	Column $II$
$(A)$ The minimum value of $\frac{x^2+2x+4}{x+2}$ for $x > -2$ is	$(p)$ $0$
$(B)$ 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 $(AB)^t=(-1)^k AB$,where $(AB)^t$ is the transpose of the matrix $AB$,then the possible values of $k$ are	$(q)$ $1$
$(C)$ Let $a=\log_3 \log_3 2$. An integer $k$ satisfying $1 < 2^{(-k+3^{-a})} < 2$,must be less than	$(r)$ $2$
$(D)$ If $\sin \theta = \cos \phi$,then the possible values of $\frac{1}{\pi}(\theta \pm \phi - \frac{\pi}{2})$ are	$(s)$ $3$

Let $z = \frac{-1 + \sqrt{3}i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1, 2, 3\}$. Let $P = \begin{bmatrix} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{bmatrix}$ and $I$ be the identity matrix of order $2$. Then the total number of ordered pairs $(r, s)$ for which $P^2 = -I$ is