Let $A = \begin{bmatrix} 3 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 3 \end{bmatrix}$. Then, the roots of the equation $\operatorname{det}(A - \lambda I_{3}) = 0$ (where $I_{3}$ is the identity matrix of order $3$) are

If each element of a second order determinant is either zero or one,what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently,each value being assumed with probability $\frac{1}{2}$).

If $A = \begin{bmatrix} 1 & a & 3 \\ b & 2 & c \\ 3 & d & 4 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 5 & b \\ -5 & 0 & -7 \\ 6 & c & 0 \end{bmatrix}$ is a skew-symmetric matrix, then $AB = $

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. $\operatorname{Tr}(A)$ denotes the sum of diagonal entries of $A$. Assume that $A^2=I$.
Statement $I$: If $A \neq I$ and $A \neq -I$,then $\operatorname{det}(A) = -1$.
Statement $II$: If $A \neq I$ and $A \neq -I$,then $\operatorname{Tr}(A) \neq 0$.

List $I$	List $II$
$P.$ Let $y(x)=\cos \left(3 \cos ^{-1} x\right), x \in[-1,1], x \neq \pm \frac{\sqrt{3}}{2}$. Then $\frac{1}{y(x)}\left\{\left(x^2-1\right) \frac{d^2 y(x)}{d x^2}+x \frac{d y(x)}{d x}\right\}$ equals	$1. \ 1$
$Q.$ Let $A_1, A_2, \ldots, A_n(n>2)$ be the vertices of a regular polygon of $n$ sides with its centre at the origin. Let $\vec{a}_k$ be the position vector of the point $A_k, k=1,2, \ldots, n$. If $\left|\sum_{k=1}^{n-1}\left(\vec{a}_k \times \vec{a}_{k+1}\right)\right|=\left|\sum_{k=1}^{n-1}\left(\vec{a}_k \cdot \vec{a}_{k+1}\right)\right|$,then the minimum value of $n$ is	$2. \ 2$
$R.$ If the normal from the point $P(h, 1)$ on the ellipse $\frac{x^2}{6}+\frac{y^2}{3}=1$ is perpendicular to the line $x+y=8$,then the value of $h$ is	$3. \ 8$
$S.$ Number of positive solutions satisfying the equation $\tan ^{-1}\left(\frac{1}{2 x+1}\right)+\tan ^{-1}\left(\frac{1}{4 x+1}\right)=\tan ^{-1}\left(\frac{2}{x^2}\right)$ is	$4. \ 9$

Codes: $P \quad Q \quad R \quad S$

The number of solutions of the matrix equation $X^2 = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}$ is