Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M=\left(a_{i j}\right)_{3 \times 3}$, define $R_i=a_{i 1}+a_{i 2}+a_{i 3}$ and $C_j=a_{1 j}+a_{2 j}+a_{3 j}$ for $i=1,2,3$ and $j=1,2,3$.

Match each entry in List-I to the correct entry in List-II.

List-I	List-II
(P) The number of matrices $ M = (a_{ij})_{3x3} $ with all entries in $ T $ such that $ R_i = C_j = 0 $ for all $ i, j $, is	(1) 1
(Q) The number of symmetric matrices $ M = (a_{ij})_{3x3} $ with all entries in $ T $ such that $ C_j = 0 $ for all $ j $, is	(2) 12
(R) Let $ M = (a_{ij})_{3x3} $ be a skew symmetric matrix such that $ a_{ij} \in T $ for $ i > j $. Then the number of elements in the set $ \left\{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} : x, y, z \in \mathbb{R}, M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_{12} \\ 0 \\ a_{13} \end{pmatrix} \right\} $ is	(3) infinite
(S) Let $ M = (a_{ij})_{3x3} $ be a matrix with all entries in $ T $ such that $ R_i = 0 $ for all $ i $. Then the absolute value of the determinant of $ M $ is	(4) 6

The correct option is

Let the straight line $y=2 x$ touch a circle with center $(0, \alpha), \alpha>0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1 B_1$ is a diameter of the circle. Let $\alpha+r=5+\sqrt{5}$.

Match each entry in List-I to the correct entry in List-II.

List-I	List-II
(P) $\alpha$ equals	(1) $(-2, 4)$
(Q) $r$ equals	(2) $\sqrt{5}$
(R) $A_1$ equals	(3) $(-2, 6)$
(S) $B_1$ equals	(4) $5$
	(5) $(2, 4)$

The correct option is

Let $\gamma \in \mathbb{R}$ be such that the lines $L_1: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O=(0,0,0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_1$ and $L_2$.

Match each entry in List-I to the correct entry in List-II.

List-I	List-II
(P) $\gamma$ equals	(1) $-\hat{i} - \hat{j} + \hat{k}$
(Q) A possible choice for $\hat{n}$ is	(2) $\sqrt{\frac{3}{2}}$
(R) $\overrightarrow{OR_1}$ equals	(3) $1$
(S) A possible value of $\overrightarrow{OR_1} \cdot \hat{n}$ is	(4) $\frac{1}{\sqrt{6}} \hat{i} - \frac{2}{\sqrt{6}} \hat{j} + \frac{1}{\sqrt{6}} \hat{k}$
	(5) $\sqrt{\frac{2}{3}}$

The correct option is :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by

$$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & \text { otherwise } .\end{cases}\right. $$

Let $a, b, c, d \in \mathbb{R}$. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ by

$$ h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R} . $$

Match each entry in List-I to the correct entry in List-II.

List-I	List-II
(P) If $a = 0$, $b = 1$, $c = 0$, and $d = 0$, then	(1) $h$ is one-one.
(Q) If $a = 1$, $b = 0$, $c = 0$, and $d = 0$, then	(2) $h$ is onto.
(R) If $a = 0$, $b = 0$, $c = 1$, and $d = 0$, then	(3) $h$ is differentiable on $\mathbb{R}$.
(S) If $a = 0$, $b = 0$, $c = 0$, and $d = 1$, then	(4) the range of $h$ is $[0, 1]$.
	(5) the range of $h$ is $\{0, 1\}$.

The correct option is