Let $X$ be a random variable, and let $P(X=x)$ denote the probability that $X$ takes the value $x$. Suppose that the points $(x, P(X=x)), x=0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P(X=x)=0$ for all $x \in \mathbb{R}-\{0,1,2,3,4\}$. If the mean of $X$ is $\frac{5}{2}$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is _____________.
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 :