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
A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0{ }^\beta h^\gamma c^\delta$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by :