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 :
An infinitely long wire, located on the $z$-axis, carries a current $I$ along the $+z$-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B} \cdot \overrightarrow{d l}$ along a straight line from the point $(-\sqrt{3} a, a, 0)$ to $(a, a, 0)$ is given by
[ $\mu_0$ is the magnetic permeability of free space.]
Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by
[ $\varepsilon_0$ is the permittivity of free space.]