A light disc made of aluminum (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

Statement 1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and Statement 2 The electrical potential of a sphere of radius \(R\) with charge \(Q\) uniformly distributed on the surface is given by \(\frac{Q}{4 \pi \varepsilon_0 R}\).

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.

The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
If the total energy of the particle is \(E\), it will perform periodic motion only if

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(R_{100}, R_{60}\) and \(R_{40}\), respectively, the relation between these resistances is

List-$\mathrm{I}$ shows different radioactive decay processes and List-$\mathrm{II}$ provides possible emitted particles. Match each entry in List-$\mathrm{I}$ with an appropriate entry from List-$\mathrm{II}$, and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	${{}^{238}}_{92}U{{\to }^{234}}_{91}Pa$	$1$	one $\alpha$ particle and one ${\beta }^{+}$ particle
$\mathrm{Q}$	${{}^{214}}_{82}Pb{{\to }^{210}}_{82}Pb$	$2$	three ${\beta }^{-}$ particles and one $\alpha$ particle
$\mathrm{R}$	${{}^{210}}_{81}Tl{{\to }^{206}}_{82}Pb$	$3$	two ${\beta }^{-}$ particles and one $\alpha$ particle
$\mathrm{S}$	${{}^{228}}_{91}Pa{{\to }^{224}}_{88}Ra$	$4$	one $\alpha$ particle and one ${\beta }^{-}$ particle
		$5$	one $\alpha$ particle and two ${\beta }^{+}$ particles

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1 \mathrm{m}$. It is maintained at temperature $360 \mathrm{K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{m}$ and height $h=9 \mathrm{m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density ${\rho }_a=1.2 \mathrm{kg} {\mathrm{m}}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and $\pi =3.14$]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm} {\mathrm{s}}^{-1}$.

Consider a hydrogen atom with its electron in the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is $\left(\mathrm{h}\mathrm{c}=1242\mathrm{ }\mathrm{e}\mathrm{V}\mathrm{ }\mathrm{n}\mathrm{m}\right)$

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $\mathrm{V}\mathrm{ }\mathrm{m}{\mathrm{m}}^3$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is 10mm, then the value of $\frac{\mathrm{V}}{250\mathrm{\pi }}\mathrm{ }$ is

The total number of lone pairs of electrons in ${\mathrm{N}}_2{\mathrm{O}}_3$ is

For a prism of prism angle $\theta =60°$, the refractive indices of the left half and the right half are, respectively, $n_1$ and $n_2$, $\left(n_2\ge n_1\right)$ as shown in the figure. The angle of incidence $i$ is chosen such that the incident light rays will have minimum deviation if $n_1=n_2=n=1.5.$ For the case of unequal refractive indices, $n_1=n$ and $n_2=n+∆n$ (where $∆n<<n$), the angle of emergence $e=i+∆e$. Which of the following statement(s) is(are) correct?

Consider the following compounds in the liquid form:
\(O_2, HF, H_2 O, NH_3, H_2 O_2, CCl_4, CHCl_3,\) \(C_6 H_6, C_6 H_5 Cl .\)
When a charged comb is brought near their flowing stream, how many of them show deflection as per the following figure?

In the given circuit, the \(\mathrm{AC}\) source has \(\omega=100 \mathrm{rad} / \mathrm{s}\). Considering the inductor and capacitor to be ideal, the correct choice(s) is (are)

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\left\{x:f\left(x\right)=0\right\}, Y=\left\{x:f^{\text{'}}\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\}, W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
List- $\mathrm{I}$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

	List-  I		List-  II
$\left(I\right)$	$X$	$\left(P\right)$	$\supseteq \left\{\frac{\pi }{2},\frac{3\pi }{2}, 4\pi , 7\pi \right\}$
$\left(II\right)$	$Y$	$\left(Q\right)$	an arithmetic progression
$\left(III\right)$	$Z$	$\left(R\right)$	NOT an arithmetic progression
$\left(IV\right)$	$W$	$\left(S\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\right\}$
		$\left(T\right)$	$\supseteq \left\{\frac{\pi }{3},\frac{2\pi }{3}, \pi \right\}$
		$\left(U\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{3\pi }{4}\right\}$

Which of the following is the only correct combination?