Two soap bubbles \(A\) and \(B\) are kept in a closed chamber where the air is maintained at pressure \(8 \mathrm{Nm}^{-2}\). The radii of bubbles \(A\) and \(B\) are \(2 \mathrm{~cm}\), respectively. Surface tension of the soap-water used to make bubbles is \(0.04\) \(\mathrm{Nm}^{-1}\). Find the ratio \(\frac{n_B}{n_A}\), where \(n_A\) and \(n_B\) are the number of moles of air in bubbles \(A\) and \(B\), respectively. [Neglect the effect of gravity]

A moving coil galvanometer has 50 turns and each turn has an area $2\times {10}^{-4} {\mathrm{m}}^2$ . The magnetic field produced by the magnet inside the galvanometer is $0.02 \mathrm{T}$ . The torsional constant of the suspension wire is ${10}^{-4} \mathrm{N}\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{rad}}^{-1}$ . When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $0.2 \mathrm{rad}$ . The resistance of the coil of the galvanometer is $50 \mathrm{\Omega }$ . This galvanometer is to be converted into an ammeter capable of measuring current in the range $0-1.0 \mathrm{A}$ . For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is __________.

An audio transmitter \((T)\) and a receiver \((R)\) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_0=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) ) is \(\qquad\)

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.
 

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

If the tension in each string is ${\mathrm{T}}_0,$ the correct match for the highest fundamental frequency in ${\mathrm{f}}_0$ units will be,

Parallel rays of light of intensity $\mathrm{I}=912\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}$ are incident on a spherical black body kept in surroundings of temperature 300 K. Take Stefan-Boltzmann constant $\mathrm{\sigma }=5.7\times {10}^{-8}\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}\mathrm{ }{\mathrm{K}}^{-4}$ and assume that the energy exchange with the surroundings is only through radiation. The final steady state temperature of the black body is close to

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A small block of mass \(M\) moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from \(60^{\circ}\) to \(30^{\circ}\) at point \(B\). The block is initially at rest at \(A\). Assume that collisions between the block and the incline are totally inelastic \(\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)\)

Question:
The speed of the block at point \(B\) immediately after it strikes the second incline is

A spherical soap bubble inside an air chamber at pressure \(P_0=10^5 \mathrm{~Pa}\) has a certain radius so that the excess pressure inside the bubble is \(\Delta P=144 \mathrm{~Pa}\). Now, the chamber pressure is reduced to \(8 P_0 / 27\) so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure \(\Delta P\) in both the cases to be much smaller than the chamber pressure. The new excess pressure \(\Delta P\) in \(\mathrm{Pa}\) is ________ .

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures \(2 T\) and \(3 T\) respectively. The temperature of the middle (i.e. second) plate under steady state condition is

A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=\mathrm{}\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case will the particle move in a straight line with a constant velocity?

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Read the following passage and answer the questions.
Let \(A B C D\) be a square of side length 2 units. \(C_2\) is the circle through vertices \(A, B, C, D\) and \(C_1\) is the circle touching all the sides of square \(A B C D\). \(L\) is the line through \(A\).Question:
If \(P\) is a point on \(C_1\) and \(Q\) is a point on \(C_2\), then \(\frac{P A^2+P B^2+P C^2+P D^2}{Q A^2+Q B^2+Q C^2+Q D^2}\) is equal to

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

Among \(\left[ I _3\right]^{+},\left[ SiO _4\right]^{4-}, SO _2 Cl _2, XeF _2, SF _4, ClF _3,\) \(Ni ( CO )_4, XeO _2 F_2,\left[ PtCl _4\right]^{2-}, XeF _4\), and \(SOCl _2\), the total number of species having \(sp ^3\) hybridised central atom is

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Let \(S=S_{1} \cap S_{2} \cap S_{3}\), where

\(S_{1}=\{z \in \mathbb{C}:|z|<4\}, \quad S_{2}=\left\{z \in \mathbb{C}: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\}\) and \(S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}\).


Question:

Area of \(S=\)