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Two trains \(A\) and \(B\) are moving with speeds \(20 \mathrm{~m} / \mathrm{s}\) and \(30 \mathrm{~m} / \mathrm{s}\), respectively in the same direction on the same straight track, with \(B\) ahead of \(A\). The engines are at the front ends. The engine of train \(A\) blows a long whistle.
Assume that the sound of the whistle is composed of components varying in frequency from \(f_1=800 \mathrm{~Hz}\) to \(f_2=1120 \mathrm{~Hz}\), as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus \(320 \mathrm{~Hz}\). The speed of sound in still air is \(340 \mathrm{~m} / \mathrm{s}\).
Question:
The spread of frequency as observed by the passengers in train \(B\) is

Two bodies, each of mass $M$, are kept fixed with a separation of $2L$. A particle of mass $m$ is projected from the midpoint of the line joining their centers, perpendicular to the line. The gravitational constant is $G$. The correct statement (s) is (are) :

A conducting solid sphere of radius \(R\) and mass \(M\) carries a charge \(Q\). The sphere is rotating about an axis passing through its center with a uniform angular speed \(\omega\). The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as \(\alpha \frac{Q}{2 M}\). The value of \(\alpha\) is _______ .

Four point charges, each of \(+q\), are rigidly fixed at the four corners of a square planar soap film of side \(a\). The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and \(a=k\left[\frac{q^2}{\gamma}\right]^{1 / N}\), where \(k\) is a constant. Then, \(N\) is

One mole of a monatomic ideal gas undergoes the cyclic process \(\mathrm{J} \rightarrow \mathrm{K} \rightarrow \mathrm{L} \rightarrow \mathrm{M} \rightarrow \mathrm{J}\), as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
[\(\mathcal{R}\) is the gas constant.]

List-I	List-II
(P) Work done in the complete cyclic process	(1) \(R T_0-4 R T_0 \ln 2\)
(Q) Change in the internal energy of the gas in the process JK	(2) \(0\)
(R) Heat given to the gas in the process KL	(3) \(3 K T_0\)
(S) Change in the internal energy of the gas in the process MJ	(4) \(-2 R T_0 \ln 2\)
	\((5)-3 R T_0 \ln 2\)

There are two vernier calipers both of which have $1\mathrm{cm}$ divided into $10$ equal divisions on the main scale. The Vernier scale of one of the calipers $\left(C_1\right)$ has $10$ equal divisions that correspond to $9$ main scale divisions. The Vernier scale of the other caliper $\left(C_2\right)$ has $10$ equal divisions that correspond to $11$ main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in $\mathrm{cm}$) by calipers $C_1$ and $C_2$ respectively, are

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch S₁ is pressed first to fully charge the capacitor C₁ and the released. The switch S₂ is then pressed to charge the capacitor C₂. After some time, S₂ is released and then is S₃ pressed. After some time,

A Young's double slit interference arrangement with slits ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ is immersed in water (refractive index $=\frac{4}{3}$ ) as shown in the figure. The positions of maxima on the surface of water are given by ${\mathrm{x}}^2={\mathrm{p}}^2{\mathrm{m}}^2{\mathrm{\lambda }}^2-{\mathrm{d}}^2$ , where $\mathrm{\lambda }$ is the wavelength of light in air (refractive index = 1), $2\mathrm{d}$ is the separation between the slits and m is an integer. The value of p is

Consider a thin square sheet of side \(L\) and thickness \(t\), made of a material of resistivity \(\rho\). The resistance between two opposite faces, shown by the shaded areas in the figure is

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Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
What is the advantage of this system?

An extended object is placed at point $O$, $10 \mathrm{cm}$ in front of a convex lens $L_1$ and a concave lens $L_2$ is placed $10 \mathrm{cm}$ behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are $20 \mathrm{cm}$. The refractive index of both the lenses is $1.5$. The total magnification of this lens system is:

A football of radius $R$ is kept on a hole of radius $r(r<R)$ made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $\theta$ from the horizontal as shown in the figure below. The maximum value of $\theta$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is