An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding P-V diagrams in column 3 of the table. Consider only the path from state 1 to state 2. W denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic process. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is n.

Column – 1	Column – 2	Column – 3
(I) \(W_{1 \rightarrow 2}=\frac{1}{\gamma-1}\) \(\left(P_2 V_2-P_1 V_1\right)\)	(i) Isothermal	(P)
(II) \(W_{1 \rightarrow 2}=\) \(-P V_2+P V_1\)	(ii) Isochoric	(Q)
(III) \(W_{1 \rightarrow 2}=0\)	(iii) Isobaric	(R)
(IV) \(W_{1 \rightarrow 2}=\) \(-n R T \ln \left(\frac{V_2}{V_1}\right)\)	(iv) Adiabatic	(S)

Which one of the following options is the correct combination?

There is a rectangular plate of mass \(M \mathrm{~kg}\) of dimensions \((a \times b)\). The plate is held in horizontal position by striking \(n\) small balls each of mass \(m\) per unit area per unit time. These are striking in the shaded half region of the plate. The balls are colliding elastically with velocity \(v\). What is \(v\) ? It is given \(n=100, M=3 \mathrm{~kg}, m=0.01 \mathrm{~kg}\); \(b=2 \mathrm{~m} ; a=1 \mathrm{~m} ; g=10 \mathrm{~m} / \mathrm{s}^2\).

Two parallel wires in the plane of the paper are distance ${\mathrm{X}}_0$ apart. A point charge is moving with speed u between the wires in the same plane at a distance ${\mathrm{X}}_1$ from one of the wires. When the wires carry current of magnitude $\mathrm{ }\mathrm{I}$  in the same direction, the radius of curvature of the path of the point charge is ${\mathrm{R}}_1$ . In contrast, if the currents $\mathrm{ }\mathrm{I}$ in the two wires have directions opposite to each other, the radius of curvature of the path is ${\mathrm{R}}_2$ . If $\frac{{\mathrm{X}}_0}{{\mathrm{X}}_1}=3$ , value of $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}$ is

Two vectors $\overrightarrow{A} \mathrm{a}\mathrm{n}\mathrm{d} \overrightarrow{B}$ are defined as $\overrightarrow{A}=a\widehat{i} and \overrightarrow{B}=a\left(\cos \omega t \widehat{i}+\sin \omega t\widehat{j}\right)$, where a is a constant and $\omega =\frac{\pi }{6} rad s^{-1}.$ If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{3}\left|\overrightarrow{A}-\overrightarrow{B}\right|$ at time $t=\tau$ for the first time, the value of $\tau$ in seconds, 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:
Which force causes the train to elevate up?

An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true?

A thin and uniform rod of mass $\mathrm{M}$ and length $\mathrm{L}$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle ${60}^{\mathrm{o}}$ with vertical?
[ $\mathrm{g}$ is the acceleration due to gravity]

When benzene sulphonic acid and \(p\)-nitrophenol are treated with \(\mathrm{NaHCO}_3\), the gases released respectively are

Among P, Q, R and S, the aromatic compound(s) is/are

A solid sphere of radius \(R\) has a charge \(Q\) distributed in its volume with a charge density \(\rho=k r^a\), where \(k\) and \(a\) are constants and \(r\) is the distance from its centre. If the electric field at \(r=\frac{R}{2}\) is \(\frac{1}{8}\) times that at \(r=R\), find the value of \(a\).

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An aqueous solution of a mixture of two inorganic salts, when treated with dilute \(H C\) ?, gave a precipitate \((\boldsymbol{P})\) and a filtrate \((\boldsymbol{Q})\). The precipitate \(\boldsymbol{P}\) was found to dissolve in hot water. The filtrate \((\boldsymbol{Q})\) remained unchanged, when treated with \(\mathrm{H}_{2} \mathrm{~S}\) in a dilute mineral acid medium. However, it gave a precipitate \((\boldsymbol{R})\) with \(H_{2} S\) in an ammoniacal medium. The precipitate \(\boldsymbol{R}\) gave a coloured solution \((\boldsymbol{S})\), when treated with \(\mathrm{H}_{2} \mathrm{O}_{2}\) in an aqueous \(\mathrm{NaOH}\) medium.
Question:
The coloured solution \(\boldsymbol{S}\) contains

The correct statement(s) regarding,
(i) $\mathrm{H}\mathrm{C}\mathrm{l}\mathrm{O}$
(ii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_2$
(iii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_3$ and
(iv) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_4$ is /are

Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s) is (are)