A massless rod \(B D\) is suspended by two identical massless strings \(A B\) and \(C D\) of equal lengths. A block of mass ' \(m\) ' is suspended point \(P\) such that \(B P\) is equal to ' \(x\) ', if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of \(x\) is

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
The buoyancy force acting on the gas bubble is (Assume \(R\) is the universal gas constant)

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The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

When \(d \approx a\) but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height \(h\) above the loop. In that case

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\mathrm{\lambda }$ are kept parallel to each other. In their resulting electric field, point charges $\mathrm{q}$ and $-\mathrm{q}$ are kept in equilibrium between them. The point charges are confined to move in the $\mathrm{x}$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statements(s) is (are)

A double slit setup is shown in the figure. One of the slits is in medium $2$ of refractive index $n_2$. The other slit is at the interface of this medium with another medium $1$ of refractive index $n_1\left(\ne n_2\right)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium $1$. A detector is placed in medium $2$ at a large distance from the slits, and at an angle $\theta$ from the line joining them, so that $\theta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.
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Which of the following statement(s) is(are) correct?

One mole of a monatomic ideal gas is taken along two cyclic processes $\text{E}\to \text{F}\to \text{G}\to \text{E} and \text{E}\to \text{F}\to \text{H}\to \text{E}$ as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic.

Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using the codes given below the lists.
$\begin{array}{ccccc}& \text{List I}& & & \text{List II}\\ \text{A.}& \text{G}\mathrm{\to }\text{E }& & \mathrm{P}\text{.}& 16\mathrm{0 }{\text{P}}_0{\text{V}}_0\text{ ln}2\\ \text{B.}& \text{G}\mathrm{\to }\text{H }& & \mathrm{Q}\text{.}& 36{\text{ P}}_0{\text{V}}_0\\ \text{C.}& \text{F}\mathrm{\to }\text{H }& & \mathrm{R}\text{.}& 24{\text{ P}}_0{\text{V}}_0\\ \text{D.}& \text{F}\mathrm{\to }\text{G }& & \mathrm{S}\text{.}& 31{\text{ P}}_0{\text{V}}_0\end{array}$

A series \(R-C\) combination is connected to an \(\mathrm{AC}\) voltage of angular frequency \(\omega=500 \mathrm{rad} / \mathrm{s}\). If the impedance of the \(R-C\) circuit is \(R \sqrt{1.25}\), the time constant (in millisecond) of the circuit is

Match the chemical substance in Column I with type of polymers/type of bond in Column II. Indicate your answer by darkening the appropriate bubbles of \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. Cellulose	p. Natural polymer
B. Nylon-6,6	q. Synthetic polymer
C. Protein	r. Amide linkage
D. Sucrose	s. Glycoside linkage

Extra pure \(\mathrm{N}_2\) can be obtained by heating

If \(f(x)=\int_{0}^{x} e^{t^{2}}(t-2)(t-3) d t\) for all \(x \in(0, \infty)\), then

Let S be the set of all twice differentiable functions f from $\mathrm{ℝ}$ to $\mathrm{ℝ}$ such that $\frac{d^{2}f}{d{x}^2}\left(x\right)>0$ for all $x\in \left(-1,1\right)$. For $f\in S$, let $X_f$ be the number of points $x\in \left(-1,1\right)$ for which $f\left(x\right)=x$. Then which of the following statements is(are) true?

The number of stereoisomers obtained by bromination of trans-2-butene is

Two spherical stars $A$ and $B$ have densities ${\rho }_A$ and ${\rho }_B$, respectively. $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B=2M_A$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains ${\rho }_A$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being ${\rho }_A$. If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_B}{v_A}=\sqrt{\frac{10n}{{15}^{{\displaystyle \frac{1}{3}}}}}$. The value of $n$ is