Two concentric circular loops, one of radius $R$ and the other of radius $2R$, lie in the $xy$-plane with the origin as their common center, as shown in the figure. The smaller loop carries current $I_1$ in the anti-clockwise direction and the larger loop carries current $I_2$ in the clockwise direction, with $I_2>2I_1. \overrightarrow{B}\left(x,y\right)$ denotes the magnetic field at a point $\left(x,y\right)$ in the $xy$-plane. Which of the following statement(s) is (are) correct?

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question :

Which one of the following statements is correct?

Two loudspeakers M and N are located 20m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let $\upsilon \left(t\right)$ represent the beat frequency measured by a person sitting in the car at time t. Let ${\nu }_p, {\nu }_Q and {\nu }_R$ be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is $330 m{s}^{-1}.$ Which of the following statement(s) is(are) true regarding the sound heard by the person ?

If the wavelength of the \(n^{\text {th }}\) line of Lyman series is equal to the de-Broglie wavelength of electron in initial orbit of a hydrogen like element \((Z=11)\). Find the value of \(n\).

A lamina is made by removing a small disc of diameter \(2 R\) from a bigger disc of uniform mass density and radius \(2 R\), as shown in the figure. The moment of inertia of this lamina about axes passing though \(O\) and \(P\) is \(I_{O}\) and \(I_{P}\) respectively. Both these axes are perpendicular to the plane of the lamina. The ratio \(I_{P} / I_{O}\) to the nearest integer is

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

Two rectangular blocks, having identical dimensions, can be arranged either in configuration$\text{I}$ or in configuration $\text{I}\text{I}$ as shown in the figure. One of the blocks has thermal conductivity $\text{K}$ and the other $\text{2 K}$. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes $\text{9 s}$ to transport a certain amount of heat from the hot end to cold end in the configuration $\text{I}$. The time to transport the same amount of heat in the configuration $\text{II}$ is

The volume (in \(\mathrm{mL}\) ) of \(0.1 \mathrm{M} \mathrm{AgNO}_3\) required for complete precipitation of chloride ions present in \(30 \mathrm{~mL}\) of \(0.01 \mathrm{M}\) solution of \(\left[\mathrm{Cr}\left(\mathrm{H}_2 \mathrm{O}_{)_5} \mathrm{Cl}\right] \mathrm{Cl}_2\right.\), as silver chloride is close to

For the reaction, \(\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)\), the plot of \(\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}\) versus \(\frac{10^{4}}{T}\) is given below (in solid line), where \(p_{\mathbf{Z}}\) is the pressure (in bar) of the gas \(\mathbf{Z}\) at temperature \(T\) and \(p^{\theta}=1\) bar.

(Given, \(\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}\), where the equilibrium constant, \(K=\frac{p_{z}}{p^{\theta}}\) and the gas constant, \(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) )
The value of standard enthalpy, $∆{\mathrm{H}}^{\mathrm{o}}$ (in $\mathrm{kJ} {\mathrm{mol}}^{-1}$) for the given reaction is ___.

Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$ . Then the number of such distinct arrangements $\left(n_1, n_2, n_3, n_4, n_5\right)$ is _________

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of $D$ is

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude \(p\), oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance \(2 r\) apart along the \(x\) direction. The midpoint of the line joining the two dipoles is \(X\). The possible resultant electric fields \(\vec{E}\) at \(X\) are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)	\(\vec{E}=0\)
\((Q)\)		\((2)\)	\(\vec{E}=-\frac{p}{2 \pi \epsilon_0 \mathrm{r}^3} \hat{\mathrm{j}}\)
\((R)\)		\((3)\)	\(\vec{E}=-\frac{p}{4 \pi \epsilon_0 \mathrm{r}^3}(\hat{\mathrm{i}}-\hat{\mathrm{j}})\)
\((S)\)		\((4)\)	\(\vec{E}=\frac{p}{4 \pi \epsilon_0 \mathrm{r}^3}(2 \hat{\mathrm{i}}-\hat{\mathrm{j}})\)
		\((5)\)	\(\vec{E}=\frac{p}{\pi \epsilon_0 \mathrm{r}^3} \hat{\mathrm{i}}\)

Two moles of ideal helium gas are in a rubber balloon at \(30^{\circ} \mathrm{C}\). The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to \(35^{\circ} \mathrm{C}\). The amount of heat required in raising the temperature is nearly (take \(R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}\) )