A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is

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 thin ring of mass \(2 \mathrm{~kg}\) and radius \(0.5 \mathrm{~m}\) is rolling without slipping on a horizontal plane with velocity 1 \(\mathrm{m} / \mathrm{s}\). A small ball of mass \(0.1 \mathrm{~kg}\), moving with velocity \(20 \mathrm{~m} / \mathrm{s}\) in the opposite directions, hits the ring at a height of \(0.75 \mathrm{~m}\) and goes vertically up with velocity 10 \(\mathrm{m} / \mathrm{s}\). Immediately after the collision

Two point charges \(-Q\) and \(+Q / \sqrt{3}\) are placed in the \(x y\)-plane at the origin \((0,0)\) and a point \((2,0)\), respectively, as shown in the figure. This results in an equipotential circle of radius \(R\) and potential \(V=0\) in the \(x y\)-plane with its center at \((b, 0)\). All lengths are measured in meters.

The value of $b$ is ________ meter.

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The speed of the particle that can take discrete values is proportional to

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $\mathrm{L},$ which of the following statement (s) is/are correct?

Match the chemical conversions in List I with the appropriate reagents in List II and select the correct answer using the code given below the lists :
 

List - I		List - II
(A)		(P)	$\left(\mathrm{i}\right) \mathrm{Hg}{\left(\mathrm{OAc}\right)}_2; \left(\mathrm{ii}\right) {\mathrm{NaBH}}_4$
(B)	​	(Q)	$\mathrm{NaOEt}$
(C)		(R)	$\mathrm{Et}-\mathrm{Br}$
(D)		(S)	$\left(\mathrm{i}\right) {\mathrm{BH}}_3; \left(\mathrm{ii}\right) {\mathrm{H}}_2{\mathrm{O}}_2/\mathrm{NaOH}$

The dissociation constant of a substituted benzoic acid at \(25^{\circ} \mathrm{C}\) is \(1.0 \times 10^{-4}\). The \(\mathrm{pH}\) of \(0.01 \mathrm{M}\) solution of its sodium salt is

Let the function \(g:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) be given by \(g(u)=2 \tan ^{-1}\left(e^u\right)-\frac{\pi}{2}\). Then, \(g\) is

Paragraph II: Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=0\), the value of \(v_{\mathrm{cm}} /(a \omega)\) will be ________ .

The densities of two solid spheres $\mathrm{A}$ and $\mathrm{B}$ of the same radii $\mathrm{R}$ vary with radial distance $\mathrm{r}$ as ${\mathrm{\rho }}_{\mathrm{A}}\left(\mathrm{r}\right)=\mathrm{k}\left(\frac{\mathrm{r}}{\mathrm{R}}\right)$ and ${\mathrm{\rho }}_{\mathrm{B}}\left(\mathrm{r}\right)=\mathrm{k}{\left(\frac{\mathrm{r}}{\mathrm{R}}\right)}^5$ , respectively, where $\mathrm{k}$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ , respectively. If $\frac{{\mathrm{I}}_{\mathrm{B}}}{{\mathrm{I}}_{\mathrm{A}}}=\frac{\mathrm{n}}{10}$ , the value of $\mathrm{n}$ is

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question :The relation between $\left[E\right]$ and $\left[B\right]$ is