A planar structure of length $\mathrm{L}$ and width $\mathrm{W}$ is made of two different optical media of refractive indices ${\mathrm{n}}_1=1.5$ and ${\mathrm{n}}_2=1.44$ as shown in figure. If $\mathrm{L}>>\mathrm{W},$ a ray entering from end $\mathrm{A}\mathrm{B}$ will emerge from end $\mathrm{C}\mathrm{D}$ only if the total internal reflection condition is met inside the structure. For $\mathrm{L}=9.6\mathrm{}\mathrm{m},$ if the incident angle $\mathrm{\theta }$ is varied, the maximum time taken by a ray to exit the plane $\mathrm{C}\mathrm{D}$ is $\mathrm{t}\times {10}^{-9}\mathrm{s},$ where $\mathrm{t}$ is ____.
[Speed of light $\mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s}]$

A student is performing the experiment of resonance column. The diameter of the column tube is \(4 \mathrm{~cm}\). The frequency of the tuning fork is \(512 \mathrm{~Hz}\). The air temperature is \(38^{\circ} \mathrm{C}\) in which the speed of sound is \(336 \mathrm{~m} / \mathrm{s}\). The zero of the meter scale coincides with the top end of the resonance column tube. When the first resonance occurs, the reading of the water level in the column is

A thin spherical insulating shell of radius $\mathrm{R}$ carries a uniformly distributed charge such that the potential at its surface is ${\mathrm{V}}_0$ . A hole with a small area $\mathrm{\alpha }4\mathrm{\pi }{\mathrm{R}}^2\left(\mathrm{\alpha }\ll 1\right)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

When two progressive waves \(y_1=4 \sin (2 x-6 t) \quad\) and \(y_2=3 \sin \left(2 x-6 t-\frac{\pi}{2}\right) \quad\) are superimposed, the amplitude of the resultant wave is

The circular scale of a screw gauge has 50 divisions and pitch of \(0.5 \mathrm{~mm}\). Find the diameter of sphere. Main scale reading is 2 .

A planet of mass M, has two natural satellites with masses $m_1\mathrm{a}\mathrm{n}\mathrm{d}{m}_2$ . The radii of their circular orbits are $R_1 \mathrm{a}\mathrm{n}\mathrm{d} R_2$ respectively. Ignore the gravitational force between the satellites. Define $v_1,L_1,K_1 \mathrm{a}\mathrm{n}\mathrm{d} T_1$ to be, respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $1; \mathrm{a}\mathrm{n}\mathrm{d} v_2,L_2,K_2\mathrm{a}\mathrm{n}\mathrm{d} T_2$ and to be the corresponding quantities of satellite 2. Given $m_1/m_2=2 \mathrm{a}\mathrm{n}\mathrm{d} R_1/R_2=\frac{1}{4}$ , match the ratios in List-I to the numbers in List-II.

	LIST -I		LIST-II
A)	$\frac{v_1}{v_2}$	P)	$\frac{1}{8}$
B)	$\frac{L_1}{L_2}$	Q)	1
C)	$\frac{K_1}{K_2}$	R)	2
D)	$\frac{T_1}{T_2}$	S)	8

Paragraph :
\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
While the piston is at a distance \(2 L\) from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

The number of water molecule(s) directly bonded to the metal centre in \(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\) is

Let $f :\left[0, 4\pi \right]\to \left[0, \pi \right]$ be defined by $f\left(x\right)={\cos }^{-1}\left(\cos x\right).$ The number of points $x \in \left[0, 4\pi \right]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$ is_________

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .

The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed in a perpendicular
magnetic field in the direction going into the plane of the figure. The magnitude of the field increases with time. \(I_1\) and \(I_2\) are the currents in the segments \(a b\) and \(c d\). Then,

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