A right-angled prism of refractive index ${\mu }_1$ is placed in a rectangular block of refractive index ${\mu }_2$, which is surrounded by a medium of refractive index ${\mu }_3$, as shown in the figure. A ray of light $e$ enters the rectangular block at normal incidence. Depending upon the relationships between ${\mu }_1\text{, }{\mu }_2$ and ${\mu }_3$, it takes one of the four possible paths $ef, eg, eh$ or $ei$.

Match the paths in List $\mathrm{I}$ with conditions of refractive indices in List $\mathrm{II}$ and select the correct answer using the codes given below the lists:

	List I		List II
A.	$e\to f$	P.	${\mu }_1>\sqrt{2}{\mu }_2$
B.	$e\to g$	Q.	${\mu }_2>{\mu }_1\text{and}{\mu }_2>{\mu }_3$
C.	$e\to h$	R.	${\mu }_1={\mu }_2$
D.	$e\to i$	S.	${\mu }_2<{\mu }_1<\sqrt{2}{\mu }_2\text{and}{\mu }_2>{\mu }_3$

Two ideal batteries of emf ${\mathrm{V}}_1$ and ${\mathrm{V}}_2$ and three resistances ${\mathrm{R}}_1,\mathrm{}{\mathrm{R}}_2$ and ${\mathrm{R}}_3$ are connected as shown in the figure. The current in resistance ${\mathrm{R}}_2$ would be zero if

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{Q}{R{T}_0}$ is:

A thin convex lens is made of two materials with refractive indices ${\mathrm{n}}_1$ and ${\mathrm{n}}_2,$ as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. $\mathrm{f}$ is the focal length of the lens when ${\mathrm{n}}_1={\mathrm{n}}_2=\mathrm{n}.$ The focal length is $\mathrm{f}+\mathrm{\Delta }\mathrm{f}$ when ${\mathrm{n}}_1=\mathrm{n}$ and ${\mathrm{n}}_2=\mathrm{n}+\mathrm{\Delta }\mathrm{n}.$ Assuming $\mathrm{\Delta }\mathrm{n}<<\left(\mathrm{n}-1\right)$ and $1<\mathrm{n}<2,$ the correct statement(s) is/are:

A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$. When the acceleration due to gravity is $g$, the range of the projectile is $d$. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\text{'}}=\frac{g}{0.81}$, then the new range is $d^{\text{'}}=nd.$ The value of $n$ is

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The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:

The correct statement is

A small object of uniform density rolls up a curved surface with an initial velocity \(v\). It reaches up to a maximum height of \(\frac{3 v^2}{4 g}\) with respect to the initial\(
\text { position. The object is }
\)

In a constant volume calorimeter, \(3.5 \mathrm{~g}\) of a gas with molecular weight \(=28\) was burnt in excess oxygen at \(298.0 \mathrm{~K}\). The temperature of the calorimeter was found to increases from \(298.0 \mathrm{~K}\) to \(298.45 \mathrm{~K}\) due to the combustion process. Given, that the heat capacity of the calorimeter is \(2.5\) \(\mathrm{kJ} \mathrm{K}^{-1}\), the numerical value for the enthalpy of combustion of the gas in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is

In terms of potential difference $V$ , electic current $I$ , permittivity ${\epsilon }_0$ , permeability ${\mu }_0$ and speed of light $c$ , the dimensionally correct equation(s) is (are)

Match each of the reactions given in Column I with the corresponding products(s) given in Column II.

The total number of distinct $x\in \left[0, 1\right]$ for which, ${\int }_0^x\frac{t^2}{1+t^4}dt=2x-1$ is

Let $\text{z}=\frac{-1+\sqrt{3}\text{i}}{2}$ , where $\text{i}=\sqrt{-1}$ , and $\text{r},\text{s}\in \left\{1,2,3\right\}.$ Let $\text{P}=\left[\begin{array}{cc}{\left(-\mathrm{z}\right)}^{\text{r}}& {\text{z}}^{2\text{s}}\\ {\text{z}}^{2\text{s}}& {\text{z}}^{\text{r}}\end{array}\right]$ and $\text{I}$ be the identity matrix of order 2. Then the total number of ordered pairs $(\text{r},\text{s})$ for which ${\text{P}}^2=-\text{I}$ is

All the energy released from the reaction $\mathrm{X}⟶\mathrm{Y},\mathrm{ }{\mathrm{\Delta }}_{\mathrm{r}}{\mathrm{G}}^{\mathrm{o}}=-193\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$ is used for oxidizing ${\mathrm{M}}^{+}$ as ${\mathrm{M}}^{+}⟶{\mathrm{M}}^{3+}+2{\mathrm{e}}^{-}\mathrm{ },\mathrm{ }{\mathrm{E}}^{\mathrm{o}}=\mathrm{ }-0.25\mathrm{ }\mathrm{V}$
Under standard conditions, the number of moles of ${\mathrm{M}}^{+}$ oxidized when one mole of X is converted to Y is
$[\mathrm{F}=96500\mathrm{ }\mathrm{C}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}]$