2. Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures $2 T$ and $3 T$ respectively. The temperature of the middle (i.e. second) plate under steady state condition is\\
(A) $\left( \frac { 65 } { 2 } \right) ^ { \frac { 1 } { 4 } } T$\\
(B) $\left( \frac { 97 } { 4 } \right) ^ { \frac { 1 } { 4 } } T$\\
(C) $\left( \frac { 97 } { 2 } \right) ^ { \frac { 1 } { 4 } } T$\\
(D) $( 97 ) ^ { \frac { 1 } { 4 } } T$

\section*{ANSWER : C}
\begin{enumerate}
  \setcounter{enumi}{2}
  \item Consider a thin spherical shell of radius $R$ with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $| \vec { E } ( r ) |$ and the electric potential $V ( r )$ with the distance $r$ from the centre, is best represented by which graph?
\end{enumerate}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(A)}
  \includegraphics[alt={},max width=\textwidth]{1235d492-3555-402b-9a20-f8b0c32f22a0-02_446_619_1314_447}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(B)}
  \includegraphics[alt={},max width=\textwidth]{1235d492-3555-402b-9a20-f8b0c32f22a0-02_437_628_1312_1303}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(C)}
  \includegraphics[alt={},max width=\textwidth]{1235d492-3555-402b-9a20-f8b0c32f22a0-02_436_626_1868_447}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(D)}
  \includegraphics[alt={},max width=\textwidth]{1235d492-3555-402b-9a20-f8b0c32f22a0-02_437_624_1865_1309}
\end{center}
\end{figure}

ANSWER : D

\section*{PHYSICS}
\begin{enumerate}
  \setcounter{enumi}{3}
  \item In the determination of Young's modulus $\left( Y = \frac { 4 M L \mathrm {~g} } { \pi l d ^ { 2 } } \right)$ by using Searle's method, a wire of length $L = 2 \mathrm {~m}$ and diameter $d = 0.5 \mathrm {~mm}$ is used. For a load $M = 2.5 \mathrm {~kg}$, an extension $l = 0.25 \mathrm {~mm}$ in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm . The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the $Y$ measurement\\
(A) due to the errors in the measurements of $d$ and $l$ are the same.\\
(B) due to the error in the measurement of $d$ is twice that due to the error in the measurement of $l$.\\
(C) due to the error in the measurement of $l$ is twice that due to the error in the measurement of $d$.\\
(D) due to the error in the measurement of $d$ is four times that due to the error in the measurement of $l$.
\end{enumerate}

\section*{ANSWER : A}
\begin{enumerate}
  \setcounter{enumi}{4}
  \item A small block is connected to one end of a massless spring of un-stretched length 4.9 m . The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at $t = 0$. It then executes simple harmonic motion with angular frequency $\omega = \frac { \pi } { 3 } \mathrm { rad } / \mathrm { s }$. Simultaneously at $t = 0$, a small pebble is projected with speed $v$ from point $P$ at an angle of $45 ^ { \circ }$ as shown in the figure. Point $P$ is at a horizontal distance of 10 m from $O$. If the pebble hits the block at $t = 1 \mathrm {~s}$, the value of $v$ is (take $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ )\\
\includegraphics[max width=\textwidth, alt={}, center]{1235d492-3555-402b-9a20-f8b0c32f22a0-04_386_877_777_352}\\
(A) $\sqrt { 50 } \mathrm {~m} / \mathrm { s }$\\
(B) $\sqrt { 51 } \mathrm {~m} / \mathrm { s }$\\
(C) $\sqrt { 52 } \mathrm {~m} / \mathrm { s }$\\
(D) $\sqrt { 53 } \mathrm {~m} / \mathrm { s }$
\end{enumerate}

\section*{ANSWER : A}
\begin{enumerate}
  \setcounter{enumi}{5}
  \item Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are $\beta _ { G } , \beta _ { R }$ and $\beta _ { B }$, respectively. Then,\\
(A) $\beta _ { G } > \beta _ { B } > \beta _ { R }$\\
(B) $\beta _ { B } > \beta _ { G } > \beta _ { R }$\\
(C) $\beta _ { R } > \beta _ { B } > \beta _ { G }$\\
(D) $\beta _ { R } > \beta _ { G } > \beta _ { B }$
\end{enumerate}

\section*{ANSWER: D}
\section*{PHYSICS}
\begin{enumerate}
  \setcounter{enumi}{6}
  \item A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in the figure. The mass is undergoing circular motion in the $x - y$ plane with centre at $O$ and constant angular speed $\omega$. If the angular momentum of the system, calculated about $O$ and $P$ are denoted by $\vec { L } _ { O }$ and $\vec { L } _ { P }$ respectively, then\\
\includegraphics[max width=\textwidth, alt={}, center]{1235d492-3555-402b-9a20-f8b0c32f22a0-05_471_288_605_201}\\
(A) $\vec { L } _ { O }$ and $\vec { L } _ { P }$ do not vary with time.\\
(B) $\vec { L } _ { O }$ varies with time while $\vec { L } _ { P }$ remains constant.\\
(C) $\vec { L } _ { O }$ remains constant while $\vec { L } _ { P }$ varies with time.\\
(D) $\vec { L } _ { O }$ and $\vec { L } _ { P }$ both vary with time.
\end{enumerate}

ANSWER : C\\