42. Four point charges, each of $+ q$, are rigidly fixed at the four corners of a square planar soap film of side ' $a$ '. The surface tension of the soap film is $\gamma$. The system of charges and planar film are in equilibrium, and $a = k \left[ \frac { q ^ { 2 } } { \gamma } \right] ^ { 1 / N }$, where ' $k$ ' is a constant. Then $N$ is

\section*{ANSWER: 3}
\begin{enumerate}
  \setcounter{enumi}{42}
  \item Steel wire of length ' L ' at $40 ^ { \circ } \mathrm { C }$ is suspended from the ceiling and then a mass ' m ' is hung from its free end. The wire is cooled down from $40 ^ { \circ } \mathrm { C }$ to $30 ^ { \circ } \mathrm { C }$ to regain its original length ' L '. The coefficient of linear thermal expansion of the steel is $10 ^ { - 5 } / { } ^ { \circ } \mathrm { C }$, Young's modulus of steel is $10 ^ { 11 } \mathrm {~N} / \mathrm { m } ^ { 2 }$ and radius of the wire is 1 mm . Assume that $L \gg$ diameter of the wire. Then the value of ' $m$ ' in kg is nearly
\end{enumerate}

\section*{ANSWER:3}
\begin{enumerate}
  \setcounter{enumi}{43}
  \item The activity of a freshly prepared radioactive sample is $10 ^ { 10 }$ disintegrations per second, whose mean life is $10 ^ { 9 } \mathrm {~s}$. The mass of an atom of this radioisotope is $10 ^ { - 25 } \mathrm {~kg}$. The mass (in mg ) of the radioactive sample is
\end{enumerate}

\section*{ANSWER: 1}
\begin{enumerate}
  \setcounter{enumi}{44}
  \item A long circular tube of length 10 m and radius 0.3 m carries a current $I$ along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $I = I _ { 0 } \cos ( 300 t )$ where $I _ { 0 }$ is constant. If the magnetic moment of the loop is $N \mu _ { 0 } I _ { 0 } \sin ( 300 t )$, then ' $N$ ' is
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{cdfce4af-7f3d-4d66-ab09-7da7ecb927ff-21_626_287_602_944}
\captionsetup{labelformat=empty}
\caption{ANSWER: 6}
\end{center}
\end{figure}

\begin{enumerate}
  \setcounter{enumi}{45}
  \item Four solid spheres each of diameter $\sqrt { 5 } \mathrm {~cm}$ and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm . The moment of inertia of the system about the diagonal of the square is $\mathrm { N } \times 10 ^ { - 4 } \mathrm {~kg} - \mathrm { m } ^ { 2 }$, then N is
\end{enumerate}

ANSWER: 9

\section*{PART III : MATHEMATICS}
\section*{SECTION-1 (Total Marks : 21)}
\section*{(Single Correct Answer Type)}
This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.\\