16. An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R / 2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2 R$ from the axis of the cylinder, is given by the expression $\frac { 23 \rho R } { 16 k \varepsilon _ { 0 } }$. The value of $k$ is\\
\includegraphics[max width=\textwidth, alt={}, center]{1235d492-3555-402b-9a20-f8b0c32f22a0-10_776_666_906_285}

\section*{ANSWER : 6}
\begin{enumerate}
  \setcounter{enumi}{16}
  \item A cylindrical cavity of diameter $a$ exists inside a cylinder of diameter $2 a$ as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point $P$ is given by $\frac { N } { 12 } \mu _ { 0 } a J$, then the value of $N$ is
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{1235d492-3555-402b-9a20-f8b0c32f22a0-10_447_526_2165_296}
\captionsetup{labelformat=empty}
\caption{ANSWER : 5}
\end{center}
\end{figure}

\begin{enumerate}
  \setcounter{enumi}{17}
  \item A lamina is made by removing a small disc of diameter $2 R$ from a bigger disc of uniform mass density and radius $2 R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I _ { O }$ and $I _ { P }$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac { I _ { P } } { I _ { O } }$ to the nearest integer is\\
\includegraphics[max width=\textwidth, alt={}, center]{1235d492-3555-402b-9a20-f8b0c32f22a0-11_414_451_491_225}
\end{enumerate}

\section*{ANSWER : 3}
\begin{enumerate}
  \setcounter{enumi}{18}
  \item A circular wire loop of radius $R$ is placed in the $x - y$ plane centered at the origin $O$. A square loop of side $a ( a \ll R )$ having two turns is placed with its center at $z = \sqrt { 3 } R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of $45 ^ { \circ }$ with respect to the $z$-axis. If the mutual inductance between the loops is given by $\frac { \mu _ { 0 } a ^ { 2 } } { 2 ^ { p / 2 } R }$, then the value of $p$ is
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{1235d492-3555-402b-9a20-f8b0c32f22a0-11_688_618_1568_225}
\captionsetup{labelformat=empty}
\caption{ANSWER : 7}
\end{center}
\end{figure}

\begin{enumerate}
  \setcounter{enumi}{19}
  \item A proton is fired from very far away towards a nucleus with charge $Q = 120 e$, where $e$ is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm ) of the proton at its start is: (take the proton mass, $m _ { p } = ( 5 / 3 ) \times 10 ^ { - 27 } \mathrm {~kg}$; $\mathrm { h } / e = 4.2 \times 10 ^ { - 15 } \mathrm {~J} . \mathrm { s } / \mathrm { C } ; \frac { 1 } { 4 \pi \varepsilon _ { 0 } } = 9 \times 10 ^ { 9 } \mathrm {~m} / \mathrm { F } ; 1 \mathrm { fm } = 10 ^ { - 15 } \mathrm {~m}$ )
\end{enumerate}

\section*{PART II : CHEMISTRY}
\section*{SECTION 1 : Single Correct Answer Type}
This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.\\