A point mass of \(1 \mathrm{~kg}\) collides elastically with a stationary point mass of \(5 \mathrm{~kg}\). After their collision, the \(1 \mathrm{~kg}\) mass reverses its direction and moves with a speed of \(2 \mathrm{~ms}^{-1}\). Which of the following statement(s) is/are correct for the system of these two masses?

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4V_1$. After this, the gas expands adiabatically and its volume becomes $32V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\mathrm{iso}}$ and $W_{\text{adia}},$ $\frac{W_{\mathrm{iso}}}{W_{\mathrm{adia}}}=f\ln \left(2\right)$, then $f$ is _________ .

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

An infinitely long wire, located on the \(z\)-axis, carries a current \(I\) along the \(+z\)-direction and produces the magnetic field \(\vec{B}\). The magnitude of the line integral \(\int \vec{B} \cdot \overrightarrow{d l}\) along a straight line from the point \((-\sqrt{3} a, a, 0)\) to \((a, a, 0)\) is given by
[ \(\mu_0\) is the magnetic permeability of free space.]

Match the temperature of a black body given in List-$\mathrm{I}$ with an appropriate statement in List-$\mathrm{II}$, and choose the correct option.
[Given: Wien’s constant as $2.9\times {10}^{-3} \mathrm{m}-\mathrm{K}$ and $\frac{hc}{e}=1.24\times {10}^{-6} \mathrm{V}-\mathrm{m}$]

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	$2000 \mathrm{K}$	$1$	The radiation at peak wavelength can lead to emission of photoelectrons from a metal of work function $4 \mathrm{eV}$
$\mathrm{Q}$	$3000 \mathrm{K}$	$2$	The radiation at peak wavelength is visible to human eye
$\mathrm{R}$	$5000 \mathrm{K}$	$3$	The radiation at peak emission wavelength will result in the widest central maximum of a single slit diffraction
$\mathrm{S}$	$10000 \mathrm{K}$	$4$	The power emitted per unit area is $\frac{1}{16}$ of that emitted by a blackbody at temperature $6000 \mathrm{K}$
		$5$	The radiation at peak emission wavelength can be used to image human bones

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}_7{}^{16}\mathrm{N}+{}_2{}^4\mathrm{He}\to {}_1{}^1\mathrm{H}+{}_8{}^{19}\mathrm{O}$ in a laboratory frame is $n$ (in $\mathrm{MeV}$). Assume that ${}_7{}^{16}\mathrm{N}$ is at rest in the laboratory frame. The masses of ${}_7{}^{16}\mathrm{N}, {}_2{}^4\mathrm{He}, {}_1{}^1\mathrm{H}$ and ${}_8{}^{19}\mathrm{O}$ can be taken to be $16.006 \mathrm{u}, 4.003 \mathrm{u}$, $1.008 \mathrm{u}$ and $19.003 \mathrm{u}$, respectively, where $1 \mathrm{u}=930 \mathrm{MeV} {\mathrm{c}}^{-2}$. The value of $n$is
If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

Shown in the figure is a semicircular metallic strip that has thickness $t$ and resistivity $\rho$. Its inner radius is $R_1$ and outer radius is $R_2$. If a voltage $V_0$ is applied between its two ends, a current $I$ flows in it. In addition, it is observed that a transverse voltage $\mathrm{\Delta }V$ develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

An audio transmitter \((T)\) and a receiver \((R)\) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_0=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) ) is \(\qquad\)

The correct statement(s) regarding defects in solids is (are)

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Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
The vapour pressure of the solution \(M\) is

Paragraph:

Football teams \(T_{1}\) and \(T_{2}\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \(T_{1}\) winning, drawing and losing a game against \(T_{2}\) are \(\frac{1}{2}, \frac{1}{6}\) and \(\frac{1}{3}\), respectively. Each team gets \(3\) points for a win, \(1\) point for a draw and \(0\) point for a loss in a game. Let \(X\) and \(Y\) denote the total points scored by teams \(T_{1}\) and \(T_{2}\), respectively, after two games.


Question:

\(P(X>Y)\) is

The graph between object distance \(u\) and image distance \(v\) for a lens is given below. The focal length of the lens is

Let $P$ be the image of the point$(3,1,7)$ with respect to the plane $x-y+z=3.$ Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is