A thin conducting rod $MN$ of mass $20 \mathrm{gm}$, length $25 \mathrm{cm}$ and resistance $10\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0=4 \mathrm{T}$ directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t=0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$, and choose the correct option. [Given: The acceleration due to gravity $g=10 {\mathrm{ms}}^{-2}$ and $e^{-1}=0.4$]  

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	At $t=0.2 \mathrm{s}$, the magnitude of the induced emf in Volt	$1$	$0.07$
$\mathrm{Q}$	At $t=0.2 \mathrm{s}$, the magnitude of the magnetic force in Newton	$2$	$0.14$
$\mathrm{R}$	At $t=0.2 \mathrm{s}$, the power dissipated as heat in Watt	$3$	$1.20$
$\mathrm{S}$	The magnitude of terminal velocity of the rod in $\mathrm{m} {\mathrm{s}}^{-1}$	$4$	$0.12$
		$5$	$2.00$

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If \(\rho_{c}\) is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$is released on the slide from rest at a height $h$from the top of the terrace. The ball leaves the slide with a velocity ${\overrightarrow{u}}_0=u_0\stackrel{\wedge }{x}$and falls on the ground at a distance $d$from the building making an angle $\theta$with the horizontal. It bounces off with a velocity $\overrightarrow{v}$and reaches a maximum height $h_1$. The acceleration due to gravity is $g$and the coefficient of restitution of the ground is $\frac{1}{\sqrt{3}}$. Which of the following statement(s) is(are) correct?

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct?

A mixture of ideal gas containing $5$ moles of monatomic gas and $1$ mole of rigid diatomic gas is initially at pressure ${\mathrm{P}}_0,$ volume ${\mathrm{V}}_0$ and temperature ${\mathrm{T}}_0.$ If the gas mixture is adiabatically compressed to a volume $\frac{{\mathrm{V}}_0}{4}$, then the correct statement(s) is/are,
(Given $2^{1.2}=\mathrm{ }2.3\text{;} 2^{3.2}=9.2\text{;}\mathrm{ }\mathrm{R}$ is gas constant)

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of \(\sigma_0\). The separation between any two consecutive sheets is \(1 \mu \mathrm{~m}\). The various regions between the sheets are denoted as \(1,2,3,4\) and 5. If \(\sigma_0=9 \mu \mathrm{C} / \mathrm{m}^2\), then which of the following statements is/are correct: (Take permittivity of free space \(\epsilon_0=9 \times 10^{-12} \mathrm{~F} / \mathrm{m}\)):

In an aluminum (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure. The electrical resistivities of Al and Fe are $2.7\times {10}^{-8}\mathrm{\Omega }\mathrm{}\mathrm{m}$ and $1.0\times {10}^{-7}\mathrm{\Omega }\mathrm{}\mathrm{m}$ , respectively. The electrical resistance between the two faces P and Q of the composite bar is

Let $z$ be a complex number satisfying ${\left|z\right|}^3+2z^2+4\overline{z}-8=0$, where $\overline{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.

List - I	List - II
(P) \(|z|^2\) is equal to	(1) 12
(Q) \(|z-\bar{z}|^2\) is equal to	(2) 4
(R) \(|z|^2+|z+\bar{z}|^2\) is equal to	(3) 8
(S) \(|z+1|^2\) is equal to	(4) 10
	(5) 7

The correct option 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:

Consider a branch of the hyperbola \(x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0\) with vertex at the point \(A\). Let \(B\) be one of the end points of its latus rectum. If \(C\) is the focus of the hyperbola nearest to the point \(A\), then the area of the \(\triangle A B C\) is

A uniform wooden stick of mass 1.6 kg and length $\mathcal{l}$ rests in an inclined manner on a smooth, vertical wall of height h $\left(<\mathcal{l}\right)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of ${30}^o$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio h/ $\mathcal{l}$ and the frictional force f at the bottom of the stick are: $\left(g=10 m{s}^{-2}\right)$

Paragraph:
Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of \({ }^{14} \mathrm{C}\) by neutron capture in the upper atmosphere.
\(
{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1
\)
\({ }^{14} \mathrm{C}\) is absorbed by living organisms during photosynthesis. The \({ }^{14} \mathrm{C}\) content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of \({ }^{14} \mathrm{C}\) in the dead being, falls to the decay which \({ }^{14} \mathrm{C}\) undergoes.
\(
{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}
\)
The half life period of \({ }^{14} \mathrm{C}\) is 5770 years. The decay constant \((\lambda)\) can be calculated by using the following formula \(\lambda=\frac{0.693}{t_{1 / 2}}\).
The comparison of the \(\beta^{-}\)activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) in living matter is \(1: 10^{12}\).
Question:
Which of the following option is correct?

Paragraph:
Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately \(6.023 \times 10^{23}\) ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.
A \(4.0\) molar aqueous solution of \(\mathrm{NaCl}\) is prepared and \(500 \mathrm{~mL}\) of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: \(\mathrm{Na}=23, \mathrm{Hg}=200 ; 1\) faraday \(=96500\) coulombs).
Question:
The total number of moles of chlorine gas evolved is