Three identical capacitors $C_1,C_2\mathrm{a}\mathrm{n}\mathrm{d}{C}_3$ have a capacitance of $1.0\mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity ${ϵ}_r$ . The cell electromotive force (emf) $V_0=8V$ . First the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\mu C$ . The value of ${ϵ}_r$ =____________.

A diatomic ideal gas is compressed adiabatically to \(\frac{1}{32}\) of its initial volume. If the initial temperature of the gas is \(T_i\) \(T_i\) (in kelvin) and the final temperature is \(a T_i\), the value of \(a\) is

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When a particle of mass \(m\) moves on the \(x\)-axis in a potential of the form \(V(x)=k x^2\), it performs simple harmonic motion.

The corresponding time period is proportional to \(\sqrt{\frac{m}{k}}\), as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of \(x=0\) in a way different from \(k x^2\) and its total energy is such that the particle does not escape to infinity. Consider a particle of mass \(\mathrm{m}\) moving on the \(x\)-axis. Its potential energy is \(V(x)=\alpha x^4(\alpha>0)\) for \(|x|\) near the origin and becomes a constant equal to \(V_0\) for \(|x| \geq X_0\) (see figure).
Question :
If the total energy of the particle is \(E\), it will perform periodic motion only if

A parallel plate capacitor \(C\) with plates of unit area and separation \(d\) is filled with a liquid of dielectric constant \(K=2\). The level of liquid is \(\frac{d}{3}\) initially. Suppose the liquid level decreases at a constant speed \(v\), the time constant as a function of time \(t\) is

A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $P_i={10}^5 Pa and volume V_i={10}^{-3} m^3$ changes to a final state at $P_f=\left(\frac{1}{32}\right)\times {10}^5 Pa and V_f=8\times {10}^{-3} m^3$ in an adiabatic quasi - static process, such that $P^3{V}^5=$ constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $P_i$ followed by an isochoric (isovolumetric) process at volumes $V_f$ . The amount of heat supplied to the system in the two step process is approximately

Six point charges are kept at the vertices of a regular hexagon of side \(\mathrm{L}\) and centre \(O\), as shown in the figure.
Given that \(K=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{L^{2}}\), which of the following statement(s) is (are) correct?

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The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:

The correct statement is

Let \(\vec{w}=\hat{i}+\hat{j}-2 \hat{k}\), and \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\) be two vectors such that \(\vec{u} \times \vec{v}=\vec{w}\) and \(\vec{v} \times \vec{w}=\vec{u}\). Let \(\alpha, \beta, \gamma\) and \(t\) be real numbers such that \(\vec{u}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k},-t \alpha+\beta+\gamma=0, \alpha-t \beta+\gamma=0\), and \(\alpha+\beta-t \gamma=0\).
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	\(|\vec{v}|^2\) is equal to	(1)	0
(Q)	If \(\alpha=\sqrt{3}\),then \(\gamma^2\) is equal to	(2)	1
(R)	If \(\alpha=\sqrt{3}\),then \((\beta+\gamma)^2\) is equal to	(3)	2
(S)	If \(\alpha=\sqrt{2}\),then \(t+3\) is equal to	(4)	3
		(5)	5

A vernier calipers has \(1 \mathrm{~mm}\) marks on the main scale. It has 20 equal divisions on the vernier scale which match with 16 main scale divisions. For this vernier calipers, the least count is

The correct statement(s) pertaining to the adsorption of a gas on a solid surface is(are)

Consider the following molecules : ${\mathrm{Br}}_3{\mathrm{O}}_8, {\mathrm{F}}_2\mathrm{O},{\mathrm{H}}_2 {\mathrm{S}}_4{\mathrm{O}}_6,{\mathrm{H}}_2 {\mathrm{S}}_5{\mathrm{O}}_6$, and ${\mathrm{C}}_3{\mathrm{O}}_2$. Count the number of atoms existing in their zero oxidation state in each molecule. Their sum is

Match the reactions in Column I with appropriate types of steps/reactive intermediate involved in these reactions as given in Column II.

The conductance of $0.0015\mathrm{ }\mathrm{M}$ aqueous solution of a weak monobasic acid was determined by using a conductivity cell consisting of platinized $\mathrm{Pt}$electrodes. The distance between the electrodes is $120\mathrm{ }\mathrm{cm}$ with an area of cross-section of $1 {\mathrm{cm}}^2$. The conductance of this solution was found to be $5\times {10}^{-7}\mathrm{S}$ . The $\mathrm{pH}$of the solution is $4$. The value of limiting molar conductivity of this weak monobasic acid in aqueous solution is $\mathrm{ }\mathrm{Z}\mathrm{ }\times \mathrm{ }{10}^2\mathrm{S}\mathrm{ }{\mathrm{cm}}^{-1}{\mathrm{mol}}^{-1}.$ The value of $\mathrm{Z}$is.