For a radioactive material, its activity $\mathrm{A}$ and rate of change of its activity $\mathrm{R}$ are defined as $\mathrm{A}=-\frac{\mathrm{d}\mathrm{N}}{\mathrm{d}\mathrm{t}}$ and $\mathrm{R}=-\frac{\mathrm{d}\mathrm{A}}{\mathrm{d}\mathrm{t}}$ , where $\mathrm{N}\left(\mathrm{t}\right)$ is the number of nuclei at time $\mathrm{t}$ . Two radioactive sources $\mathrm{P}$ (mean life $\mathrm{\tau }$ ) and $\mathrm{Q}$ (mean life $2\mathrm{\tau }$ ) have the same activity at $\mathrm{t}=0$ . Their rates of change of activities at $\mathrm{t}=2\mathrm{\tau }$ are ${\mathrm{R}}_{\mathrm{P}}$ and ${\mathrm{R}}_{\mathrm{Q}}$ , respectively. If $\frac{{\mathrm{R}}_{\mathrm{P}}}{{\mathrm{R}}_{\mathrm{Q}}}=\frac{\mathrm{n}}{\mathrm{e}}$ , then the value of $\mathrm{n}$ is

An open-ended U-tube of uniform cross-sectional area contains water (density ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$ ). Initially the water level stands at $0.29\mathrm{m}$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800\mathrm{kg}{\mathrm{m}}^{-3}$ is added to the left arm until its length is $0.1\mathrm{m}$, as shown in the schematic figure below. The ratio $\left(\frac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms is

Consider an $\mathrm{LC}$ circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C={10}^{-3} \mathrm{F}$, kept on a plane. The area of the circuit is $1 {\mathrm{m}}^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_0+\beta \mathrm{t}$ with $\beta =0.04 \mathrm{T} {\mathrm{s}}^{-1}$. The maximum magnitude of the current in the circuit is____$\mathrm{m} \mathrm{A}$.

In the circuit shown below, the switch \(\mathrm{S}\) is connected to position \(\mathrm{P}\) for a long time so that the charge on the capacitor becomes \(q_{1} \mu \mathrm{C}\). Then \(\mathrm{S}\) is switched to position \(\mathrm{Q}\). After a long time, the charge on the capacitor is \(q_{2} \mu \mathrm{C}\).

The magnitude of $q_1$ is _______.

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The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
The electric field within the nucleus is generally observed to be linearly dependent on \(r\). This implies

Two infinitely long straight wires lie in the xy - plane along the lines $x=\pm R$. The wire located at $x = +R$ carries a constant current $I_1$ and the wire located at $x =-R$ carries a constant current $I_2$. A circular loop of radius R is suspended with its centre at $\left(0, 0\sqrt{3}R\right)$ and in a plane parallel to the xy - plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\widehat{j}$ direction. Which of the following statements regarding the magnetic field $\overrightarrow{B}$ is (are) true?

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

A circular disc with a groove along its diameter is placed horizontally. A block of mass \(1 \mathrm{~kg}\) is placed as shown. The coefficient of friction between the block and all surfaces of groove in contact is \(\mu=2 / 5\). The disc has an acceleration of \(25 \mathrm{~m} / \mathrm{s}^2\). Find the acceleration of the block with respect to disc.

Match the chemical substance in Column I with type of polymers/type of bond in Column II. Indicate your answer by darkening the appropriate bubbles of \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. Cellulose	p. Natural polymer
B. Nylon-6,6	q. Synthetic polymer
C. Protein	r. Amide linkage
D. Sucrose	s. Glycoside linkage

Assuming $2\mathrm{s}\mathrm{ }–\mathrm{ }2\mathrm{p}$ mixing is NOT operative, the paramagnetic species among the following is

The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions, is

List - I	List - II
(P) Passing \(H _2 S\) in the presence of \(NH _4 OH\)	(1) \(Cu ^{2+}\)
(Q) \(\left( NH _4\right)_2 CO _3\) in the presence of \(NH _4 OH\)	(2) \(Al ^{3+}\)
(R) \(NH _4 OH\) in the presence of \(NH _4 Cl\)	(3) \(Mn ^{2+}\)
(S) Passing \(H _2 S\) in the presence of dilute HCl	(4) \(Ba ^{2+}\)
	(5) \(Mg ^{2+}\)

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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:
What should be the age of fossil for meaningful determination of its age?

A ray \(O P\) of monochromatic light is incident on the face \(A B\) of prism \(A B C D\) near vertex \(B\) at an incident angle of \(60^{\circ}\) (see figure). If the refractive index of the material of the prism is \(\sqrt{3}\), which of the following is (are) correct?