A metal rod \(A B\) of length \(10 x\) has its one end \(A\) in ice at \(0^{\circ} \mathrm{C}\) and the other end \(B\) in water at \(100^{\circ} \mathrm{C}\). If a point \(P\) on the rod is maintained at \(400^{\circ} \mathrm{C}\), then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is \(540 \mathrm{calg}^{-1}\) and latent heat of melting of ice is \(80 \mathrm{calg}^{-1}\). If the point \(P\) is at a distance of \(\lambda x\) from the ice end \(A\), find the value of \(\lambda\). [Neglect any heat loss to the surrounding.]

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{T_R}{T_0}$ 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 at \(r=R\) is

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is

Consider a hydrogen-like ionized atom with atomic number $Z$ with a single electron. In the emission spectrum of this atom, the photon emitted in the $n=2 \mathrm{t}\mathrm{o} n= 1$ transition has energy $74.8 \mathrm{eV}$ higher than the photon emitted in the $n=3 \mathrm{t}\mathrm{o} n=2$ transition. Given that the ionization energy of the hydrogen atom is $13.6 \mathrm{eV}$, what is the value of $Z$?

Two moles of ideal helium gas are in a rubber balloon at \(30^{\circ} \mathrm{C}\). The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to \(35^{\circ} \mathrm{C}\). The amount of heat required in raising the temperature is nearly (take \(R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}\) )

An ideal gas undergoes a four step cycle as shown in the $P-V$ diagram below. During this cycle, heat is absorbed by the gas in:

Perpendiculars are drawn from points on the line $\frac{\text{x}+2}{2}=\frac{\text{y}+1}{-1}=\frac{\text{z}}{3}$  to the plane $x+y+z=3$. The feet of perpendiculars lie on the line

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by $E=\frac{3}{5}\frac{Z\left(Z-1\right)e^2}{4\pi {\epsilon }_{0}R}$ The measured masses of the neutron, ${}_1{}^{1}H, {}_8{}^{15}O$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u respectively. Given that the radii of both the ${}_7{}^{15}N and {}_8{}^{15}O$ nuclei are same, 1u = 931.5 Me V/ $c^2$ (c is the speed of light) and $\frac{e^2}{\left(4\pi {ϵ}_0\right)}=1.44 MeV fm.$ Assuming that the difference between the binding energies of ${}_7{}^{15}N and {}_8{}^{15}O$ is purely due to the electrostatic energy, the radius of either of the nuclei is $\left(1 fm={10}^{-15}m\right)$

Choose the correct option(s) from the following

If the normal of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ drawn at the end points of its latus rectum are tangents to the circle ${\left(\mathrm{x}-3\right)}^2+{\left(\mathrm{y}+2\right)}^2={\mathrm{r}}^2,$ then the value of ${\mathrm{r}}^2$ is

Consider the following list of reagents: Acidified ${\mathrm{K}}_2\mathrm{C}{\mathrm{r}}_2{\mathrm{O}}_7$ , alkaline $\mathrm{K}\mathrm{M}\mathrm{n}{\mathrm{O}}_4,\mathrm{C}\mathrm{u}\mathrm{S}{\mathrm{O}}_4,{\mathrm{H}}_2{\mathrm{O}}_2,\mathrm{C}{\mathrm{l}}_2,{\mathrm{O}}_3,\mathrm{F}\mathrm{e}\mathrm{C}{\mathrm{l}}_3,\mathrm{H}\mathrm{N}{\mathrm{O}}_3$ and $\mathrm{N}{\mathrm{a}}_2{\mathrm{S}}_2{\mathrm{O}}_3$ . The total number of reagents that can oxidise aqueous iodide to iodine is