One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $\frac{T_A}{T_B}$?

A composite block is made of slabs \(A, B, C, D\) and \(E\) of different thermal conductivities (given in terms of a constant, \(K\) ) and sizes (given in terms of length, \(L\) ) as shown in the figure. All slabs are of same width. Heat \(Q\) flows only from left to right through the blocks. Then, in steady state

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l.$ A point dipole of dipole moment $\overrightarrow{p}$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times \left(mgh\right)$, where $g$ is the acceleration due to gravity, then the value of $N$ is ____________ (Note that for three coplanar forces keeping a point mass in equilibrium, $\frac{\mathrm{F}}{\sin \theta }$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

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In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).

Question :
Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$.
[Given: The speed of sound in air is $324 \mathrm{m} {\mathrm{s}}^{-1}$]

Consider both sources emitting sound. When $S_2$ is at $R$ and $S_1$ approaches the detector with a speed $4 \mathrm{m} {\mathrm{s}}^{-1}$, the beat frequency measured by the detector is _____ $\mathrm{Hz}$.

A monochromatic light is travelling in a medium of refractive index $n=1.6$ . It enters a stack of glass layers from the bottom side at an angle $\theta =30°.$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_m=n–m\mathrm{\Delta }n$ where $n_m$ is the refractive index of the $m^{th}$ slab and $\mathrm{\Delta }n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the ${\left(m–1\right)}^{th}$ and $m^{th}$ slabs from the right side of the stack. What is the value of m?

Steel wire of length \(L\) at \(40^{\circ} \mathrm{C}\) is suspended from the ceiling and then a mass \(m\) is hung from its free end. The wire is cooled down from \(40^{\circ} \mathrm{C}\) to \(30^{\circ} \mathrm{C}\) to regain its original length \(L\). The coefficient of linear thermal expansion of the steel is \(10^{-5} /{ }^{\circ} \mathrm{C}\), Young's modulus of steel is \(10^{11} \mathrm{~N} / \mathrm{m}^2\) and radius of the wire is \(1 \mathrm{~mm}\). Assume that \(L>\) diameter of the wire. Then, the value of \(m\) in \(\mathrm{kg}\) is nearly

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 \(R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}\) )

In a triangle $PQR$ , let $\overrightarrow{a}=\overrightarrow{QR}, \overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$ . If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}.\left(\overrightarrow{c}-\overrightarrow{b}\right)}{\overrightarrow{c}.\left(\overrightarrow{a}-\overrightarrow{b}\right)}=\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}$, then the value of ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ is _______

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right], P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \& F=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]$ If $Q$ is a nonsingular matrix of order $3\times 3$, then which of the following statements is (are) TRUE ?

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

Let $b$ be a nonzero real number. Suppose $f:\mathrm{ℝ}\to \mathrm{ℝ}$ is a differentiable function such that $f\left(0\right)=1$. If the derivative $f^{\text{'}}$ of $f$ satisfies the equation $f^{\text{'}}\left(x\right)=\frac{f\left(x\right)}{b^2+x^2}$ for all $x\in \mathrm{ℝ},$ then which of the following statements is/are TRUE?

The amount of water produced ($\mathrm{in} \mathrm{g}$) in the oxidation of $1$ $\mathrm{mole}$of rhombic sulphur by conc. ${\mathrm{H}\mathrm{N}\mathrm{O}}_3$ to a compound with the highest oxidation state of sulphur is
(Given data: Molar mass of water$=18\mathrm{ }\mathrm{g}\mathrm{ }{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ )