One mole of an ideal monatomic gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100\hspace{0.17em}\mathrm{K}$ and the universal gas constant $R=8.0\mathrm{ J} \mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{ K}}^{-1}$, then how much is the decrease in its internal energy (in $\mathrm{J}$) ?

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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 :
For periodic motion of small amplitude \(A\), the time period \(T\) of this particle is proportional to

A football of radius $R$ is kept on a hole of radius $r(r<R)$ made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $\theta$ from the horizontal as shown in the figure below. The maximum value of $\theta$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

Two soap bubbles \(A\) and \(B\) are kept in a closed chamber where the air is maintained at pressure \(8 \mathrm{Nm}^{-2}\). The radii of bubbles \(A\) and \(B\) are \(2 \mathrm{~cm}\), respectively. Surface tension of the soap-water used to make bubbles is \(0.04\) \(\mathrm{Nm}^{-1}\). Find the ratio \(\frac{n_B}{n_A}\), where \(n_A\) and \(n_B\) are the number of moles of air in bubbles \(A\) and \(B\), respectively. [Neglect the effect of gravity]

A rocket is moving in a gravity free space with a constant acceleration of $2\mathrm{}\mathrm{m}{\mathrm{s}}^{-2}$ along + x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in + x direction with a speed of $0.3\mathrm{}\mathrm{m}{\mathrm{s}}^{-1}$ relative to the rocket. At the same time, another ball is thrown in - x direction with a speed of $0.2\mathrm{}\mathrm{m}{\mathrm{s}}^{-1}$ from its right end relative to the rocket. The time in seconds when the two balls hit each other for the first time is

In an insulated vessel, \(0.05 \mathrm{~kg}\) steam at \(373 \mathrm{~K}\) and \(0.45 \mathrm{~kg}\) of ice at \(253 \mathrm{~K}\) are mixed. Find the final temperature of the mixture (in kelvin).
\( \text { Given, }L_{\text {fusion }} =80 \mathrm{cal} / \mathrm{g}=336 \mathrm{~J} / \mathrm{g},\) \( L_{\text {vaporization }}=540 \mathrm{cal} / \mathrm{g}=2268 \mathrm{~J} / \mathrm{g}, \)
\( S_{\text {ice }} =2100 \mathrm{~J} / \mathrm{kg}, K=0.5 \mathrm{cal} / \mathrm{gK} \text { and } S_{\text {water }}=\) \(4200 \mathrm{~J} / \mathrm{kg}, K=1 \mathrm{cal} / \mathrm{gK} .\)

The potential energy of a particle of mass m at a distance r from a fixed point O is given by $V\left(r\right)=k{r}^2/2$, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

A compound ${\mathrm{H}}_2\mathrm{X}$ with molar weight of 80g is dissolved in a solvent having density of $0.4\mathrm{ }\mathrm{g}\mathrm{m}{\mathrm{l}}^{–1}$ . Assuming no change in volume upon dissolution, the molality of a 3.2 molar solution is

A bar of mass $M=1.00 \mathrm{kg}$ and length $L=0.20 \mathrm{m}$ is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $m=0.10 \mathrm{kg}$ is moving on the same horizontal surface with $5.00 \mathrm{m} {\mathrm{s}}^{–1}$ speed on a path perpendicular to the bar. It hits the bar at a distance $\frac{L}{2}$ from the pivoted end and returns back on the same path with speed $v$. After this elastic collision, the bar rotates with an angular velocity $\omega$. Which of the following statement is correct?

For a polynomial $g\left(x\right)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g\left(x\right)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S=\left\{{\left(x^2-1\right)}^2\left(a_0+a_{1}x+a_2{x}^2+a_3{x}^3\right) :a_0, a_1, a_2, a_3\in R\right\}$. For a polynomial $f$, let $f\text{'}$ and $f^{″}$ denote its first and second order derivatives respectively. Then the minimum possible value of $\left(m_{f^{\prime }}+m_{f^{″}}\right)$, where $f\in S$, is ____

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Given that for each \(a \in(0,1)\),

\(\lim _{h \rightarrow 0^{+}} \int_{h}^{1-h} t^{-a}(1-t)^{a-1} d t\)

exists. Let this limit be \(g(a)\). In addition, it is given that the function \(g(a)\) is differentiable on \((0,1)\).


Question:

The value of \(g^{\prime}\left(\frac{1}{2}\right)\) is

Match the gases under specified conditions listed in Column I with their properties/laws 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. Hydrogen gas \((P=200 atm,\) \(T=273 K)\)	p. Compressibility factor \(\neq 1\)
B. Hydrogen gas \((P \sim 0,\) \(T=273 K)\)	q. Attractive forces are dominant
C. \(CO _2(P=1 atm,\) \(T=273 K)\)	r. \(P V=n R T\)
D. Real gas with very large molar volume	s. \(P(V-n b)=n R T\)

A cubical solid aluminum (bulk modulus \(=-\text V \frac{\text{dP}}{\text{dV}}=70\text{ GPa}\)) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be \(10^3\text{ kg m} ^{-3}\) and \(10\text{ ms} ^{-2}\), respectively, the change in the edge length of the block in mm is _____________ .