Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4V_1$. After this, the gas expands adiabatically and its volume becomes $32V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\mathrm{iso}}$ and $W_{\text{adia}},$ $\frac{W_{\mathrm{iso}}}{W_{\mathrm{adia}}}=f\ln \left(2\right)$, then $f$ is _________ .

A charged shell of radius $\mathrm{R}$ carries a total charge $\mathrm{Q}.$ Given $\mathrm{\Phi }$ as the flux of electric field through a closed cylindrical surface of height $\mathrm{h},$ radius $\mathrm{r}$ and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct? [ ${\in }_0$ is the permittivity of free space]

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
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Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

Statement I If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic \(X\)-rays do not change.
Statement II When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

A mixture of 2 moles of helium gas (atomic mass \(=4 \mathrm{amu}\) ) and 1 mole of argon gas (atomic mass \(=40 \mathrm{amu}\) ) is kept at \(300 \mathrm{~K}\) in a container. The ratio of the rms speeds \(\left(\frac{v_{\mathrm{rms}}(\text { helium })}{v_{\mathrm{rms}}(\operatorname{argon})}\right)\) is

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The net external force acting on the disk when its centre of mass is at displacement \(x\) with respect to its equilibrium position is

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate $\left({\mathrm{KMnO}}_4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\mathrm{\Delta }n=\left(n_1-n_2\right)\ll n_1.$ When they are connected by a tube of small length $l$ and cross-sectional area $S,{\mathrm{KMnO}}_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order ${\left(\mathrm{\Delta }n\right)}^2$. Which of the following is/are correct? ($k_B$ is the Boltzmann constant)

In the following carbocation, \(\mathrm{H} / \mathrm{CH}_3\) that is most likely to migrate to the positively charged carbon is

The number of distinct real values of \(\lambda\), for which the vectors \(-\lambda^2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\), \(\hat{\mathbf{i}}-\lambda^2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\lambda^2 \hat{\mathbf{k}}\) are coplanar, is

Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.

List - I	List - II
(P) \(P _2 O _3\) \(+~3 H _2 O \rightarrow\)	(1) \(P ( O )\left( OCH _3\right)\) \(Cl _2\)
(Q) \(P _4+3 NaOH\) \(+~3 H _2 O \rightarrow\)	(2) \(H _3 PO _3\)
(R) \(PCl _5+ CH _3\) \(COOH \rightarrow\)	(3) \(PH _3\)
(S) \(H _3 PO _2\) \(+~2 H _2 O ~+\) \(4 AgNO _3 \rightarrow\)	(4) \(POCL _3\)
	(5) \(H _3 PO _4\)

The atomic masses of $\mathrm{He}$ and $\mathrm{Ne}$ are $4$ and $20 \mathrm{amu}$, respectively. The value of the de Broglie wavelength of $\mathrm{He}$ gas at $-73° \mathrm{C}$ is "M" times that of the de Broglie wavelength of $\mathrm{Ne}$ at $727° \mathrm{C}$. M is:

A stationary source is emitting sound at a fixed frequency \(f_0\), which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is \(1.2 \%\) of \(f_0\). What is the difference in the speeds of the cars (in km per hour) to the nearest integer ? The cars are moving at constant speeds much smaller than the speed of sound which is \(330 \mathrm{~ms}^{-1}\).

Let \(S=\{1,2,3,4\}\). The total number of unordered pairs of disjoint subsets of \(S\) is equal to