One end of a taut string of length 3m along the x axis is fixed at x = 0. The speed of the waves in the string is $100\mathrm{ }\mathrm{m}{\mathrm{s}}^{-1}$ . The other end of the string is vibrating in the y direction so that stationary waves are set up in the string. The possible waveform(s) of these stationary waves is (are)

A light disc made of aluminum (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1 \mathrm{m}$. It is maintained at temperature $360 \mathrm{K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{m}$ and height $h=9 \mathrm{m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density ${\rho }_a=1.2 \mathrm{kg} {\mathrm{m}}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and $\pi =3.14$]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm} {\mathrm{s}}^{-1}$.

Paragraph :
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the piston is pushed at a speed of \(5 \mathrm{~mms}^{-1}\), the air comes out of the nozzle with a speed of

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

Paragraph
The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

An $\alpha$-particle (mass $4$ amu) and a singly charged Sulphur ion (mass $32$ amu) are initially at rest. They are accelerated through a potential $V$ and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $\alpha$-particle and the sulphur ion move in circular orbits of radii $R_{\alpha }$ and $R_S$, respectively. The ratio $\left(\frac{R_S}{R_{\alpha }}\right)$ is __________ .

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]

Paragraph:
A tertiary alcohol \(H\) upon acid catalysed dehydration gives a product \(I\). Ozonolysis of I leads to compounds \(\mathrm{J}\) and \(\mathrm{K}\). Compound \(\mathrm{J}\) upon reaction with
\(\mathrm{KOH}\) gives benzyl alcohol and a compound \(\mathrm{L}\), where \(\mathrm{K}\) on reaction with \(\mathrm{KOH}\) gives only \(M\).

Question:
The structures of compounds \(J, K\) and \(L\), respectively, are

Let \(f, g\) and \(h\) be real-valued functions defined on the interval \([0,1]\) by \(f(x)=e^{x^2}+e^{-x^2}, \quad g(x)=x e^{x^2}+e^{-x^2}\) and \(h(x)=x^2 e^{x^2}+e^{-x^2}\). If \(a, b\) and \(c\) denote respectively, the absolute maximum of \(f, g\) and \(h\) on \([0,1]\), then

Two vehicles, each moving with speed u on  the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f₁. An observer in the other vehicle hears the frequency of the whistle to be f₂. The speed of sound in still air is V. The correct statement (s) is (are) :

Addition of excess aqueous ammonia to a pink coloured aqueous solution of $MC{l}_2.6H_{2}O \left(X\right)$ and $N{H}_{4}Cl$ gives an octahedral complex Y in the presence of air. In aqueous solution, complex Y behaves as 1 : 3 electrolyte. The reaction of X with excess HCl at room temperature results in the formation of a blue coloured complex Z. The calculated spin only magnetic moment of X and Z is 3.87 B.M., whereas it is zero for complex Y.
Among the following options, which statement(s) is(are) correct ?

The entropy versus temperature plot for phases $\alpha$ and $\beta$ at $1$ bar pressure is given. ${\mathrm{S}}_{\mathrm{T}}$ and ${\mathrm{S}}_0$ are entropies of the phases at temperatures $\mathrm{T}$ and $0 \mathrm{K}$, respectively. The transition temperature for $\alpha$ to $\beta$ phase change is $600 \mathrm{K}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}=1 \mathrm{J} {\mathrm{mol}}^{-1} {\mathrm{K}}^{-1}$. Assume $\left({\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}-{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}\right)$ is independent of temperature in the range of $200$ to $700 \mathrm{K}.{\mathrm{C}}_{\mathrm{p},\mathrm{\alpha }}$ and ${\mathrm{C}}_{\mathrm{p},\mathrm{\beta }}$ are heat capacities of $\alpha$ and $\beta$ phases, respectively.
The value of entropy change, ${\mathrm{S}}_{\beta }-{\mathrm{S}}_{\alpha }$ (in ${\mathrm{Jmol}}^{-1} {\mathrm{K}}^{-1}$ ), at $300 \mathrm{K}$ is [Use: $\ln 2=0.69$0. Given: ${\mathrm{S}}_{\beta }-{\mathrm{S}}_{\alpha }=0$ at $0 \mathrm{K}$]