Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$ . Which of the following statements is (are) true?

A thin spherical insulating shell of radius $\mathrm{R}$ carries a uniformly distributed charge such that the potential at its surface is ${\mathrm{V}}_0$ . A hole with a small area $\mathrm{\alpha }4\mathrm{\pi }{\mathrm{R}}^2\left(\mathrm{\alpha }\ll 1\right)$ is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Paragraph II: Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=0\), the value of \(v_{\mathrm{cm}} /(a \omega)\) will be ________ .

Three glass cylinders of equal height $\mathrm{H}=30\mathrm{}\mathrm{c}\mathrm{m}$ and same refractive index $\mathrm{n}=1.5$ are placed on a horizontal surface shown in figure. Cylinder $\mathrm{I}$ has a flat top, cylinder $\mathrm{I}\mathrm{I}$ has a convex top and cylinder $\mathrm{I}\mathrm{I}\mathrm{I}$ has a concave top. The radii of curvature of the two curved tops are same $\left(\mathrm{R}=3\mathrm{m}\right).$ If ${\mathrm{H}}_1,{\mathrm{H}}_2$ and ${\mathrm{H}}_3$ are the apparent depths of a point $\mathrm{X}$ on the bottom of the three cylinders, respectively, the correct statement(s) is/are

A police car with a siren of frequency \(8 \mathrm{kHz}\) is moving with uniform velocity \(36 \mathrm{~km} / \mathrm{h}\) towards a tall building which reflects the sound waves. The speed of sound in air is \(320 \mathrm{~m} / \mathrm{s}\). The frequency of the siren heard by the car driver is

A student is performing the experiment of resonance column. The diameter of the column tube is \(4 \mathrm{~cm}\). The frequency of the tuning fork is \(512 \mathrm{~Hz}\). The air temperature is \(38^{\circ} \mathrm{C}\) in which the speed of sound is \(336 \mathrm{~m} / \mathrm{s}\). The zero of the meter scale coincides with the top end of the resonance column tube. When the first resonance occurs, the reading of the water level in the column is

Paragraph :
\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
While the piston is at a distance \(2 L\) from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top 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

The product of all positive real values of $x$ satisfying the equation $x^{\left(16{\left({\log }_{5}x\right)}^3-68{\log }_{5}x\right)}=5^{-16}$ is

A circuit with an electrical load having impedance \(Z\) is connected with an AC source as shown in the diagram. The source voltage varies in time as \(V(t)=300 \sin (400 t) \mathrm{V}\), where \(t\) is time in s. List-I shows various options for the load. The possible currents \(i(t)\) in the circuit as a function of time are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)
\((Q)\)		\((2)\)
\((R)\)		\((3)\)
\((S)\)		\((4)\)
		\((5)\)

Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum temperature (in K ) at which the reduction of cassiterite by coke would take place.

\(298 K: \Delta_{ f } H ^0\left( SnO _2(s)\right)=-581.0 kJ mol ^{-1}\), \(\Delta_{ f } H ^0\left( CO _2(g)\right)=-394.0 kJ mol ^{-1}\)

\(S ^0\left( SnO _2(s)\right)=56.0 JK ^{-1} mol^{-1}, S^0( Sn ( s ))\) \(=52.0 JK ^{-1} mol^{-1}\)

\(S ^0( C ( s ))=6.0 JK ^{-1} mol^{-1}, S^0\left( CO _2(g)\right)\) \(=210.0 JK ^{-1} mol^{-1}\)

Assume that the enthalpies and the entropies are temperature independent.

When ${\mathrm{O}}_2$ is adsorbed on a metallic surface, electron transfer occurs from the metal to ${\mathrm{O}}_2$ . The TRUE statement(s) regarding this adsorption is(are)

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ and dielectric constant $21$. If the angle between the strings remains the same after the immersion, then