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A wooden cylinder of diameter \(4 r\), height \(H\) and density \(\rho / 3\) is kept on a hole of diameter \(2 r\) of a tank, filled with liquid of density \(\rho\) as shown in the figure.

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If height \(h_2\) of water level is further decreased, then

A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height \(y\) (in m ) of the elevator, from the ground, with time \(t\) (in s ) is given by \(y=8\left[1+\sin \left(\frac{2 \pi t}{T}\right)\right]\), where \(T=40 \pi \mathrm{~s}\). Taking acceleration due to gravity, \(g=10 \mathrm{~m} / \mathrm{s}^2\), the maximum variation of the object's weight (in N ) as observed in the experiment is _______ .

The instantaneous voltages at three terminals marked $X, Y$ and $Z$ are given by
$V_X=V_0\sin \omega t$
$V_Y=V_0\sin \left(\omega t+\frac{2\pi }{3}\right)$ and
$V_Z=V_0\sin \left(\omega t+\frac{4\pi }{3}\right).$
An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $X$ and $Y$ and then between $Y$ and $Z$ . The reading(s) of the voltmeter will be

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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 \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).


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Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

A vibrating string of certain length \(l\) under a tension \(T\) resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length \(75 \mathrm{~cm}\) inside a tube closed at one end. The string also generates 4 beat/s when excited along with a tuning fork of frequency \(n\). Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be \(340 \mathrm{~m} / \mathrm{s}\), the frequency \(n\) of the tuning fork (in \(\mathrm{Hz}\) ) is

The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed in a perpendicular
magnetic field in the direction going into the plane of the figure. The magnitude of the field increases with time. \(I_1\) and \(I_2\) are the currents in the segments \(a b\) and \(c d\). Then,

A light beam is travelling from Region I to Region IV (refer figure). The refractive index in Region I, II, III and IV are \(n_0, \frac{n_0}{2}, \frac{n_0}{6}\) and \(\frac{n_0}{8}\), respectively. The angle of incidence \(\theta\) for which the beam just misses entering Region IV is

The quadratic equation $\mathrm{p}\left(\mathrm{x}\right)\mathrm{ }=\mathrm{ }0$ with real coefficients has purely imaginary roots. Then the equation $\mathrm{p}\left(\mathrm{p}\right(\mathrm{x}\left)\right)\mathrm{ }=\mathrm{ }0$ has

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The figure shows a surface \(X Y\) separating two transparent media, medium-1 and medium-2. The lines \(a b\) and \(c d\) represent wavefronts of a light wave travelling in medium-1 and incident on \(X Y\). The lines ef and \(g h\) represent wavefronts of the light wave in medium-2 after refraction.
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Speed of light is

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20\%$ and plant $T_2$ produces $80\%$ of the total computers produced. $7\%$ of computers produced in the factory turn out to be defective. It is known that,  $P$(computer turns out to be defective given that it is produced in plant $T_1$) $=10P$(computer turns out to be defective given that it is produced in plant $T_2$). Where, $P\left(E\right)$ denotes the probability of an event $E.$ A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is

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Read the following passage and answer the questions.
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The value of \(|U|\) is

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\(P_{14-16}\) : 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 \(p_0\).

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The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is \(\rho\). In equilibrium, the height \(H\) of the water column in the cylinder satisfies

Let \(\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.\) then \(\int\) is