A beaker of radius $r$ is filled with water$\left(\text{refractive index}\frac{4}{3}\right)$ up to a height $H$ as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is $h\left(h\ll H\text{,}h\ll r\right)$, as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature $R.$ Which of the following is/are correct? ($g$ is the acceleration due to gravity)

The circuit shown in the figure contains an inductor \(L\), a capacitor \(C_0\), a resistor \(R_0\) and an ideal battery. The circuit also contains two keys \(\mathrm{K}_1\) and \(\mathrm{K}_2\). Initially, both the keys are open and there is no charge on the capacitor. At an instant, key \(K_1\) is closed and immediately after this the current in \(R_0\) is found to be \(I_1\). After a long time, the current attains a steady state value \(I_2\). Thereafter, \(\mathrm{K}_2\) is closed and simultaneously \(\mathrm{K}_1\) is opened and the voltage across \(C_0\) oscillates with amplitude \(V_0\) and angular frequency \(\omega_0\).

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{T_R}{T_0}$ is:

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .

Consider a system of three connected strings, \(S_1, S_2\) and \(S_3\) with uniform linear mass densities \(\mu \mathrm{kg} / \mathrm{m}\), \(4 \mu \mathrm{~kg} / \mathrm{m}\) and \(16 \mu \mathrm{~kg} / \mathrm{m}\), respectively, as shown in the figure. \(S_1\) and \(S_2\) are connected at the point \(P\), whereas \(S_2\) and \(S_3\) are connected at the point \(Q\), and the other end of \(S_3\) is connected to a wall. A wave generator O is connected to the free end of \(S_1\). The wave from the generator is represented by \(y=y_0\) \(\cos (\omega t-k x) \mathrm{cm}\), where \(y_0, \omega\) and \(k\) are constants of appropriate dimensions. Which of the following statements is/are correct:

Statement I The formula connecting \(u, v\) and \(f\) for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature.
Statement II Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces.

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1=300\mathrm{K}\mathrm{a}\mathrm{n}\mathrm{d}{T}_2=100\mathrm{K}$as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1\mathrm{a}\mathrm{n}\mathrm{d}{K}_2$, respectively. If the temperature at the junction of the two cylinders in the steady state is $200\mathrm{}\mathrm{K}$, then $K_1/K_2$is

The dimensions of a cone are measured using a scale with a least count of \(2 \mathrm{~mm}\). The diameter of the base and the height are both measured to be \(20.0 \mathrm{~cm}\). The maximum percentage error in the determination of the volume is _______ .

The equation of a plane passing through the line of intersection of the planes \(x+2 y+3 z=2\) and \(x-y+z=3\)

and at a distance \(\frac{2}{\sqrt{3}}\) from the point \((3,1,-1)\) is

The reaction sequence given below is carried out with 16 moles of \(\mathbf{X}\). The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of \(\mathbf{S}\) produced is ______ .

Use: Atomic mass (in amu) : \(\mathrm{H}=1, \mathrm{C}=\) \(12, \mathrm{O}=16, \mathrm{Br}=80\)

If the distance between the plane \(A x-2 y+z=d\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then \(|d|\) is

Let \(\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) be three given vectors. If \(\mathbf{r}\) is a vector such that \(\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\) and \(\mathbf{r} \cdot \mathbf{a}=0\), then the value of \(\mathbf{r} \cdot \mathbf{b}\) is

The number of aldol reaction(s) that occurs in the given transformation is :