A biconvex lens of focal length \(f\) forms a circular image of radius \(r\) of sun in focal plane. Then, which option is correct?

One mole of a monatomic ideal gas is taken along two cyclic processes $\text{E}\to \text{F}\to \text{G}\to \text{E} and \text{E}\to \text{F}\to \text{H}\to \text{E}$ as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic.

Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using the codes given below the lists.
$\begin{array}{ccccc}& \text{List I}& & & \text{List II}\\ \text{A.}& \text{G}\mathrm{\to }\text{E }& & \mathrm{P}\text{.}& 16\mathrm{0 }{\text{P}}_0{\text{V}}_0\text{ ln}2\\ \text{B.}& \text{G}\mathrm{\to }\text{H }& & \mathrm{Q}\text{.}& 36{\text{ P}}_0{\text{V}}_0\\ \text{C.}& \text{F}\mathrm{\to }\text{H }& & \mathrm{R}\text{.}& 24{\text{ P}}_0{\text{V}}_0\\ \text{D.}& \text{F}\mathrm{\to }\text{G }& & \mathrm{S}\text{.}& 31{\text{ P}}_0{\text{V}}_0\end{array}$

The densities of two solid spheres $\mathrm{A}$ and $\mathrm{B}$ of the same radii $\mathrm{R}$ vary with radial distance $\mathrm{r}$ as ${\mathrm{\rho }}_{\mathrm{A}}\left(\mathrm{r}\right)=\mathrm{k}\left(\frac{\mathrm{r}}{\mathrm{R}}\right)$ and ${\mathrm{\rho }}_{\mathrm{B}}\left(\mathrm{r}\right)=\mathrm{k}{\left(\frac{\mathrm{r}}{\mathrm{R}}\right)}^5$ , respectively, where $\mathrm{k}$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ , respectively. If $\frac{{\mathrm{I}}_{\mathrm{B}}}{{\mathrm{I}}_{\mathrm{A}}}=\frac{\mathrm{n}}{10}$ , the value of $\mathrm{n}$ is

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:

A student is performing an experiment using a resonance column and a tuning fork of frequency \(244 s^{-1}\). He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is \((0.350 \pm 0.005) m\), the gas in the tube is (Useful information: \(\sqrt{167\text{ RT}}=640\text{ J}^{\frac{1}{2}}\text{ mol}^{-\frac{1}{2}} ;\) \(\sqrt{140\text{ RT}}=590\text{ J}^{\frac{1}{2}}\text{ mole}^{-\frac{1}{2}}\) . The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{\text M }}\) for each gas as given there.)

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question :The relation between $\left[E\right]$ and $\left[B\right]$ is

The largest wavelength in the ultraviolet region of the hydrogen spectrum is \(122 \mathrm{~nm}\). The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3}a\cos x+2b\sin x=c, x\in \left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha +\beta =\frac{\pi }{3}$. Then, the value of $\frac{b}{a}$ is

Let \(S_n=\sum_{k=0}^n \frac{n}{n^2+k n+k^2}\) and \(T_n=\sum_{k=0}^{n-1} \frac{n}{n^2+k n+k^2}\), for \(n=1,2,3, \ldots\), then

In the given reaction scheme, $\mathbf{P}$ is a phenyl alkyl ether, $\mathbf{Q}$  is an aromatic compound; $\mathbf{R}$ and $\mathbf{S}$ are the major products.

The correct statement about $\mathbf{S}$ is

Paragraph:
Consider the lines: \(L_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}, L_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}\)Question:
The shortest distance between \(L_1\) and \(L_2\) is

Choose the correct reason(s) for the stability of the lyophobic colloidal particles.

Paragraph:
A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is