An infinitely long thin wire, having a uniform charge density per unit length of \(5 \mathrm{nC} / \mathrm{m}\), is passing through a spherical shell of radius \(1 \mathrm{~m}\), as shown in the figure. A \(10 \mathrm{nC}\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points \(\mathrm{P}\) and \(\mathrm{R}\), in Volt, is _______ .
[Given: In SI units \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9, \ln 2=0.7\). Ignore the area pierced by the wire.]

A monochromatic light is incident from air on a refracting surface of a prism of angle ${75}^{\mathrm{o}}$ and refractive index ${\mathrm{n}}_0=\sqrt{3}.$ The other refracting surface of a prism is coated by a thin film of material of refractive index $\mathrm{n}$ as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of $\mathrm{\theta }\le {60}^{\mathrm{o}}.$ The value of ${\mathrm{n}}^2$ is_______.

A hydrogen atom in its ground state is irradiated by light of wavelength $970 Å.$ Taking hc/e = $1.237\times {10}^{-6} eV$ m and the ground state energy of hydrogen atom as $-13.6 eV$ , the number of lines present in the emission spectrum is

For the circuit shown in the figure

Paragraph I: In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of \(0.8 \mathrm{~mm}\). The distance between the slits at time \(t\) is given by \(d=(0.8+0.04 \sin \omega t) \mathrm{mm}\), where \(\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}\). The distance of the screen from the slits is \(1 \mathrm{~m}\) and the wavelength of the light used to illuminate the slits is \(6000 Ã…\). The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point \(O\).

The \(8^{\text {th }}\) bright fringe above the point \(\mathrm{O}\) oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer \((\mu \mathrm{m})\), is ___________ .

Two rectangular blocks, having identical dimensions, can be arranged either in configuration$\text{I}$ or in configuration $\text{I}\text{I}$ as shown in the figure. One of the blocks has thermal conductivity $\text{K}$ and the other $\text{2 K}$. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes $\text{9 s}$ to transport a certain amount of heat from the hot end to cold end in the configuration $\text{I}$. The time to transport the same amount of heat in the configuration $\text{II}$ is

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

The length of the strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4},$ respectively. Strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are vibrated at their $1^{\mathrm{s}\mathrm{t}},\mathrm{ }{3}^{\mathrm{r}\mathrm{d}},\mathrm{ }{5}^{\mathrm{t}\mathrm{h}}$ and ${14}^{\mathrm{t}\mathrm{h}}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.

\[
\text { Match the statements/expressions in Column I with the open intervals in Column II. }
\]

Paragraph :
\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.
Question :
When disc \(B\) is brought in contact with disc \(A\), they acquire a common angular velocity in time \(t\). The average frictional torque on one disc by the other during this period is

The correct option(s) regarding the complex ${\left[Co\left(en\right){\left(N{H}_3\right)}_3\left(H_{2}O\right)\right]}^{3+}$ :-
(en $= H_{2}NC{H}_{2}C{H}_{2}N{H}_2$ ) is (are)

A physical quantity $\overrightarrow{S}$ is defined as $\overrightarrow{S}=\frac{\left(\overrightarrow{E}\times \overrightarrow{B}\right)}{{\mu }_0}$. where $\overrightarrow{E}$ is electric field, $\overrightarrow{B}$ is magnetic field and ${\mu }_0$ is the permeability of free space. The dimensions of $\overrightarrow{S}$ are the same as the dimensions of which of the following quantity/quantities?

In a radioactive decay chain reaction, ${}_{90}^{230}\mathrm{Th}$ nucleus decays into ${}_{84}{}^{214}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of ${\beta }^{-}$ particles emitted in this process is _______.

Match the following