In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10\pm 0.1 \mathrm{cm}$ and the distance of its real image from the lens is $20\pm 0.2 \mathrm{cm}$. The error in the determination of focal length of the lens is $n\%$. The value of $n$ is _____.

Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures \(\mathrm{T}_{\mathrm{P}}\) and \(\mathrm{T}_{\mathrm{Q}}\), respectively, with \(\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}\), as shown in Fig. 1. The radiated power transferred per unit area from P to Q is \(W_0\). Subsequently, two more plates, identical to P and Q, are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from P to Q (Fig. 2) in the steady state is \(W_S\), then the ratio \(\frac{W_0}{W_S}\) is ________.

For the resistance network shown in the figure, choose the correct option(s)

Let ${\mathrm{E}}_1\left(\mathrm{r}\right),\mathrm{ }{\mathrm{E}}_2\left(\mathrm{r}\right)$ and ${\mathrm{E}}_3\left(\mathrm{r}\right)$ be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $\mathrm{\lambda }$ , and an infinite plane with uniform surface charge density $\mathrm{\sigma }$ . If ${\mathrm{E}}_1\left({\mathrm{r}}_0\right)={\mathrm{E}}_2\left({\mathrm{r}}_0\right)={\mathrm{E}}_3\left({\mathrm{r}}_0\right)$ at a given distance ${\mathrm{r}}_0$ , then

A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is C while that of the portion with dielectric in between is ${\mathrm{C}}_1$ . When the capacitor is charged, the plate area covered by the dielectric gets charge ${\mathrm{Q}}_1$ and the rest of the area gets charge ${\mathrm{Q}}_2$ . The electric field in the dielectric is ${\mathrm{E}}_1$ and that in the other portion is ${\mathrm{E}}_2$ . Choose the correct option/options, ignoring edge effects.

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in $\mathrm{m} {\mathrm{s}}^{-1}$) at time $t=\frac{\pi }{3} \mathrm{s}$.

	List-I		List-II
(I)	$A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{m}$ with uniform angular speed $\omega =1 \mathrm{rad} {\mathrm{s}}^{-1}$. The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta =0$ and $\theta =\frac{\pi }{2}$ respectively.	(P)	$\frac{\sqrt{3}+1}{2}$
(II)	Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{s}$ respectively, with the same speed $v=\frac{5\pi }{\sqrt{2}} \mathrm{m} {\mathrm{s}}^{-1}$ and at $45°$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide, $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$.	(Q)	$\frac{\left(\sqrt{3}-1\right)}{\sqrt{2}}$
(III)	Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_A=x_0\sin \frac{t}{t_0}$ and $x_B=x_0\sin \left(\frac{t}{t_0}+\frac{\pi }{2}\right)$ respectively, starting from $t=0$. Take $x_0=1 \mathrm{m},t_0=1 \mathrm{s}$.	(R)	$\sqrt{10}$
(IV)	Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{m}$ on the $xy$ plane, with constant angular speed $\omega =1 \mathrm{rad} {\mathrm{s}}^{-1}$. Particle $B$ is moving up at a constant speed $3 \mathrm{m} {\mathrm{s}}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S)	$\sqrt{2}$
		(T)	$\sqrt{25{\pi }^2+1}$

Which one of the following options is correct?

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Two trains \(A\) and \(B\) are moving with speeds \(20 \mathrm{~m} / \mathrm{s}\) and \(30 \mathrm{~m} / \mathrm{s}\), respectively in the same direction on the same straight track, with \(B\) ahead of \(A\). The engines are at the front ends. The engine of train \(A\) blows a long whistle.
Assume that the sound of the whistle is composed of components varying in frequency from \(f_1=800 \mathrm{~Hz}\) to \(f_2=1120 \mathrm{~Hz}\), as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus \(320 \mathrm{~Hz}\). The speed of sound in still air is \(340 \mathrm{~m} / \mathrm{s}\).
Question:
The speed of sound of the whistle is

On decreasing the $\mathrm{pH}$ from $7$ to $2$, the solubility of a sparingly soluble salt $\left(\mathrm{MX}\right)$ of a weak acid $\left(\mathrm{HX}\right)$ increased from ${10}^{-4} \mathrm{mol} {\mathrm{L}}^{-1}$ to ${10}^{-3} \mathrm{mol} {\mathrm{L}}^{-1}$. The ${\mathrm{pK}}_{\mathrm{a}}$ of $\mathrm{HX}$ is

In a neutral or faintly alkaline solution, $8$ moles of permanganate anion quantitatively oxidize thiosulphate anions to produce $\mathrm{X}$ moles of a sulphur containing product. The magnitude of $\mathrm{X}$ is:

In the following reaction sequence in aqueous solution, the species $\text{X,}\text{Y}$ and $\text{Z}$ respectively, are
\(S _2 O _3^{2-} \xrightarrow{ Ag ^{+}} \underset{\substack{\text { Clear } \\ \text { Solution }}}{ X } \xrightarrow{ Ag ^{+}} \underset{\substack{\text { White } \\ \text { Precipitate }}}{ Y }\) \(\xrightarrow{\text { With time }} \underset{\substack{\text { Black } \\ \text { Precipitate }}}{ Z }\)

A student skates up a ramp that makes an angle $30°$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity)

The difference in the oxidation numbers of the two types of sulphur atoms in \(\mathrm{Na}_2 \mathrm{~S}_4 \mathrm{O}_6\) is

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )