A uniform magnetic field B exists in the region between $x=0$ and $x=\frac{3R}{2}$ (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge $+Q$ and momentum p directed along $x$ -axis enters region 2 from region 1 at point $P_1\left(y=-R\right).$ Which of the following option(s) is/are correct?

Paragraph:
In a mixture of \(\mathrm{H}-\mathrm{H}^{+}\)gas \(\left(\mathrm{He}^{+}\right.\)is singly ionised \(\mathrm{He}\) atom), \(\mathrm{H}\) atoms and \(\mathrm{He}^{+}\)ions are excited to their respective first excited states. Subsequently, \(\mathrm{H}\) atoms transfer their total excitation energy to \(\mathrm{He}^{+}\)ions (by collisions). Assume that the Bohr model of atom is exactly valid.
Question:
The ratio of the kinetic energy of the \(n=2\) electron for the \(\mathrm{H}\) atom to that of \(\mathrm{He}^{+}\) ion is

An audio transmitter \((T)\) and a receiver \((R)\) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the \(X\) axis. They are pulled from the equilibrium position in opposite directions along the \(X\) axis by a small angular amplitude \(\theta_0=\cos ^{-1}(0.9)\) and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the maximum variation in the frequency (in Hz ) as measured by the receiver (Take the acceleration due to gravity \(g=10 \mathrm{~m} / \mathrm{s}^2\) ) is \(\qquad\)

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $xy$-plane with the center of the loop always on the axis $\hat{\mathrm{n}}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counter clockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{\mathrm{n}}$ direction. List-I gives time dependences of $\hat{\mathrm{n}}$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t=\frac{\pi }{6\omega }$, Let $\alpha =\frac{A^2{\mu }_0^2{m}^2{I}^2\omega }{2R}$.

	List-I		List-II
(i)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{j}}+\cos \omega t\hat{\mathrm{k}}\right)$	(p)	$0$
(ii)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{i}}+\cos \omega t\hat{\mathrm{j}}\right)$	(q)	$-\frac{\alpha }{4}\hat{\mathrm{i}}$
(iii)	$\frac{1}{\sqrt{2}}\left(\sin \omega t\hat{\mathrm{i}}+\cos \omega t\hat{\mathrm{k}}\right)$	(r)	$\frac{3\alpha }{4}\hat{\mathrm{i}}$
(iv)	$\frac{1}{\sqrt{2}}\left(\cos \omega t\hat{\mathrm{j}}+\sin \omega t\hat{\mathrm{k}}\right)$	(s)	$\frac{\alpha }{4}\hat{\mathrm{j}}$
		(t)	$-\frac{3\alpha }{4}\hat{\mathrm{i}}$

Which one of the following options is correct?

A point charge $+Q$ is placed just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct?

Statement 1 For an observer looking out through the window of a fast moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary.
and Statement 2 If the observer and the object are moving at velocities \(\overrightarrow{\mathbf{v}}_{\mathbf{1}}\) and \(\overrightarrow{\mathbf{v}_2}\) respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is \(\overrightarrow{\mathbf{v}_2}-\overrightarrow{\mathbf{v}_1}\).

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $Y$ is _____.

When benzene sulphonic acid and \(p\)-nitrophenol are treated with \(\mathrm{NaHCO}_3\), the gases released respectively are

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if } x=0\end{cases}\)
Then which of the following statements is (are) TRUE?

Paragraph:
Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.

Question:
Which one of the following reagents is used the above reaction?

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially the gas is at temperature ${\mathrm{T}}_1$ , pressure ${\mathrm{P}}_1$ and volume ${\mathrm{V}}_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature ${\mathrm{T}}_2$ , pressure ${\mathrm{P}}_2$ and volume ${\mathrm{V}}_2$ . During this process the piston moves out by a distance $\mathrm{x}$ . Ignoring the friction between the piston the cylinder, the correct statement(s) is (are)

Paragraph:
An aqueous solution of a mixture of two inorganic salts, when treated with dilute \(H C\) ?, gave a precipitate \((\boldsymbol{P})\) and a filtrate \((\boldsymbol{Q})\). The precipitate \(\boldsymbol{P}\) was found to dissolve in hot water. The filtrate \((\boldsymbol{Q})\) remained unchanged, when treated with \(\mathrm{H}_{2} \mathrm{~S}\) in a dilute mineral acid medium. However, it gave a precipitate \((\boldsymbol{R})\) with \(H_{2} S\) in an ammoniacal medium. The precipitate \(\boldsymbol{R}\) gave a coloured solution \((\boldsymbol{S})\), when treated with \(\mathrm{H}_{2} \mathrm{O}_{2}\) in an aqueous \(\mathrm{NaOH}\) medium.

Question:
The precipitate \(\boldsymbol{P}\) contains

In the following reactions, P, Q, R and S are the major products.

The correct statement(s) about P, Q, R and S is(are)