In a Young's double slit experiment, the slit separation $\mathrm{d}$ is $0.3\mathrm{}\mathrm{m}\mathrm{m}$ and the screen distance $\mathrm{D}$ is $1\mathrm{m}.$ A parallel beam of light of wavelength $600\mathrm{}\mathrm{n}\mathrm{m}$ is incident on the slits at angle $\alpha$ as shown in figure. On the screen, the point $\mathrm{O}$ is equidistant from the slits and distance $\mathrm{P}\mathrm{O}$ is $11.0\mathrm{}\mathrm{m}\mathrm{m}.$ Which of the following statement(s) is/are correct?

In the circuit shown below, the switch \(\mathrm{S}\) is connected to position \(\mathrm{P}\) for a long time so that the charge on the capacitor becomes \(q_{1} \mu \mathrm{C}\). Then \(\mathrm{S}\) is switched to position \(\mathrm{Q}\). After a long time, the charge on the capacitor is \(q_{2} \mu \mathrm{C}\).

The magnitude of $q_2$ is _____.

Statement 1 The sensitivity of a moving coil galvanometer is increased by placing a suitable magnetic material as a core inside the coil.
and Statement 2 Soft iron has a high magnetic permeability and cannot be easily magnetized or demagnetized.

A ball is thrown from the location \(\left(x_0, y_0\right)=(0,0)\) of a horizontal playground with an initial speed \(v_0\) at an angle \(\theta_0\) from the \(+x\)-direction. The ball is to be hit by a stone, which is thrown at the same time from the location \(\left(x_1, y_1\right)=(L, 0)\). The stone is thrown at an angle \(\left(180-\theta_1\right)\) from the \(+x\)-direction with a suitable initial speed. For a fixed \(v_0\), when \(\left(\theta_0, \theta_1\right)=\left(45^{\circ}, 45^{\circ}\right)\), the stone hits the ball after time \(T_1\), and when \(\left(\theta_0, \theta_1\right)=\left(60^{\circ}, 30^{\circ}\right)\), it hits the ball after time \(T_2\). In such a case, \(\left(T_1 / T_2\right)^2\) is _______ .

A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case will the particle describe a helical path with axis along positive z direction?

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The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:
The kinetic energy (in \(k e V\) ) of the alpha particle, when the nucleus \({ }_{84}^{210} P o\) at rest undergoes alpha decay, is

Using the expression $2d \sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angle $\text{θ}$ in the range $0$ to $90°$. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all the values of $\theta$. As $\text{θ}$ increases from $0°$,

Let \(f:(-1,1) \rightarrow I R\) be such that \(f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}\) for \(\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\). Then the value (s) of \(f\left(\frac{1}{3}\right)\) is (are)

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Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function. Suppose that \(F(1)=0, F(3)=-4\) and \(F^{\prime}(x)<0\) for all \(x \in(1 / 2,3)\). Let \(f(x)=x F(x)\) for all \(x \in \mathbb{R}\).


Question:

The correct statement(s) is(are)

For the process ${\mathrm{H}}_2\mathrm{O}\mathrm{ }\left(\mathcal{l}\right)\to {\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)\mathrm{ }$ at $\mathrm{T}={100}^{\mathrm{o}}\mathrm{C}\mathrm{ }$ and 1 atmosphere pressure, the correct choice is

Four point charges, each of \(+q\), are rigidly fixed at the four corners of a square planar soap film of side \(a\). The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and \(a=k\left[\frac{q^2}{\gamma}\right]^{1 / N}\), where \(k\) is a constant. Then, \(N\) is

Let $f :\left[0, 4\pi \right]\to \left[0, \pi \right]$ be defined by $f\left(x\right)={\cos }^{-1}\left(\cos x\right).$ The number of points $x \in \left[0, 4\pi \right]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$ is_________

Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then