In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch S₁ is pressed first to fully charge the capacitor C₁ and the released. The switch S₂ is then pressed to charge the capacitor C₂. After some time, S₂ is released and then is S₃ pressed. After some time,

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The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
The electric field at \(r=R\) is

A particle of mass M and positive charge Q, moving with a constant velocity ${\overrightarrow{\text{u}}}_1=4\hat{\text{i}}{\text{ms}}^{-1}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity ${\overrightarrow{\text{u}}}_2=2\left(\sqrt{3}\hat{\text{i}}+\hat{\text{j}}\right){\text{ms}}^{-1}$. The correct statement (s) is (are)

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)

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_______.

The circular scale of a screw gauge has 50 divisions and pitch of \(0.5 \mathrm{~mm}\). Find the diameter of sphere. Main scale reading is 2 .

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The allowed energy for the particle for a particular value of \(n\) is proportional to

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
\(T_r\) is always

A particle of mass $M=0.2 \mathrm{kg}$ is initially at rest in the $xy$-plane at a point $\left(x=-l, y=-h\right)$, where $l=10 \mathrm{m}$ and $h=1 \mathrm{m}$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10 \mathrm{m} {\mathrm{s}}^{-2}$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\overrightarrow{L}$ and  $\overrightarrow{\tau }$, respectively. $\widehat{\mathrm{i}}, \widehat{\mathrm{j}}$ and $\widehat{\mathrm{k}}$ are unit vectors along the positive $x,y$ and $z$-directions, respectively. If $\widehat{\mathrm{k}}=\widehat{\mathrm{i}}\times \widehat{\mathrm{j}}$ then which of the following statement(s) is (are) correct?

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

In an experiment to determine the focal length \((f)\) of a concave mirror by the \(u-v\) method, a student places the object pin \(A\) on the principal axis at a distance \(x\) from the pole \(P\). The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then

For the reaction, \(\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)\), the plot of \(\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}\) versus \(\frac{10^{4}}{T}\) is given below (in solid line), where \(p_{\mathbf{Z}}\) is the pressure (in bar) of the gas \(\mathbf{Z}\) at temperature \(T\) and \(p^{\theta}=1\) bar.

(Given, \(\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}\), where the equilibrium constant, \(K=\frac{p_{z}}{p^{\theta}}\) and the gas constant, \(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) )
The value of standard enthalpy, $∆{\mathrm{H}}^{\mathrm{o}}$ (in $\mathrm{kJ} {\mathrm{mol}}^{-1}$) for the given reaction is ___.

Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be a continuous function such that $f\left(0\right)=1$ and ${\int }_0^{\frac{\pi }{3}}f\left(t\right)dt=0$ Then which of the following statements is (are) TRUE?