You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of \(50 \mathrm{~Hz}\) frequency (the next three circuits) in different ways as shown in Column II. When a current \(I\) (steady state for DC or rms for \(A C\) ) flows through the circuit, the corresponding voltage \(V_1\) and \(V_2\) (indicated in circuits) are related as shown in Column I. Match the two

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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:
If the mass of the particle is \(m=1.0 \times 10^{-30} \mathrm{~kg}\) and \(a=6.6 \mathrm{~nm}\), the energy of the particle in its ground state is closest to

Consider an elliptically shaped rail PQ in the vertical plane with $\mathrm{O}\mathrm{P}\mathrm{}=\mathrm{}3\mathrm{}\mathrm{m}$ and $\mathrm{O}\mathrm{Q}\mathrm{}=\mathrm{}4\mathrm{}\mathrm{m}$ . A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is $(\mathrm{n}\times 10)$ Joules. The value of n is (take acceleration due to gravity = $10\mathrm{}\mathrm{m}{\mathrm{s}}^{–2}$ )

Light of wavelength ${\lambda }_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\varphi$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is ${\lambda }_e,$ which of the following statement(s) is(are) true ?

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 $10\mathrm{}\mathrm{c}\mathrm{m}$ long perfectly conducting wire $\mathrm{P}\mathrm{Q}$ is moving, with a velocity $1\mathrm{}\mathrm{c}\mathrm{m}/\mathrm{s}$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $\mathrm{L}=1\mathrm{}\mathrm{m}\mathrm{H}$ and a resistance $\mathrm{R}=1\mathrm{\Omega }$ as shown in figure. The horizontal rails, $\mathrm{L}$ and $\mathrm{R}$ lie in the same plane with a uniform magnetic field $\mathrm{B}=1\mathrm{}\mathrm{T}$ perpendicular to the plane. If the key $\mathrm{S}$ is closed at certain instant, the current in the circuit after $1$ milli second is $\mathrm{x}\times {10}^{-3}\mathrm{A},$ where the value of $\mathrm{x}$ is_______.
[Assume the velocity of wire $\mathrm{P}\mathrm{Q}$ remains constant $\left(1\mathrm{c}\mathrm{m}/\mathrm{s}\right)$ after key $\mathrm{S}$ is closed. Given: ${\mathrm{e}}^{-1}=0.37,$ where $\mathrm{e}$ is base of the natural logarithm]

A piece of wire is bent in the shape of a parabola \(y=k x^2\) (y-axis vertical) with a bead of mass \(m\) on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the \(x\)-axis with the constant acceleration \(a\). The distance of the new equilibrium position of the bead, where the bead can stays at rest with respect to the wire, from the \(y\)-axis is

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 _____.

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is \(0.5 \mathrm{~mm}\) and there are 50 divisions on the circular scale. The reading on the main scale is \(2.5 \mathrm{~mm}\) and that on the circular scale is 20 divisions. If the measured mass of the ball has a relative error of \(2 \%\), the relative percentage error in the density is

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$ . Which of the following statements is (are) true?

Among the following, the coloured compound is

The positon vector \(\vec{r}\) of a particle of mass m is given by the following equation \(\vec{r}(t)=\alpha t^3 \hat{i} +\beta t^2 \hat{j}\), where \(\alpha=\frac{10}{3} m s^{-3}, \beta=5 m s^{-2}\) and \(m=0.1 kg\). At \(t=1 s\), which of the following statements (s) is (are) true about the particle?

A Young's double slit interference arrangement with slits ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ is immersed in water (refractive index $=\frac{4}{3}$ ) as shown in the figure. The positions of maxima on the surface of water are given by ${\mathrm{x}}^2={\mathrm{p}}^2{\mathrm{m}}^2{\mathrm{\lambda }}^2-{\mathrm{d}}^2$ , where $\mathrm{\lambda }$ is the wavelength of light in air (refractive index = 1), $2\mathrm{d}$ is the separation between the slits and m is an integer. The value of p is