A source of constant voltage $\mathrm{V}$ is connected to a resistance$\mathrm{R}$ and two ideal inductors ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ through a switch $\mathrm{S}$ as shown. There is no mutual inductance between the two inductors. The switch $\mathrm{S}$ is initially open. At $\mathrm{t}=0$ , the switch is closed and current begins to flow. Which of the following options is/are correct?

In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?

A pendulum consists of a bob of mass \(m=0.1 \mathrm{~kg}\) and a massless inextensible string of length \(L=1.0 \mathrm{~m}\). It is suspended from a fixed point at height \(H=0.9 \mathrm{~m}\) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \(P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}\) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \(J \mathrm{~kg}-\mathrm{m}^{2} / \mathrm{s}\). The kinetic energy of the pendulum just after the liftoff is \(K\) Joules.
The value of $J$ is ____.

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A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \(N\) be the number density of free electrons, each of mass \(m\). When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions.If the electric field becomes zero, the electrons being to oscillate about the positive ions with a natural angular frequency \(\omega_p\), which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency \(\omega\), where a part of the energy is absorbed and a part of it is reflected. As \(\omega\) approaches \(\omega_p\), all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Question :
Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons \(N \approx 4 \times 10^{27} \mathrm{~m}^{-1}\). Take \(\varepsilon_0=10^{-11}\) and \(m=10^{-30}\), where these quantities are in proper SI units.

Two solid cylinders \(P\) and \(Q\) of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder \(P\) has most of its mass concentrated near its surface, while \(Q\) has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas $\left(\gamma =\frac{5}{3}\right)$ and one mole of an ideal diatomic gas $\left(\gamma =\frac{7}{5}\right)$. Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of $66$ Joule when heated at constant pressure. The change in its internal energy is _____ Joule.

Students I, II and III perform an experiment for measuring the acceleration due to gravity \((g)\) using a simple pendulum. They use different lengths of the pendulum and/or record time for different number of oscillations. The observations are shown in the table.
Least count for length \(=0.1 \mathrm{~cm}\)
Least count for time \(=0.1 \mathrm{~s}\)

If \(E_{\mathrm{I}}, E_{\mathrm{II}}\) and \(E_{\mathrm{III}}\) are the percentage errors in \(g\), i.e., \(\left(\frac{\Delta g}{g} \times 100\right)\) for students I, II and III, respectively.

Let $\overline{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of ${\left(\overline{z}\right)}^2+\frac{1}{z^2}$ are integers, then which of the following is/are possible value(s) of $\left|z\right|$ ?

Statement 1 The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.
and Statement 2 In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.

A thermodynamic system is taken from an initial state $\mathrm{i}$ with internal energy ${\mathrm{U}}_{\mathrm{i}}\mathrm{}=100\mathrm{}\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are ${\mathrm{w}}_{\mathrm{a}\mathrm{f}}=\mathrm{}200\mathrm{}\mathrm{J},\mathrm{}{\mathrm{W}}_{\mathrm{i}\mathrm{b}}\mathrm{}=\mathrm{}50\mathrm{}\mathrm{J}$ and ${\mathrm{W}}_{\mathrm{b}\mathrm{f}}\mathrm{}=\mathrm{}100\mathrm{}\mathrm{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are iaf ib ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}},\mathrm{}{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}$ and ${\mathrm{Q}}_{\mathrm{b}\mathrm{f}}$ respectively. If the internal energy of the system in the state b is ${\mathrm{U}}_{\mathrm{b}}=200\mathrm{}\mathrm{J}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}}=500\mathrm{}\mathrm{J}$ , the ratio $\frac{{\mathrm{Q}}_{\mathrm{b}\mathrm{f}}}{{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}}\mathrm{}$ is

A thin conducting rod $MN$ of mass $20 \mathrm{gm}$, length $25 \mathrm{cm}$ and resistance $10\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0=4 \mathrm{T}$ directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t=0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$, and choose the correct option. [Given: The acceleration due to gravity $g=10 {\mathrm{ms}}^{-2}$ and $e^{-1}=0.4$]  

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	At $t=0.2 \mathrm{s}$, the magnitude of the induced emf in Volt	$1$	$0.07$
$\mathrm{Q}$	At $t=0.2 \mathrm{s}$, the magnitude of the magnetic force in Newton	$2$	$0.14$
$\mathrm{R}$	At $t=0.2 \mathrm{s}$, the power dissipated as heat in Watt	$3$	$1.20$
$\mathrm{S}$	The magnitude of terminal velocity of the rod in $\mathrm{m} {\mathrm{s}}^{-1}$	$4$	$0.12$
		$5$	$2.00$

A factory has a total of three manufacturing units, \(M_1, M_2\), and \(M_3\), which produce bulbs independent of each other. The units \(M_1, M_2\), and \(M_3\) produce bulbs in the proportions of 2:2:1, respectively. It is known that \(20 \%\) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \(M_1, 15 \%\) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \(M_2\) is \(\frac{2}{5}\).
If a bulb is chosen randomly from the bulbs produced by \(M_3\), then the probability that it is defective is ____.

The reagent(s) for the following conversion,

is/are