In Circuit-$1$ and Circuit-$2$ shown in the figures, $R_1=1\mathrm{\Omega },R_2=2\mathrm{\Omega }$ and $R_3=3\mathrm{\Omega }$.
$P_1$ and $P_2$ are the power dissipations in Circuit-$1$ and Circuit-$2$ when the switches $S_1$ and $S_2$ are in open conditions, respectively.
$Q_1$ and $Q_2$ are the power dissipations in Circuit-$1$ and Circuit-$2$ when the switches $S_1$ and $S_2$ are in closed conditions, respectively.

Which of the following statement(s) is(are) correct?

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 $K$ is ____.

A geostationary satellite above the equator is orbiting around the earth at a fixed distance \(r_1\) from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance \(r_2\) from the center of the earth, such that \(r_1=1.21 r_2\). The time period of the second satellite as measured from the geostationary satellite is \(\frac{24}{p}\) hours. The value of \(p\) is

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
The buoyancy force acting on the gas bubble is (Assume \(R\) is the universal gas constant)

The \(\beta\)-decay process, discovered around 1900 , is basically the decay of a neutron \((n)\). In the laboratory, a proton \((p)\) and an electron \(\left(e^{-}\right)\)are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has continuous spectrum. Considering a three-body decay process, i.e.
\(n \rightarrow p+e^{-}+\bar{v}_{e}\), around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino \(\left(\bar{v}_{e}\right)\) to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is \(0.8 \times 10^{6} \mathrm{eV}\). The kinetic energy carried by the proton is only the recoil energy.

Question:
If the anti-neutrino had a mass of \(3 \mathrm{eV} / \mathrm{c}^{2}\) (where \(\mathrm{c}\) is the speed of light) instead of zero mass, what should be the range of the kinetic energy, \(K\), of the electron?

Six point charges are kept at the vertices of a regular hexagon of side \(\mathrm{L}\) and centre \(O\), as shown in the figure.
Given that \(K=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{L^{2}}\), which of the following statement(s) is (are) correct?

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 $X$is ____.

Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures \(\mathrm{T}_{\mathrm{P}}\) and \(\mathrm{T}_{\mathrm{Q}}\), respectively, with \(\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}\), as shown in Fig. 1. The radiated power transferred per unit area from P to Q is \(W_0\). Subsequently, two more plates, identical to P and Q, are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from P to Q (Fig. 2) in the steady state is \(W_S\), then the ratio \(\frac{W_0}{W_S}\) is ________.

Among the species given below, the total number of diamagnetic species is _____________.
\(H\) atom, \(N O_2\) monomer, \(O_2^{-}\)(superoxide), dimeric sulphur in vapour phase,
\(Mn_3 O_4, \left(NH_4\right)_2\left[FeCl_4\right], \left(NH_4\right)_2\left[NiCl_4\right],\) \(K_2 MnO_4, K_2 CrO_4\)

The IUPAC name(s) of the following compound is(are)

Gravitational acceleration on the surface of a planet is \(\frac{\sqrt{6}}{11} \mathrm{~g}\), where \(g\) is the gravitational acceleration on the surface of the earth. The average mass density of the planet is \(\frac{2}{3}\) times that of the earth. If the escape speed on the surface of the earth is taken to be 11 \(\mathrm{kms}^{-1}\), the escape speed on the surface of the planet in \(\mathrm{kms}^{-1}\) will be

Statement I If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.
Statement II The linear momentum of an isolated system remains constant.

Sunlight of intensity $1.3 kW m^{-2}$ is incident normally on a thin convex lens of focal length $20 cm$ . Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $kW m^{-2}$ , at a distance $22 cm$ from the lens on the other side is __________.