A small object is placed at the centre of a large evacuated hollow spherical container. Assume the container is maintained at $0 \mathrm{K}$. At time $t=0$, the temperature of the object is $200 \mathrm{K}$. The temperature of the object becomes $100 \mathrm{K}$ at $t=t_1$ and $50 \mathrm{K}$ at $t=t_2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left(\frac{t_2}{t_1}\right)$ _________ .

Two small particles of equal masses start moving in opposite directions from a point \(A\) in a horizontal circular orbit. Their tangential velocities are \(v\) and \(2 v\) respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at \(A\), these two particles will again reach the point \(A\) ?

A positive point charge of \(10^{-8} \mathrm{C}\) is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm . The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2\) (where \(\epsilon_0\) is the permittivity of free space), which of the following
statements is/are correct: :

Paragraph:
Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Assume that two deuteron nuclei in the core of fusion reactor at temperature \(T\) are moving towards each other, each with kinetic energy \(1.5 \mathrm{kT}\), when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature \(T\) required for them to reach a separation of \(4 \times 10^{-15} \mathrm{~m}\) is in the range

Three identical capacitors $C_1,C_2\mathrm{a}\mathrm{n}\mathrm{d}{C}_3$ have a capacitance of $1.0\mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity ${ϵ}_r$ . The cell electromotive force (emf) $V_0=8V$ . First the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\mu C$ . The value of ${ϵ}_r$ =____________.

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $P,Q$ and $R$ are $E_P, E_Q$ and $E_R$, respectively, and they are related by $E_P=2E_Q=2E_R$. In this experiment, the same source of monochromatic light is used for metals $P$ and $Q$ while a different source of monochromatic light is used for the metal $R$. The work functions for metals $P,Q$ and $R$ are $4.0 \mathrm{eV}, 4.5 \mathrm{eV}$ and $5.5 \mathrm{eV}$, respectively. The energy of the incident photon used for metal $R$, in $eV$ is ___.

Paragraph :
\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.
Question :
When disc \(B\) is brought in contact with disc \(A\), they acquire a common angular velocity in time \(t\). The average frictional torque on one disc by the other during this period is

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

The sum of the spin only magnetic moment values (in B.M.) of \(\left[\mathrm{Mn}(\mathrm{Br})_6\right]^{3-}\) and \(\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-}\) is _____.

Paragraph:
Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The function \(f^{\prime}(x)\) is

Statement 1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and Statement 2 The electrical potential of a sphere of radius \(R\) with charge \(Q\) uniformly distributed on the surface is given by \(\frac{Q}{4 \pi \varepsilon_0 R}\).

Let \(f\) be a real-valued function defined on the interval \((-1,1)\) such that \(e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad\) for all \(x \in(-1,1)\) and let \(f^{-1}\) be the inverse function of \(f\). Then \(\left(f^{-1}\right)^{\prime}(2)\) is equal to

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right-hand limit $\underset{x\to 0^{+}}{\lim }\frac{{\left(1-x\right)}^{\frac{1}{\chi }}-e^{-1}}{x^a}$ is equal to a non-zero real number, is