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A heavy nucleus \( N \), at rest, undergoes fission \( N \rightarrow P + Q \), where \( P \) and \( Q \) are two lighter nuclei. Let \( \delta = M_{N} - M_{P} - M_{Q} \), where \( M_{P}, M_{Q} \) and \( M_{N} \) are the masses of \( P, Q \) and \( N \), respectively. \( E_{P} \) and \( E_{Q} \) are the kinetic energies of \( P \) and \( Q \), respectively. The speeds of \( P \) and \( Q \) are \( v_{P} \) and \( v_{Q} \), respectively. If \( c \) is the speed of light, which of the following statement(s) is (are) correct?

In a radioactive decay process, the activity is defined as \( A = \dfrac{-dN}{dt} \), where \( N(t) \) is the number of radioactive nuclei at time \( t \). Two radioactive sources, \( S_1 \) and \( S_2 \) have same activity at time \( t = 0 \). At a later time, the activities of \( S_1 \) and \( S_2 \) are \( A_1 \) and \( A_2 \), respectively. When \( S_1 \) and \( S_2 \) have just completed their \( 3^{\text{rd}} \) and \( 7^{\text{th}} \) half-lives, respectively, the ratio \( \dfrac{A_1}{A_2} \) is __________.

In the List-I four optical effects are mentioned. The physical phenomena of light which are essential to describe these optical effects are given in List-II. Choose the option which describes the correct match between the entries in List-I to those in List-II.

A heavy nucleus \( Q \) of half-life \( 20 \) minutes undergoes alpha-decay with probability of \( 60\% \) and beta-decay with probability of \( 40\% \). Initially, the number of \( Q \) nuclei is \( 1000 \). The number of alpha-decays of \( Q \) in the first one hour is :

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction \( ^{16}_{7}\mathrm{N} + ^{4}_{2}\mathrm{He} \rightarrow \, ^{1}_{1}\mathrm{H} + ^{19}_{8}\mathrm{O} \) in a laboratory frame is \( n \) (in \( \mathrm{MeV} \)). Assume that \( ^{16}_{7}\mathrm{N} \) is at rest in the laboratory frame. The masses of \( ^{16}_{7}\mathrm{N} \), \( ^{4}_{2}\mathrm{He} \), \( ^{1}_{1}\mathrm{H} \) and \( ^{19}_{8}\mathrm{O} \) can be taken to be \( 16.006 \, \mathrm{u} \), \( 4.003 \, \mathrm{u} \), \( 1.008 \, \mathrm{u} \) and \( 19.003 \, \mathrm{u} \), respectively, where \( 1\,\mathrm{u} = 930 \, \mathrm{MeV}\,c^{-2} \). The value of \( n \) is ________.

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 \( 4.0 \,\text{eV} \), \( 4.5 \,\text{eV} \) and \( 5.5 \,\text{eV} \), respectively. The energy of the incident photon used for metal \( R \), in \( \text{eV} \), is _____.

Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option.
[Given: Wien’s constant as \( 2.9 \times 10^{-3} \, \text{m-K} \) and \( \dfrac{hc}{e} = 1.24 \times 10^{-6} \, \text{eV-m} \) ]

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

A Hydrogen-like atom has atomic number \( Z \). Photons emitted in the electronic transitions from level \( n = 4 \) to level \( n = 3 \) in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is \( 1.95 \, \text{eV} \). If the photoelectric threshold wavelength for the target metal is \( 310 \, \text{nm} \), the value of \( Z \) is ______.
[Given: \( hc = 1240 \, \text{eV} \! \cdot \! \text{nm} \) and \( Rhc = 13.6 \, \text{eV} \), where \( R \) is the Rydberg constant, \( h \) is the Planck’s constant and \( c \) is the speed of light in vacuum]

A particle of mass \( m \) is moving in a circular orbit under the influence of the central force \( F(r)=-kr \), corresponding to the potential energy \( V(r)=\dfrac{kr^{2}}{2} \), where \( k \) is a positive force constant and \( r \) is the radial distance from the origin. According to the Bohr’s quantization rule, the angular momentum of the particle is given by \( L=n\hbar \), where \( \hbar=\dfrac{h}{2\pi} \), \( h \) is the Planck’s constant, and \( n \) a positive integer. If \( v \) and \( E \) are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

List-I shows various functional dependencies of energy \( (E) \) on the atomic number \( (Z) \). Energies associated with certain phenomena are given in List-II.
Choose the option that describes the correct match between the entries in List-I to those in List-II.

A nuclear reactor starts producing a radioactive nuclide \(X\) from \(t = 0\), at a constant rate of \(\alpha\) per second. Each decay of \(X\) produces energy \(E_{0}\), which is utilized to heat a liquid of mass \(m\) and specific heat \(s\). Assuming no heat loss from the liquid and taking \(\lambda\) as the decay constant of \(X\), the rate of increase in the temperature of the liquid is:

Consider an electron in the \( n = 3 \) orbit of a hydrogen-like atom with atomic number \( Z \). At absolute temperature \( T \), a neutron having thermal energy \( k_B T \) has the same de Broglie wavelength as that of this electron. If this temperature is given by \( T = \dfrac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B} \), (where \( h \) is the Planck’s constant, \( k_B \) is the Boltzmann constant, \( m_N \) is the mass of the neutron and \( a_0 \) is the first Bohr radius of hydrogen atom) then the value of \( \alpha \) is ________.

Consider a hydrogen atom with \( v_{k} \), \( r_{k} \), and \( K_{k} \) denoting the velocity, orbital radius and kinetic energy of the electron in the \( k^{\text{th}} \) orbit, respectively. The electron undergoes a transition from the \( n^{\text{th}} \) orbit, emitting radiation corresponding to the Lyman series. Considering \( h \) to be the Planck’s constant and \( \epsilon_{0} \) the permittivity of the free space, the correct statement(s) is/are:

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 \( 10 \, \text{eV} \). 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 \( 5 \, \text{eV} \). 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 ______.

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be \( E_b^p \) and the binding energy of a neutron be \( E_b^n \) in the nucleus.
Which of the following statement(s) is(are) correct?

In a radioactive decay chain reaction, \( {}^{230}_{90}\mathrm{Th} \) nucleus decay into \( {}^{214}_{84}\mathrm{Po} \) nucleus. The ratio of the number of \( \alpha \) to number of \( \beta^{-} \) particles emitted in this process is _____.

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.