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JEE Mains | PYQ

If a radioactive element having half-life of \( 30 \, \text{min} \) is undergoing beta decay, the fraction of radioactive element remains undecayed after \( 90 \, \text{min} \). will be:

Consider the following radioactive decay process
\( {}^{218}_{84}A \xrightarrow{\alpha} A_1 \xrightarrow{\beta^{-}} A_2 \xrightarrow{\gamma} A_3 \xrightarrow{\alpha} A_4 \xrightarrow{\beta^{+}} A_5 \xrightarrow{\gamma} A_6 \)
The mass number and the atomic number of \( A_6 \) are given by:

A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of \( 24 \, \text{W} \). The radius of curvature of hemisphere is \( 10 \, \text{cm} \) and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is _______ \( \times 10^{-8} \, \text{N} \).

The threshold wavelength for photoelectric emission from a material is 5500 Å. Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a
A. \( 75 \, \text{W} \) infra-red lamp
B. \( 10 \, \text{W} \) infra-red lamp
C. \( 75 \, \text{W} \) ultra-violet lamp
D. \( 10 \, \text{W} \) ultra-violet lamp
Choose the correct answer from the options given below:

Match List I with List II:

Choose the correct answer from the options given below:

An \( \alpha \) – particle, a proton and an electron have the same kinetic energy. Which one of the following is correct in case of their de-Broglie wavelength:

From the photoelectric effect experiment, following observations are made. Identify which of these are correct.
I. The stopping potential depends only on the work function of the metal
II. The saturation current increase as the intensity of incident light increases
III. The maximum kinetic energy of a photo electron depends on the intensity of the incident light
IV. Photoelectric effect can be explained using wave theory of light
Choose the correct answer from the options given below:

A common example of alpha decay is: \( {}^{238}_{92}\text{U} \longrightarrow {}^{234}_{90}\text{Th} + {}^{4}_{2}\text{He} + Q \).
Given:
\( {}^{238}_{92}\text{U} = 238.05060\,\text{u}, \)
\( {}^{234}_{90}\text{Th} = 234.04360\,\text{u}, \)
\( {}^{4}_{2}\text{He} = 4.00260\,\text{u} \) and
\( 1\,\text{u} = 931.5\,\dfrac{MeV}{{C}^{2}} \).
The energy released \( (Q) \) during the alpha decay of \( {}^{238}_{92}\text{U} \) is _____ \( \text{MeV} \).

A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of \( \lambda \). An alpha particle having certain kinetic energy has the same de-Broglie wavelength \( \lambda \). The ratio of kinetic energy of proton and that of alpha particle is:

Given below are two statements: one is labelled as Assertion A and other is labelled as Reason R.
Assertion A: EM waves used for optical communication have longer wavelengths than that of microwave, employed in Radar technology.
Reason R: Infrared EM waves are more energetic than microwaves, (used in Radar)
In the light of given statements, choose the correct answer from the options given below.

Substance A has atomic mass number \( 16 \) and half life of \( 1 \, \text{day} .\) Another substance B has atomic mass number \( 32 \) and half life of \( \frac{1}{2} \, \text{day} .\) If both A and B simultaneously start undergo radio activity at the same time with initial mass \( 320 \, \text{g} \) each, how many total atoms of A and B combined would be left after \( 2 \, \text{days} .\)

The ratio of wavelength of spectral line \( H_{\alpha} \) and \( H_{\beta} \) in the Balmer series is \( \dfrac{x}{20} \). The value of \( x \) is ________.

The amplitude of \( 15\sin(1000\pi t) \) is modulated \( 10\sin(4\pi t) \) signal. The amplitude modulated signal contains frequencies of
A \( 500 \,\text{Hz} \)    B. \( 2 \,\text{Hz} \)    C. \( 250 \,\text{Hz} \)    D. \( 498 \,\text{Hz} \)    E. \( 502 \,\text{Hz} \).
Choose the correct answer from the option given below:

Electron beam used in an electron microscope, when accelerated by a voltage of \( 20 \, \text{kV} \), has a de-Broglie wavelength of \( \lambda_0 \). If the voltage is increased to \( 40 \, \text{kV} \), then the de-Broglie wavelength associated with the electron beam would be:

Nucleus \( A \) having \( Z = 17 \) and equal number of protons and neutrons has \( 1.2 \, \text{MeV} \) binding energy per nucleon. Another nucleus \( B \) of \( Z = 12 \) has total \( 26 \) nucleon and \( 1.8 \, \text{MeV} \) binding energy per nucleons. The difference of binding energy of \( B \) and \( A \) will be _______ \( \text{MeV} \).

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The binding energy per nucleon is practically independent of the atomic number for nuclei of mass number in the range \( 30 \) to \( 170 \).
Reason R: Nuclear force is short ranged.
In the light of above statements, choose the correct answer from the options given below:

A free neutron decays into a proton but a free proton does not decay into neutron. This is because

The angular momentum for the electron in Bohr’s orbit is \( L \). If the electron in assumed to revolve in second orbit of hydrogen atom, then the change in angular momentum will be:

Proton \( (P) \) and electron \( (e) \) will have same de-Broglie wavelength when the ratio of their momentum is (assume, \( m_p = 1849\, m_e \) ):

In photo electric effect
A. The photocurrent is proportional to the intensity of the incident radiation.
B. Maximum Kinetic energy with which photoelectrons are emitted depends on the intensity of incident light.
C. Max. K.E. with which photoelectrons are emitted depends on the frequency of incident light.
D. The emission of photoelectrons requires a minimum threshold intensity of incident radiation.
E. Max. K.E. of the photoelectrons is independent of the frequency of the incident light
Choose the correct answer from the options given below:

A light of energy \( 12.75 \, \text{eV} \) is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is \( \frac{x}{\pi}\times 10^{-17}\, \text{eV}\,\text{s} \). The value of \( x \) is _______. (use \( h=4.14\times 10^{-15}\, \text{eV}\,\text{s},\, c=3\times 10^{8}\, \text{m}\,\text{s}^{-1} \)).

The ratio of the de-Broglie wavelengths of proton and electron having same Kinetic energy :
(Assume \( m_{p} = m_{e} \times 1849 \) )

The kinetic energy of an electron, \( \alpha \) – particle and a proton are given as \( 4K \), \( 2K \) and \( K \) respectively. The de-Broglie wavelength associated with electron \( (\lambda_{e}) \), \( \alpha \) – particle \( (\lambda_{\alpha}) \) and the proton \( (\lambda_{p}) \) are as follows:

A small object rest, absorbs a light pulse of power \( 20 \, \text{mW} \) and duration \( 30 \, \text{ns} \). Assuming speed of light as \( 3 \times 10^{8} \, \text{m/s} \), the momentum of the object becomes equal to:

\( {}^{238}_{92}A \rightarrow {}^{234}_{90}B + {}^{4}_{2}D + Q \)
In the given nuclear reaction, the approximate amount of energy released will be: 
[Given, mass of \( {}^{238}_{92}A = 238.05079 \times 931.5 \, \text{MeV}/c^{2} \), 
Mass of \( {}^{234}_{90}B = 234.04363 \times 931.5 \, \text{MeV}/c^{2} \), 
Mass of \( {}^{4}_{2}D = 4.00260 \times 931.5 \, \text{MeV}/c^{2} \)]

Experimentally it is found that \( 12.8 \, \text{eV} \) energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is \( \dfrac{9}{x} \times 10^{-10} \, \text{m} \). The value of the \( x \) is __________.
(\( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \), \( \dfrac{1}{4 \pi \epsilon_{0}} = 9 \times 10^{9} \, \text{N m}^{2} / \text{C}^{2} \) and electronic charge \( = 1.6 \times 10^{-19} \, \text{C} \).)

A nucleus disintegrates into two smaller parts, which have their velocities in the ratio \( 3 : 2 \). The ratio of their nuclear sizes will be \( \left( \dfrac{x}{3} \right)^{\tfrac{1}{3}} \). The value of ‘\( x \)’ is:

The variation of stopping potential \( (V_0) \) as a function of the frequency \( (v) \) of the incident light for a metal is shown in figure. The work function of the surface is: 

A nucleus with mass number \( 242 \) and binding energy per nucleon as \( 7.6 \, \text{MeV} \) breaks into two fragment each with mass number \( 121 \). If each fragment nucleus has binding energy per nucleon as \( 8.1 \, \text{MeV} \), the total gain in binding energy is ______ \( \text{MeV} \).

A radioactive nucleus decays by two different process. The half life of the first process is \( 5 \, \text{minutes} \) and that of the second process is \( 30 \, \text{s} \). The effective half-life of the nucleus is calculated to be \( \frac{\alpha}{11} \, \text{s} \). The value of \( \alpha \) is __________.

The wavelength of the radiation emitted is \( \lambda_{0} \) when an electron jumps from the second excited state to the first excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the hydrogen atom, the wavelength of the radiation emitted will be \( \frac{20}{x} \lambda_{0} \). the value of \( x \) is ________

The threshold frequency of a metal is \( f_0 \). When the light of frequency \( 2 f_0 \) is incident on the metal plate, the maximum velocity of photoelectrons is \( v_1 \). When the frequency of incident radiation is increased to \( 5 f_0 \), the maximum velocity of photoelectrons emitted is \( v_2 \). The ratio of \( v_1 \) and \( v_2 \) is:

The work functions of Aluminium and Gold are \( 4.1 \, \text{eV} \) and \( 5.1 \, \text{eV} \) respectively. The ratio of the slope of the stopping potential versus frequency plot for Gold to that of Aluminium is:

An atom absorbs a photon of wavelength \( 500 \, \text{nm} \) and emits another photon of wavelength \( 600 \, \text{nm} \). The net energy absorbed by the atom in this process is \( n \times 10^{-4} \, \text{eV} \). The value of \( n \) is ________.
[Assume the atom to be stationary during the absorption and emission process]
(Take \( h=6.6\times 10^{-34} \, \text{J s} \) and \( c=3\times 10^{8} \, \text{m/s} \).)

A radioactive element \( {}^{242}_{92}X \) emits two \( \alpha \)-particles, one electron and two positrons. The product nucleus is represented by \( {}^{234}_{P}Y \). The value of \( P \) is ________.

The ratio of de-Broglie wavelength of an \( \alpha \) particle and a proton accelerated from rest by the same potential is \( \dfrac{1}{\sqrt{m}} \), the value of \( m \) is:

A small particle of mass \( m \) moves in such a way that its potential energy \( U = \dfrac{1}{2} m \omega^{2} r^{2} \) where \( \omega \) is constant and \( r \) is the distance of the particle from origin. Assuming Bohr’s quantization of momentum and circular orbit, the radius of \( n^{\text{th}} \) orbit will be proportional to:

A metallic surface is illuminated with radiation of wavelength \( \lambda \), the stopping potential is \( V_{0} \). If the same surface is illuminated with radiation of wavelength \( 2\lambda \), the stopping potential becomes \( \dfrac{V_{0}}{4} \). The threshold wavelength for this metallic surface will be:

The half-life of a radioactive substance is \( T \). The time taken for disintegrating \( {7}/{8}^{th} \) part of its original mass will be:

The energy of \( He^{+} \) ion in its first excited state is, (The ground state energy for the Hydrogen atom is: (\( -13.6 \, \text{eV} \)))

The radius of \( 2^{\text{nd}} \) orbit of \( \text{He}^{+} \) of Bohr’s model is \( r_{1} \) and that of fourth orbit of \( \text{Be}^{3+} \) is represented as \( r_{2} \). Now the ratio \( \dfrac{r_{2}}{r_{1}} \) is \( x : 1 \). The value of \( x \) is ________.

The radius of fifth orbit of the \( \mathrm{Li}^{++} \) is ________ \( \times 10^{-12} \, \text{m} \). Take: radius of hydrogen atom \( = 0.51 \, \mathring{A} \)

The radius of electron’s second stationary orbit in Bohr’s atom is \( R \). The radius of \( 3^{\text{rd}} \) orbit will be:

A radio active material is reduced to \( {1}/{8} \) of its original amount in \( 3 \, \text{days} \). If \( 8 \times 10^{-3} \, \text{kg} \) of the material is left after \( 5 \, \text{days} \) the initial amount of the material is:

The energy levels of an hydrogen atom are shown below. The transition corresponding to emission of shortest wavelength is:

Give below are two statements:
Statement I : Out of microwaves, infrared rays and ultraviolet rays, ultraviolet rays are the most effective for the emission of electrons from a metallic surface.
Statement II : Above the threshold frequency, the maximum kinetic energy of photoelectrons is inversely proportional to the frequency of the incident light.
In the light of above statements, choose the correct answer from the options given below.

A \( 12.5 \, \text{eV} \) electron beam is used to bombard gaseous hydrogen at room temperature. The number of spectral lines emitted will be:

If the metals \( A \) and \( B \) are exposed to radiation of wavelength \( 350 \, \text{nm} \). The work functions of metals \( A \) and \( B \) are \( 4.8 \, \text{eV} \) and \( 2.2 \, \text{eV} \). The choose the correct option:

The mass of proton, neutron and helium nucleus are respectively \( 1.0073 \ \text{u} \), \( 1.0087 \ \text{u} \) and \( 4.0015 \ \text{u} \). The binding energy of helium nucleus is:

A photon is emitted in transition from \( n = 4 \) to \( n = 1 \) level in hydrogen atom. The corresponding wavelength for this transition is: (given, \( h = 4 \times 10^{-15} \, \text{eVs} \))

If a source of electromagnetic radiation having power \( 15 \, \text{kW} \) produces \( 10^{16} \) photons per second, the radiation belongs to a part of spectrum is: (Take constant \( h = 6 \times 10^{-34} \, \text{J s} \) )

For a nucleus \( {}^{A}_{Z}X \) having mass number \( A \) and atomic number \( Z \).
A. The surface energy per nucleon \( (b_s) = -a_1 A^{2/3} \)
B. The Coulomb contribution to the binding energy \( b_c = -a_2 \dfrac{Z(Z-1)}{A^{4/3}} \)
C. The volume energy \( b_v = a_3 A \)
D. Decrease in the bonding energy is proportional to surface area
E. While estimating the surface energy, it is assumed that each nucleon interacts with \( 12 \) nucleons. \( (a_1, a_2 \text{ and } a_3 \text{ are constants}) \)
Choose the most appropriate answer from the options given below:

The energy released per fission of nucleus of \( {}^{240}\mathrm{X} \) is \( 200 \, \text{MeV} \). The energy released if all the atoms in \( 120 \, \text{g} \) of pure \( {}^{240}\mathrm{X} \) undergo fission is ________ \( \times 10^{25} \, \text{MeV} \).
(Given \( N_A = 6 \times 10^{23} \))

The ratio of the density of oxygen nucleus \(( ^{16}_{8}\text{O} )\) and helium nucleus \(( ^{4}_{2}\text{He} )\) is:

The de Broglie wavelength of an electron having kinetic energy \( E \) is \( \lambda \). If the kinetic energy of electron becomes \( \dfrac{E}{4} \), then its de-Broglie wavelength will be:

If \( 917 \, \mathring{A}. \) be the lowest wavelength of Lyman series then the lowest wavelength of Balmer series will be ______ \( \mathring{A}. \).

Two radioactive elements \( A \) and \( B \) initially have same number of atoms. The half life of \( A \) is same as the average life of \( B \). If \( \lambda_A \) and \( \lambda_B \) are decay constants of \( A \) and \( B \) respectively, then choose the correct relation from the given options.

An electron accelerated through a potential difference \( V_{1} \) has a de-Broglie wavelength of \( \lambda \). When the potential is changed to \( V_{2} \), its de-Broglie wavelength increases by \( 50\% \). The value of \( \left( \frac{V_{1}}{V_{2}} \right) \) is equal to:

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: The nuclear density of nuclides \( {}^{10}_{5}\mathrm{B} \), \( {}^{6}_{3}\mathrm{Li} \), \( {}^{56}_{26}\mathrm{Fe} \), \( {}^{20}_{10}\mathrm{Ne} \) and \( {}^{209}_{83}\mathrm{Bi} \) can be arranged as \( \rho^{N}_{\mathrm{Bi}} > \rho^{N}_{\mathrm{Fe}} > \rho^{N}_{\mathrm{Ne}} > \rho^{N}_{\mathrm{B}} > \rho^{N}_{\mathrm{Li}} \).
Reason R: The radius \( R \) of nucleus is related to its mass number \( A \) as \( R = R_{0} A^{1/3} \), where \( R_0 \) is a constant.
In the light of the above statements, choose the correct answer from the options given below

The energy levels of an atom is shown in figure.

Which one of these transitions will result in the emission of a photon of wavelength \( 124.1 \, \text{nm} \)?
Given \( ( \, h = 6.62 \times 10^{-34} \, \text{Js} \, ) \).

Given below are two statements:
Statement I: Stopping potential in photoelectric effect does not depend on the power of the light source.
Statement II: For a given metal, the maximum kinetic energy of the photoelectron depends on the wavelength of the incident light.
In the light of above statements, choose the most appropriate answer from the options given below:

A proton and an \( \alpha \) – particle are accelerated from rest by \( 2V \) and \( 4V \) potentials, respectively. The ratio of their de-Broglie wavelength is:

The difference between threshold wavelengths for two metal surfaces \( A \) and \( B \) having work function \( \phi_A = 9 \, \text{eV} \) and \( \phi_B = 4.5 \, \text{eV} \) in \( \text{nm} \) is: {Given, \( hc = 1242 \, \text{eV} \, \text{nm} \)}

An electron of a hydrogen like atom, having \( Z = 4 \), jumps from \( 4^{\text{th}} \) energy state to \( 2^{\text{nd}} \) energy state. The energy released in this process, will be: (Given \( R c h = 13.6 \, \text{eV} \)) Where \( R \) = Rydberg constant
\( c \) = Speed of light in vacuum
\( h \) = Plank’s constant

As per given figure \( A \), \( B \) and \( C \) are the first, second and third excited energy levels of hydrogen atom respectively. If the ratio of the two wavelengths \( \left(\text{i.e. } \dfrac{\lambda_{1}}{\lambda_{2}}\right) \) is \( \dfrac{7}{4n} \), then the value of \( n \) will be _______.

A monochromatic light is incident on a hydrogen sample in ground state. Hydrogen atoms absorb a fraction of light and subsequently emit radiation of six different wavelengths. The frequency of incident light is \( x \times 10^{15} \, \text{Hz} \). The value of \( x \) is ______.
(Given \( h = 4.25 \times 10^{-15} \, \text{eV s} \))