JEE Previous year questions (2021-22): Atom & Nuclei PDF Download

Q.1. In a radioactive decay chain reaction, nucleus decays into  

nucleus. The ratio of the number of α to β^- number of particles emitted in this process is _______.      (JEE Advance 2022)

Ans. 2

Q.2. 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_bⁿ in the nucleus.
Which of the following statement(s) is(are) correct?    (JEE Advance 2022)
(a)  is proportional to Z(Z−1) where Z is the atomic number of the nucleus.  BEbp−Ebn is proportional to A^−1/3 where A is the mass number of the nucleus.  
(c)   is positive.  
(d) E_p^b increases if the nucleus undergoes a beta decay emitting a positron.

Ans. a, b and d

Q.3. Two radioactive materials A and B have decay constants 25λ and 16λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be "e" after a time 1/aλ. The value of a is _________.      (JEE Main 2022)

Ans. 9

Q.4. In the hydrogen spectrum, λ be the wavelength of first transition line of Lyman series. The wavelength difference will be "aλ'' between the wavelength of 3rd  transition line of Paschen series and that of 2nd  transition line of Balmer series where  a =___________.      (JEE Main 2022)

Ans. 5

Q.5. x/x+4 is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its
(i) third permitted energy level to the second level and
(ii) the highest permitted energy level to the second permitted level.
The value of x will be ____________.      (JEE Main 2022)

Ans. 5

Q.6. A hydrogen atom in its first excited state absorbs a photon of energy x  10^-2 eV and excited to a higher energy state where the potential energy of electron is -1.08 eV. The value of x is ______________.      (JEE Main 2022)

Ans. 286

Q.7. A sample contains 10^-2 kg each of two substances A and B with half lives 4 s and 8 s respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is x/100. The value of x is ___________.      (JEE Main 2022)

Ans. 25

∴ Mass ratio of A and B,

Q.8. A beam of monochromatic light is used to excite the electron in Li⁺⁺ from the first orbit to the third orbit. The wavelength of monochromatic light is found to be x  10^-10 m. The value of x is ___________.      (JEE Main 2022)
[Given hc = 1242 eV nm]

Ans. 114

Q.9. The half life of a radioactive substance is 5 years. After x years a given sample of the radioactive substance gets reduced to 6.25% of its initial value. The value of x is ____________.      (JEE Main 2022)

Ans. 20

Q.10. √d₁ and √d₂  are the impact parameters corresponding to scattering angles 60 and 90 respectively, when an α particle is approaching a gold nucleus. For d₁ = x d₂, the value of x will be ____________.      (JEE Main 2022)

Ans. 3
 

Q.11. Read the following statements :      (JEE Main 2022)
(A) Volume of the nucleus is directly proportional to the mass number.
(B) Volume of the nucleus is independent of mass number.
(C) Density of the nucleus is directly proportional to the mass number.
(D) Density of the nucleus is directly proportional to the cube root of the mass number.
(E) Density of the nucleus is independent of the mass number.
Choose the correct option from the following options.
(a) (A) and (D) only.  
(b) (A) and (E) only.  
(c) (B) and (E) only.  
(d) (A) and (C) only.

Ans. b
We know,
Radius of nucleus, r=r₀A^1/3
where, A = mass number
∴ Volume (v)  = 4/3πr³

∴ Volume is directly proportional to mass number.

We know,



∴Density of nucleus is independent of mass number.

Q.12. An α particle and a proton are accelerated from rest through the same potential difference. The ratio of linear momenta acquired by above two particles will be:      (JEE Main 2022)
(a)√2:1
(b) 2√2:1
(c) 4√2:1
(d) 8:1

Ans. b
We know,

Both α particle and proton are passed through same potential difference.

Q.13. Find the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its (i) second permitted energy level to the first level, and (ii) the highest permitted energy level to the first permitted level.      (JEE Main 2022)
(a) 3:4
(b) 4:3
(c) 1:4
(d) 4:1

Ans. a

Q.14. Match List - I with List - II:      (JEE Main 2022)

Choose the correct answer from the options given below :
(a) (a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)  
(b) (a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)  
(c) (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)  
(d) (a)-(iii), (b)-(i), (c)-(ii), (d)-(iv)

Ans. b
UV - Water purification
X-rays - Diagnostic tool in medicine
Microwave - Communication, Radar
Infrared wave - Improving visibility in foggy days.

Q.15. A radioactive sample decays 7/8 times its original quantity in 15 minutes. The half-life of the sample is      (JEE Main 2022)
(a) 5 min
(b) 7.5 min
(c) 15 min
(d) 30 min

Ans. a

Q.16.  The equation can be used to find the de-Brogli wavelength of an electron. In this equation  stands for:
Where
m= mass of electron
P = momentum of electron
K = Kinetic energy of electron
V= Accelerating potential in volts for electron          (JEE Main 2022)
(a) √mK
(b) √p
(c) √K
(d) √V

Ans. d

Q.17. The activity of a radioactive material is 6.4 x 10^-4 curie. Its half life is 5 days. The activity will become 5 x 10^-6 curie after:          (JEE Main 2022)
(a) 7 days
(b) 15 days
(c) 25 days
(d) 35 days

Ans. d

Q.18. What is the half-life period of a radioactive material if its activity drops to 

1/6^th  of its initial value in 30 years?           (JEE Main 2022)
(a) 9.5 years
(b) 8.5 years
(c) 7.5 years
(d)10.5 years

Ans. c

Q.19. Mass numbers of two nuclei are in the ratio of 4:3. Their nuclear densities will be in the ratio of          (JEE Main 2022)
(a) 4 : 3
(b) (3/4)^1/3
(c) 1 : 1
(d) (4/3)^1/3

Ans. c

Q.20. A nucleus of mass M at rest splits into two parts having masses M′/3 and . The ratio of de Broglie wavelength of two parts will be:          (JEE Main 2022)
(a) 1:2
(b) 2:1
(c) 1:1
(d) 2:3

Ans. c
Linear momentum is conserved

Q.21. A parallel beam of light of wavelength  900 nm and intensity 100Wm⁻² is incident on a surface perpendicular to the beam. The number of photons crossing  1 cm² area perpendicular to the beam in one second is:          (JEE Main 2022)
(a) 3×10¹⁶ 
(b) 4.5×10¹⁶ 
(c) 4.5×10¹⁷ 
(d) 4.5×10²⁰

Ans. b

Number of photons incident per second

Q.22. The disintegration rate of a certain radioactive sample at any instant is 4250 disintegrations per minute. 10 minutes later, the rate becomes 2250 disintegrations per minute. The approximate decay constant is:          (JEE Main 2022)
(Take log₁₀ 1.88 = 0.274 )
(a) 0.02min⁻¹ 
(b) 2.7min⁻¹
(c) 0.063min⁻¹
(d) 6.3min⁻¹

Ans. c

Q.23. Hydrogen atom from excited state comes to the ground state by emitting a photon of wavelength λ. The value of principal quantum number 'n' of the excited state will be : (R: Rydberg constant)          (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

Q.24. The ratio of wavelengths of proton and deuteron accelerated by potential Vp and Vd is 1 : √2. Then the ratio of V_p to V_d will be:           (JEE Main 2022)
(a) 1 : 1
(b) √2 : 1
(c) 2 : 1
(d) 4 : 1

Ans. d

Q.25. The momentum of an electron revolving in n^th orbit is given by:
(Symbols have their usual meanings)           (JEE Main 2022)
(a)
(b)
(c)
(d)  

Ans. a

Q.26. A hydrogen atom in ground state absorbs 12.09 eV of energy. The orbital angular momentum of the electron is increased by:           (JEE Main 2022)
(a) 1.05 × 10⁻³⁴ Js
(b) 2.11 × 10⁻³⁴ Js
(c) 3.16 × 10⁻³⁴ Js 

(d) 4.22 × 10⁻³⁴ Js
Ans. b
Change in energy

Q.27. A cyclotron is working at a frequency of 10 MHz. If the radius of its dees is 60 cm. The maximum kinetic energy of accelerated proton will be:
(Take : e = 1.6 x 10^-19 C, m_p = 1.67 x 10^-27 kg)           (JEE Main 2022)
(a) 7.4 MeV
(b) 14.86 MeV  
(c) 7.4 GeV  
(d) 704 GeV

Ans. a
Given
f= 10x10⁶ Hz
r = 0.6m
Charge on proton (q) = e

We know,

Also, we know,

Cyclotron oscillation frequency should be equal to the pendulum evolution frequency.

Putting this value of eB in equation (1) we get

Q.28. Choose the correct option from the following options given below:           (JEE Main 2022)
(a) In the ground state of Rutherford's model electrons are in stable equilibrium. While in Thomson's model electrons always experience a net-force.  
(b) An atom has a nearly continuous mass distribution in a Rutherford's model but has a highly non-uniform mass distribution in Thomson's model.
(c) A classical atom based on Rutherford's model is doomed to collapse.
(d) The positively charged part of the atom possesses most of the mass in Rutherford's model but not in Thomson's model.

Ans. c
An atom based on classical theory of Rutherford's model should collapse as the electrons in continuous circular motion that is a continuously accelerated charge should emit EM waves and so should lose energy. These electrons losing energy should soon fall into heavy nucleus collapsing the whole atom.

Q.29 In Bohr's atomic model of hydrogen, let K, P and E are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level:           (JEE Main 2022)
(a) All K, P and E increase.
(b) K decreases, P and E increase.
(c) P decreases, K and E increase.  
(d) K increases, P and E decrease.

Ans. b

As electron makes transition to higher level, total energy and potential energy increases (due to negative sign) while the kinetic energy reduces.

Q.30. The ratio for the speed of the electron in the 3rd orbit of He+ to the speed of the electron in the 3rd orbit of hydrogen atom will be:           (JEE Main 2022)
(a) 1:1
(b) 1:2
(c) 4:1
(d) 2:1

Ans. d

Q.31. Which of the following figure represents the variation of  (if R = radius of a nucleus and A = its mass number)         (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

We know that

Straight line

Q.32. A proton, a neutron, an electron and an α particle have same energy. If λ_p, λ_n, λ_e and λ_a are the de Broglie's wavelengths of proton, neutron, electron and α particle respectively, then choose the correct relation from the following:           (JEE Main 2022)
(a) λ_p = λ_n > λ_e > λ_a 
(b) λ_a < λ_n < λ_p < λ_e 
(c)λ_e < λ_p = λ_n > λ_a 
(d) λ_e = λ_p = λ_n = λ_a

Ans. b

Where K : kinetic energy

Q.33.A radioactive nucleus can decay by two different processes. Half-life for the first process is 3.0 hours while it is 4.5 hours for the second process. The effective half-life of the nucleus will be:           (JEE Main 2022)
(a) 3.75 hours  
(b) 0.56 hours
(c) 0.26 hours
(d) 1.80 hours

Ans. d

Q.34. Which is the correct ascending order of wavelengths?           (JEE Main 2022)
(a) λ_visible <  λ_X-ray < λ_gamma-ray < λ_microwave
(b) λ_gamma-ray<  λ_X-ray < λ_visible  < λ_microwave
(c)  λ_X-ray < λ_gamma-ray <λ_visible  < λ_microwave
(d) λ_microwave < λ_visible < λ_gamma-ray < λ_X-ray

Ans. b
Wavelength of microwave is maximum then visible light then X-rays and then gamma rays so the correct order will be
λ_gamma-ray<  λ_X-ray < λ_visible  < λ_microwave

Q.35. An electron with speed v and a photon with speed c have the same de-Broglie wavelength. If the kinetic energy and momentum of electron are E_e and p_e and that of photon are E_ph and p_ph respectively. Which of the following is correct?           (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

Q.36. A hydrogen atom in its ground state absorbs 10.2 eV of energy. The angular momentum of electron of the hydrogen atom will increase by the value of:           (JEE Main 2022)
(Given, Planck's constant = 6.6  10^-34 Js).
(a) 2.10 × 10⁻³⁴ Js
(b) 1.05 × 10⁻³⁴ Js  
(c) 3.15 × 10⁻³⁴ Js
(d) 4.2 × 10⁻³⁴ Js

Ans. b

Q.37. An α particle and a carbon 12 atom has same kinetic energy K. The ratio of their de-Broglie wavelengths (λ_α:λ_C12) is :           (JEE Main 2022)
(a) 1:√3
(b) √3:1
(c) 3:1
(d) 2 : √3

Ans. b

Q.38. Match List-I with List-II:           (JEE Main 2022)

Choose the correct answer from the options given below :
(a) (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)
(b) (a)-(iii), (b)-(i), (c)-(ii), (d)-(iv)
(c) (a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
(d) (a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)

Ans. a
UV rays are used to sterilize surgical material.

Microwaves are used in radar system.

Infrared are used for green house effect and

X-rays are used to study crystal structure.

Q.39. The de Broglie wavelengths for an electron and a photon are λ_e and λ_p respectively. For the same kinetic energy of electron and photon, which of the following presents the correct relation between the de Broglie wavelengths of two ?           (JEE Main 2022)
(a) λ_p∝ λ_e² 
(b) λ_p∝ λ_e 
(c) λ_p∝ √λ_e
(d)

Ans. a

Q.40. Following statements related to radioactivity are given below:           (JEE Main 2022)
(a) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
(b) The number of un-decayed nuclei in the radioactive sample decays exponentially with time.
(c) Slope of the graph of loge (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time (τ).
(d) Product of decay constant (λ) and half-life time (T_1/2) is not constant.
Choose the most appropriate answer from the options given below:
(a) (A) and (B) only  
(b) (B) and (D) only  
(c) (B) and (C) only  
(d) (C) and (D) only

Ans. c
Radioactive decay is a random and spontaneous process it depends on unbalancing of nucleus.

Q.41. Let K₁ and K₂ be the maximum kinetic energies of photo-electrons emitted when two monochromatic beams of wavelength λ₁ and λ₂, respectively are incident on a metallic surface. If λ₁ = 3λ₂ then:           (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b

from (i) and (ii) we can say

Q.42. In the following nuclear reaction, 

Mass number of D is 182 and atomic number is 74. Mass number and atomic number of D₄ respectively will be _________.            (JEE Main 2022)
(a) 174 and 71
(b) 174 and 69
(c) 172 and 69
(d) 172 and 71

Ans. a
Equivalent reaction can be written as

⇒ Mass number of D₄ = Mass number of D  - 2 x 4
= 182 -  8 = 174
⇒ Atomic number of D₄

= Atomic number of D - 2 x 2 + 1

= 74  4 + 1 = 71

Q.43. The activity of a radioactive material is 2.56 x 10^-3 Ci. If the half life of the material is 5 days, after how many days the activity will become 2 x 10^-5 Ci ?           (JEE Main 2022)
(a) 30 days
(b)  35 days
(c)  40 days
(d)  25 days

Ans. b

Q.44. 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            (JEE Advance 2021)
(a) 50
(b) 75
(c) 350
(d) 525

Ans. d

Out of 1000 nuclei of Q, 60% may go α-decay

 600 nuclei may have α-decay

t = 1 h = 60 min

Using,

N = 75

75 nuclei are left after one hour

So, number of nuclei decayed = 600  75 = 525

Q.45. A moving proton and electron have the same de-Broglie wavelength. If K and P denote the K.E. and momentum respectively. Then choose the correct option:      (JEE Main 2021)
(a) K_p < K_e and P_p = P_e 
(b) K_p = K_e and P_p = P_e
(c)K_p < K_e an P_p < P_e 
(d) K_p > K_e and P_p = P_e

Ans. a

Q.46. A sample of a radioactive nucleus A disintegrates to another radioactive nucleus B, which in turn disintegrates to some other stable nucleus C. Plot of a graph showing the variation of number of atoms of nucleus B versus time is:
(Assume that at t = 0, there are no B atoms in the sample)       (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. d
A → B → C (stable)
Initially no. of atoms of B = 0 at t = 0, no. of atoms of B will starts increasing & reaches maximum value when rate of decay of B = rate of formation of B.
After that maximum value, no. of atoms will starts decreasing as growth & decay both are exponential functions, so best possible graph is (d).

Q.47. There are 1010 radioactive nuclei in a given radioactive element, its half-life time is 1 minute. How many nuclei will remain after 30 seconds?  (√2 = 1.414)       (JEE Main 2021)
(a) 2 x 10¹⁰
(b) 7 x 10⁹
(c) 10⁵
(d)  4 x 10¹⁰

Ans. b

Q.48. At time t = 0, a material is composed of two radioactive atoms A and B, where N_A(0) = 2N_B(0). The decay constant of both kind of radioactive atoms is λ . However, A disintegrates to B and B disintegrates to C. Which of the following figures represents the evolution of N_B(t) / N_B(0) with respect to time t?
[N_A (0) = No. of A atoms at t = 0
N_B (0) = No. of B atoms at t = 0]       (JEE Main 2021)
(a)
(b)  
(c)  
(d)  

Ans. c

Q.49. A particular hydrogen like ion emits radiation of frequency 2.92  x10¹⁵ Hz when it makes transition from n = 3 to n = 1. The frequency in Hz of radiation emitted in transition from n = 2 to n = 1 will be:       (JEE Main 2021)
(a) 0.44 × 10¹⁵ 
(b) 6.57 × 10¹⁵
(c) 4.38 × 10¹⁵
(d) 2.46 × 10¹⁵

Ans. d

Q.50. Consider the following statements:       (JEE Main 2021)
A. Atoms of each element emit characteristics spectrum.
B. According to Bohr's Postulate, an electron in a hydrogen atom, revolves in a certain stationary orbit.
C. The density of nuclear matter depends on the size of the nucleus.
D. A free neutron is stable but a free proton decay is possible.
E. Radioactivity is an indication of the instability of nuclei.
Choose the correct answer from the options given below:
(a) A, B, C, D and E
(b) A, B and E only
(c) B and D only  
(d) A, C and E only

Ans. b
(A) True, atom of each element emits characteristic spectrum.

(B) True, according to Bohr's postulates  and hence electron resides into orbits of specific radius called stationary orbits.
(C) False, density of nucleus is constant.
(D) False, A free neutron is unstable decays into proton and electron and antineutrino.
(E) True, unstable nucleus show radioactivity.

Q.51. An electron and proton are separated by a large distance. The electron starts approaching the proton with energy 3 eV. The proton captures the electron and forms a hydrogen atom in second excited state. The resulting photon is incident on a photosensitive metal of threshold wavelength . What is the maximum kinetic energy of the emitted photoelectron?        (JEE Main 2021)
(a) 7.61 eV
(b) 1.41 eV
(c) 3.3 eV
(d) No photoelectron would be emitted

Ans. b

initially, energy of electron = + 3eV

finally, in 2nd excited state,

Loss in energy is emitted as photon,
So, photon energy

Now, photoelectric effect equation

Q.52. An electron moving with speed v and a photon moving with speed c, have same D-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is :        (JEE Main 2021)
(a) 3c/v
(b) v/3c
(c) v/2c
(d) 2c/v

Ans. c

Q.53. A particle of mass 4M at rest disintegrates into two particles of mass M and 3M respectively having non zero velocities. The ratio of de-Broglie wavelength of particle of mass M to that of mass 3M will be :       (JEE Main 2021)
(a) 1:3
(b) 3:1
(c) 1:√3
(d) 1:1

Ans. d

both the particles will move with momentum same in magnitude & opposite in direction.

So De-Broglie wavelength of both will be same i.e. ratio 1 : 1

Q.54. What should be the order of arrangement of de-Broglie wavelength of electron (λ_e), an α-particle (λ_a) and proton (λ_p) given that all have the same kinetic energy?       (JEE Main 2021)
(a) λe = λp = λα
(b) λe < λp < λα
(c) λe > λp > λα
(d) λe = λp > λα

Ans. c

Q.55. An electron of mass me and a proton of mass m_p are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is       (JEE Main 2021)
(a)
(b) 1
(c)
(d)

Ans. d

Q.56. A nucleus with mass number 184 initially at rest emits an α-particle. If the Q value of the reaction is 5.5 MeV, calculate the kinetic energy of the α-particle.       (JEE Main 2021)
(a) 5.5 MeV
(b) 5.0  MeV
(c) 5.38 MeV
(d) 0.12 MeV

Ans. c

Q.57. For a certain radioactive process the graph between In R and t(sec) is obtained as shown in the figure. Then the value of half life for the unknown radioactive material is approximately:       (JEE Main 2021)

(a) 9.15 sec
(b) 6.93 sec
(c) 2.62 sec
(d) 4.62 sec

Ans. d

Q.58. A nucleus of mass M emits γ -ray photon of frequency 'v'. The loss of internal energy by the nucleus is :
[Take 'c' as the speed of electromagnetic wave]        (JEE Main 2021)
(a) hv
(b)
(c)
(d) 0

Ans. b
Energy of γ-ray, E_γ = hv
and momentum of γ-ray, p_γ=h/λ .... (i)

From Eqs. (i) and (ii), we get

Since, during the emission of γ-ray photon, momentum is conserved.

Kinetic energy of decayed nuclei,

From Eqs. (iii) and (iv), we get

∴ Loss in internal energy = E_γ + KE_{decayed nuclei}

Q.59. A radioactive material decays by simultaneous emissions of two particles with half lives of 1400 years and 700 years respectively. What will be the time after which one third of the material remains ? (Take ln 3 = 1.1)        (JEE Main 2021)
(a) 740 years
(b) 1110  years
(c) 700  years
(d) 340 years

Ans. a
The given situation can be shown as

Here, radioactive material X is decayed into two particles Y and Z with their respective decay constant, λ_a and λ_b. It means that

where, t_1/2(a) = 700 yr
and t_1/2(b) = 1400 yr


Suppose the initial number of radioactive nuclei was N₀.

where, N = number of nuclei present at time = t and N₀ = number of nuclei present at time = 0


Taking log on both the sides of above equation, we get

⇒ t  ≈740 yr

Q.60. A proton and an α-particle, having kinetic energies K_p and K_α respectively, enter into a magnetic field at right angles.The ratio of the radii of trajectory of proton to that of α-particle is 2 : 1. The ratio of K_p : K_α is :        (JEE Main 2021)
(a) 8:1
(b) 4:1
(c) 1:8
(d) 1:4

Ans. b

Q.61. The decay of a proton to neutron is:        (JEE Main 2021)
(a) always possible as it is associated only with β⁺ decay
(b) possible only inside the nucleus
(c) not possible as proton mass is less than the neutron mass
(d) not possible but neutron to proton conversation is possible

Ans. b
Positron emission or Beta plus decay is a subtype of radioactive decay called Beta decay, in which a proton inside a nucleus is converted into a neutron while releasing a positron and an electron neutrino.

So, decay of a proton to neutron is possible only inside the nucleus. Free proton cannot decay to neutron as mass of proton is less compared to neutron so to decay into higher mass, proton need extra energy which free proton can’t get, only proton inside nucleus can get.

Q.62. The speed of electrons in a scanning electron microscope is 1 x 10⁷ ms^-1. If the protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of :        (JEE Main 2021)
(a) 1/1837
(b) 1837
(c) √1837
(d) 1/√1837

Ans. b
Resolving power (RP)  ∝ 1/λ
We know, de-Broglie wavelength

Q.63. A radioactive sample disintegrates via two independent decay processes having half lives respectively. The effective half-life T_1/2 of the nuclei is :        (JEE Main 2021)
(a)
(b)
(c)
(d) None of the above

Ans. b

Q.64. A particle is travelling 4 time as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is 2 : 1, the mass of the particle is:       (JEE Main 2021)
(a) 1/16 times the mass of e⁻  
(b) 8 times the mass of e⁻ 
(c) 16 times the mass of e⁻ 
(d) 1/8 times the mass of e⁻

Ans. d

Q.65. Imagine that the electron in a hydrogen atom is replaced by a muon (μ). The mass of muon particle is 207 times that of an electron and charge is equal to the charge of an electron. The ionization potential of this hydrogen atom will be:       (JEE Main 2021)
(a) 13.6 eV
(b) 28152.2 eV
(c) 331.2 eV
(d) 27.2 eV

Ans. b
m_m = 207 m_e
In hydrogen atom one electron present. Now that the electron in hydrogen atom is replaced by a muon (μ).

For muon,

Q.66. The atomic hydrogen emits a line spectrum consisting of various series. Which series of hydrogen atomic spectra is lying in the visible region?       (JEE Main 2021)
(a) Brackett series
(b) Balmer series
(c) Paschen series
(d) Lyman series

Ans. b
Balmer series of hydrogen atomic spectrum is lying in the visible region, when electron jumps from a higher energy level to n = 2 orbit.

Q.67. An electron of mass m and a photon have same energy E. The ratio of wavelength of electron to that of photon is : (c being the velocity of light)       (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. b

Q.68. Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?       (JEE Main 2021)
(a) 8
(b) 6
(c) 1
(d) 4

Ans. b

Q.69. If an electron is moving in the n^th orbit of the hydrogen atom, then its velocity (vn) for the nth orbit is given as:       (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. a
We know velocity of electron in n^th shell of hydrogen atom is given by

Q.70. The half-life of Au¹⁹⁸ is 2.7 days. The activity of 1.50 mg of Au¹⁹⁸ if its atomic weight is 198 g mol^-1 is, (Na = 6 x 10²³/mol).       (JEE Main 2021)
(a) 240 Ci
(b) 357 Ci
(c) 252  Ci
(d) 535 Ci

Ans. b
Activity, A=λN
Where, N=nN_A
N_A=6×10²³/mol

Here, A₀=λN₀
We know,

Q.71. The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of 100 V. What should nearly be the ratio of their wavelengths? (m_p = 1.00727u m_e = 0.00055u)       (JEE Main 2021)
(a) 41.4:1
(b) (1860)²:1
(c)1860:1
(d) 43:1

Ans. d

Q.72. Calculate the time interval between 33% decay and 67% decay if half-life of a substance is 20 minutes.       (JEE Main 2021)
(a) 40 minutes
(b) 60  minutes
(c) 13  minutes
(d) 20  minutes

Ans. d

Q.73. A radioactive sample is undergoing  α decay. At any time t1, its activity is A and another time t₂, the activity is A/5, What is the average life time for the sample?       (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. d
Let initial activity be A₀

Q.74. If λ₁ and λ₂ are the wavelengths of the third member of Lyman and first member of the Paschen series respectively, then the value of λ₁ : λ₂ is:       (JEE Main 2021)
(a) 7:135
(b) 7:108
(c) 1:9
(d) 1:3

Ans. a
For Lyman series

n₁ = 1, n₂ = 4

For paschen series

n₁ = 3, n₂ = 4

Q.75. An electron of mass me and a proton of mass m_p = 1836 m_e are moving with the same speed. The ratio of their de Broglie wavelength  will be:       (JEE Main 2021)
(a) 1
(b) 1836
(c) 1/1836
(d) 918

Ans. b
Given mass of electron = m_e

Mass of proton = m_p

given m_p = 1836 m_e

From de-Broglie wavelength

Q.76. Two radioactive substances X and Y originally have N₁ and N₂ nuclei respectively. Half life of X is half of the half life of Y. After three half lives of Y, number of nuclei of both are equal. The ratio  N₁/N₂ will be equal to:       (JEE Main 2021)
(a) 1/8
(b) 3/1
(c) 1/3
(d) 8/1

Ans. d

Let Half life of x = t

then half life of y = 2t

when 3 half life of y is completed then 6 half life of x is completed.

 Now x have = N₁/2⁶ nuclei
and y have = N²/2³ nuclei
From question,

Q.77. The de-Broglie wavelength of a proton and α-particle are equal. The ratio of their velocities is:       (JEE Main 2021)
(a) 4:2
(b) 4:3
(c) 4:1
(d) 1:4

Ans. c
Let λ_p, λ_α, m_p, m_α, v_p, v_α, p_p and p_α be the wavelength, mass, velocity and momentum of proton and α-particle, respectively. Given, λ_p = λ_α
As we know that

Q.78. In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E.
The transitions A, B and C respectively represent:        (JEE Main 2021)

(a) The series limit of Lyman series, second member of Balmer series and second member of Paschen series.  
(b) The ionization potential of hydrogen, second member of Balmer series and third member of Paschen series.  
(c) The series limit of Lyman series, third member of Balmer series and second member of Paschen series.  
(d) The first member of the Lyman series, third member of Balmer series and second member of Paschen series.

Ans. c
In transition A, hydrogen atom comes from higher energy state n =  to lower energy state n = 1. Hence, transition A represents series limit of Lyman series. In transition B, hydrogen atom comes from higher energy state n = 5 to lower energy state n = 2. Hence, transition B represents 3rd line of Balmer series. In transition C, hydrogen atom comes from higher energy state n = 5 to lower energy state n = 3. Hence, transition C represents 2nd line of Paschen series.
Hence, option (c) is correct.