Quarks, Leptons, and the Big Bang Problems

This section provides 100 problems to test your understanding of particle physics and cosmology, including calculations of particle charges, conservation numbers, decay products, Hubble’s law velocities, and CMB temperature shifts, as well as applications like cosmic ray effects on spacecraft. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering particle physics and cosmology, a key topic for JEE/NEET success.

Numerical Problems

1. Calculate the electric charge of a proton (uud) in units of elementary charge. Given: up quark charge $q_u=+2/3$, down quark charge $q_d=-1/3$.

   • (a) +0.99
   • (b) +1.00
   • (c) +1.01
   • (d) +1.02

2. A galaxy is at a distance of 20 Mpc from Earth. Using Hubble’s law with $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

   • (a) 1347 km/s
   • (b) 1348 km/s
   • (c) 1349 km/s
   • (d) 1350 km/s

3. Calculate the baryon number of a neutron (udd). Given: each quark has a baryon number $B_q=1/3$.

   • (a) 0.99
   • (b) 1.00
   • (c) 1.01
   • (d) 1.02

4. A spacecraft moves at $v=400\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$, $c=3\times {10}^5\text{km/s}$). Calculate the temperature shift $\mathrm{\Delta }T$ in K due to the Doppler effect ($\mathrm{\Delta }T/T\approx v/c$).

   • (a) $3.59\times {10}^{-3}$ K
   • (b) $3.60\times {10}^{-3}$ K
   • (c) $3.61\times {10}^{-3}$ K
   • (d) $3.62\times {10}^{-3}$ K

5. Calculate the lepton number in the decay ${\mu }^{-}\to e^{-}+{\overline{\nu }}_e+{\nu }_{\mu }$. Given: ${\mu }^{-}$ ($L=1$), $e^{-}$ ($L=1$), ${\overline{\nu }}_e$ ($L=-1$), ${\nu }_{\mu }$ ($L=1$).

   • (a) 0.99
   • (b) 1.00
   • (c) 1.01
   • (d) 1.02

6. A galaxy cluster is at 50 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

   • (a) 3369 km/s
   • (b) 3370 km/s
   • (c) 3371 km/s
   • (d) 3372 km/s

7. Calculate the electric charge of a ${\pi }^{+}$ meson (u$\overline{\text{d}}$) in units of elementary charge. Given: $q_u=+2/3$, $q_{\overline{\text{d}}}=+1/3$.

   • (a) +0.99
   • (b) +1.00
   • (c) +1.01
   • (d) +1.02

8. A spacecraft moves at $v=500\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K due to the Doppler effect.

   • (a) $4.49\times {10}^{-3}$ K
   • (b) $4.50\times {10}^{-3}$ K
   • (c) $4.51\times {10}^{-3}$ K
   • (d) $4.52\times {10}^{-3}$ K

9. Calculate the baryon number of a ${\pi }^0$ meson (u$\overline{\text{u}}$). Given: $B_q=1/3$, $B_{\overline{q}}=-1/3$.

   • (a) -0.01
   • (b) 0.00
   • (c) 0.01
   • (d) 0.02

10. A quasar is at 100 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 6739 km/s
    • (b) 6740 km/s
    • (c) 6741 km/s
    • (d) 6742 km/s

11. Calculate the lepton number in the decay $n\to p+e^{-}+{\overline{\nu }}_e$. Given: $n$ ($L=0$), $p$ ($L=0$), $e^{-}$ ($L=1$), ${\overline{\nu }}_e$ ($L=-1$).

    • (a) -0.01
    • (b) 0.00
    • (c) 0.01
    • (d) 0.02

12. A spacecraft moves at $v=600\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K.

    • (a) $5.39\times {10}^{-3}$ K
    • (b) $5.40\times {10}^{-3}$ K
    • (c) $5.41\times {10}^{-3}$ K
    • (d) $5.42\times {10}^{-3}$ K

13. Calculate the electric charge of a ${\mathrm{\Delta }}^{++}$ baryon (uuu). Given: $q_u=+2/3$.

    • (a) +1.99
    • (b) +2.00
    • (c) +2.01
    • (d) +2.02

14. A galaxy is at 30 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 2021 km/s
    • (b) 2022 km/s
    • (c) 2023 km/s
    • (d) 2024 km/s

15. Calculate the baryon number of a $K^{+}$ meson (u$\overline{\text{s}}$). Given: $B_q=1/3$, $B_{\overline{q}}=-1/3$.

    • (a) -0.01
    • (b) 0.00
    • (c) 0.01
    • (d) 0.02

16. A spacecraft moves at $v=300\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K.

    • (a) $2.69\times {10}^{-3}$ K
    • (b) $2.70\times {10}^{-3}$ K
    • (c) $2.71\times {10}^{-3}$ K
    • (d) $2.72\times {10}^{-3}$ K

17. Calculate the electric charge of a ${\mathrm{\Sigma }}^{-}$ baryon (dds). Given: $q_d=-1/3$.

    • (a) -0.99
    • (b) -1.00
    • (c) -1.01
    • (d) -1.02

18. A galaxy cluster is at 80 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 5391 km/s
    • (b) 5392 km/s
    • (c) 5393 km/s
    • (d) 5394 km/s

19. Calculate the lepton number in the decay ${\tau }^{-}\to {\mu }^{-}+{\overline{\nu }}_{\mu }+{\nu }_{\tau }$. Given: ${\tau }^{-}$ ($L=1$), ${\mu }^{-}$ ($L=1$), ${\overline{\nu }}_{\mu }$ ($L=-1$), ${\nu }_{\tau }$ ($L=1$).

    • (a) 0.99
    • (b) 1.00
    • (c) 1.01
    • (d) 1.02

20. A quasar is at 150 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 10109 km/s
    • (b) 10110 km/s
    • (c) 10111 km/s
    • (d) 10112 km/s

21. Calculate the electric charge of a $D^{+}$ meson (c$\overline{\text{d}}$). Given: $q_c=+2/3$, $q_{\overline{\text{d}}}=+1/3$.

    • (a) +0.99
    • (b) +1.00
    • (c) +1.01
    • (d) +1.02

22. A spacecraft moves at $v=200\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K.

    • (a) $1.79\times {10}^{-3}$ K
    • (b) $1.80\times {10}^{-3}$ K
    • (c) $1.81\times {10}^{-3}$ K
    • (d) $1.82\times {10}^{-3}$ K

23. Calculate the baryon number of a ${\mathrm{\Lambda }}^0$ baryon (uds). Given: $B_q=1/3$.

    • (a) 0.99
    • (b) 1.00
    • (c) 1.01
    • (d) 1.02

24. A galaxy is at 40 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 2695 km/s
    • (b) 2696 km/s
    • (c) 2697 km/s
    • (d) 2698 km/s

25. Calculate the lepton number in the decay $p\to n+e^{+}+{\nu }_e$ (hypothetical). Given: $p$ ($L=0$), $n$ ($L=0$), $e^{+}$ ($L=-1$), ${\nu }_e$ ($L=1$).

    • (a) -0.01
    • (b) 0.00
    • (c) 0.01
    • (d) 0.02

26. A galaxy cluster is at 60 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 4043 km/s
    • (b) 4044 km/s
    • (c) 4045 km/s
    • (d) 4046 km/s

27. Calculate the electric charge of a $B^{-}$ meson (b$\overline{\text{u}}$). Given: $q_b=-1/3$, $q_{\overline{\text{u}}}=-2/3$.

    • (a) -0.99
    • (b) -1.00
    • (c) -1.01
    • (d) -1.02

28. A spacecraft moves at $v=100\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K.

    • (a) $8.99\times {10}^{-4}$ K
    • (b) $9.00\times {10}^{-4}$ K
    • (c) $9.01\times {10}^{-4}$ K
    • (d) $9.02\times {10}^{-4}$ K

29. Calculate the baryon number of a ${\mathrm{\Xi }}^0$ baryon (uss). Given: $B_q=1/3$.

    • (a) 0.99
    • (b) 1.00
    • (c) 1.01
    • (d) 1.02

30. A quasar is at 200 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 13479 km/s
    • (b) 13480 km/s
    • (c) 13481 km/s
    • (d) 13482 km/s

31. A spacecraft detects a cosmic ray proton (uud) with energy 10 GeV. If its velocity is $v\approx c$, calculate its Lorentz factor $\gamma$ ($E=\gamma m{c}^2$, $m_p{c}^2=0.938\text{GeV}$).

    • (a) 10.65
    • (b) 10.66
    • (c) 10.67
    • (d) 10.68

32. Calculate the electric charge of a ${\mathrm{\Omega }}^{-}$ baryon (sss). Given: $q_s=-1/3$.

    • (a) -0.99
    • (b) -1.00
    • (c) -1.01
    • (d) -1.02

33. A galaxy is at 70 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 4717 km/s
    • (b) 4718 km/s
    • (c) 4719 km/s
    • (d) 4720 km/s

34. Calculate the lepton number in the decay ${\mu }^{+}\to e^{+}+{\nu }_e+{\overline{\nu }}_{\mu }$. Given: ${\mu }^{+}$ ($L=-1$), $e^{+}$ ($L=-1$), ${\nu }_e$ ($L=1$), ${\overline{\nu }}_{\mu }$ ($L=-1$).

    • (a) -1.01
    • (b) -1.00
    • (c) -0.99
    • (d) -0.98

35. A spacecraft moves at $v=700\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K.

    • (a) $6.29\times {10}^{-3}$ K
    • (b) $6.30\times {10}^{-3}$ K
    • (c) $6.31\times {10}^{-3}$ K
    • (d) $6.32\times {10}^{-3}$ K

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Conceptual Problems

36. What is the electric charge of an up quark?

    • (a) $+1/3$
    • (b) $+2/3$
    • (c) $-1/3$
    • (d) $-2/3$

37. Which force is responsible for beta decay?

    • (a) Strong force
    • (b) Electromagnetic force
    • (c) Weak force
    • (d) Gravitational force

38. What is the unit of Hubble’s constant $H_0$ in the given form ($H_0=67.4\text{km/s/Mpc}$)?

    • (a) km/s
    • (b) km/s/Mpc
    • (c) Mpc
    • (d) s${}^{-1}$

39. What happens to a particle’s lepton number in the decay ${\pi }^{+}\to {\mu }^{+}+{\nu }_{\mu }$?

    • (a) Increases
    • (b) Decreases
    • (c) Remains zero
    • (d) Becomes undefined

40. What particles make up the first generation of the Standard Model?

    • (a) u, d, e${}^{-}$, ${\nu }_e$
    • (b) c, s, ${\mu }^{-}$, ${\nu }_{\mu }$
    • (c) t, b, ${\tau }^{-}$, ${\nu }_{\tau }$
    • (d) u, d, ${\mu }^{-}$, ${\nu }_{\mu }$

41. What is the unit of the CMB temperature shift $\mathrm{\Delta }T$?

    • (a) Kelvin (K)
    • (b) Joule
    • (c) Hertz
    • (d) Watt

42. What does a positive baryon number indicate about a particle?

    • (a) It is a lepton
    • (b) It is a baryon or quark
    • (c) It is a meson
    • (d) It is a photon

43. What happens during the quark-hadron transition in the early universe?

    • (a) Quarks form protons and neutrons
    • (b) Nuclei form atoms
    • (c) Photons decouple
    • (d) Universe cools to 2.7 K

44. What is a major limitation of the Standard Model?

    • (a) Includes gravity
    • (b) Excludes gravity and dark matter
    • (c) Explains dark energy
    • (d) Predicts neutrino masses

45. What is the dimension of Hubble’s constant $H_0$ in km/s/Mpc?

    • (a) $[\text{L}{\text{T}}^{-1}{\text{L}}^{-1}]$
    • (b) $[{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

46. What does the weak force mediate in particle interactions?

    • (a) Binding of quarks
    • (b) Electromagnetic attraction
    • (c) Decays like beta decay
    • (d) Gravitational pull

47. What is the significance of the CMB temperature of 2.7 K?

    • (a) Temperature at the Big Bang
    • (b) Remnant radiation from the Big Bang
    • (c) Nucleosynthesis temperature
    • (d) Recombination temperature

48. What happens to the universe’s expansion rate due to dark energy?

    • (a) Slows down
    • (b) Accelerates
    • (c) Remains constant
    • (d) Stops

49. What does the Higgs boson do in the Standard Model?

    • (a) Mediates the strong force
    • (b) Gives particles mass via the Higgs field
    • (c) Mediates gravity
    • (d) Explains dark matter

50. How do cosmic rays affect spacecraft electronics?

    • (a) Increase efficiency
    • (b) Cause single-event upsets via particle interactions
    • (c) Reduce radiation
    • (d) Increase mass

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Derivation Problems

51. Derive the electric charge of a proton (uud) using quark charges.

52. Derive the recession velocity of a galaxy using Hubble’s law $v=H_{0}d$.

53. Derive the baryon number conservation in the decay $n\to p+e^{-}+{\overline{\nu }}_e$.

54. Derive the temperature shift $\mathrm{\Delta }T$ of the CMB due to the Doppler effect.

55. Derive the lepton number conservation in the decay ${\mu }^{-}\to e^{-}+{\overline{\nu }}_e+{\nu }_{\mu }$.

56. Derive the electric charge of a ${\pi }^{+}$ meson (u$\overline{\text{d}}$).

57. Derive the baryon number of a ${\pi }^0$ meson (u$\overline{\text{u}}$).

58. Derive the recession velocity of a galaxy cluster using Hubble’s law.

59. Derive the lepton number conservation in the decay $p\to n+e^{+}+{\nu }_e$ (hypothetical).

60. Derive the temperature shift $\mathrm{\Delta }T$ for a spacecraft moving relative to the CMB.

61. Derive the electric charge of a ${\mathrm{\Delta }}^{++}$ baryon (uuu).

62. Derive the baryon number of a ${\mathrm{\Lambda }}^0$ baryon (uds).

63. Derive the lepton number in the decay ${\tau }^{-}\to {\mu }^{-}+{\overline{\nu }}_{\mu }+{\nu }_{\tau }$.

64. Derive the Lorentz factor $\gamma$ of a cosmic ray proton using $E=\gamma m{c}^2$.

65. Derive the electric charge of a ${\mathrm{\Omega }}^{-}$ baryon (sss).

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NEET-style Conceptual Problems

66. What is the unit of the recession velocity in Hubble’s law?

    • (a) km/s
    • (b) Mpc
    • (c) Hertz
    • (d) Watt

67. What particles are produced in the decay $n\to p+e^{-}+{\overline{\nu }}_e$?

    • (a) Proton, electron, neutrino
    • (b) Proton, electron, antineutrino
    • (c) Neutron, positron, neutrino
    • (d) Proton, positron, antineutrino

68. What is the relationship between velocity $v$ and distance $d$ in Hubble’s law?

    • (a) $v\propto d$
    • (b) $v\propto 1/d$
    • (c) $v$ is independent of $d$
    • (d) $v\propto d^2$

69. What happens to the CMB temperature if a spacecraft moves toward the CMB frame?

    • (a) Increases
    • (b) Decreases
    • (c) Remains the same
    • (d) Becomes zero

70. What is the dimension of lepton number $L$?

    • (a) Dimensionless
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

71. What does the strong force do in particle interactions?

    • (a) Binds quarks into hadrons
    • (b) Causes beta decay
    • (c) Mediates electromagnetic attraction
    • (d) Affects gravity

72. What is the role of particle physics in spacecraft electronics?

    • (a) Increases efficiency
    • (b) Predicts cosmic ray effects causing upsets
    • (c) Reduces radiation
    • (d) Increases mass

73. What happens to the universe’s density during inflation?

    • (a) Increases
    • (b) Decreases rapidly
    • (c) Remains constant
    • (d) Becomes zero

74. Why does the weak force have a short range?

    • (a) Due to massless mediators
    • (b) Due to massive W and Z bosons
    • (c) Due to infinite range
    • (d) Due to strong interaction

75. What is the unit of the Lorentz factor $\gamma$?

    • (a) Dimensionless
    • (b) Joule
    • (c) Hertz
    • (d) Watt

76. What does a high recession velocity of a galaxy indicate?

    • (a) Closer distance
    • (b) Greater distance from Earth
    • (c) No expansion
    • (d) Constant velocity

77. Which particles are in the second generation of the Standard Model?

    • (a) u, d, e${}^{-}$, ${\nu }_e$
    • (b) c, s, ${\mu }^{-}$, ${\nu }_{\mu }$
    • (c) t, b, ${\tau }^{-}$, ${\nu }_{\tau }$
    • (d) u, d, ${\tau }^{-}$, ${\nu }_{\tau }$

78. What is the effect of recombination in the early universe?

    • (a) Quarks form hadrons
    • (b) Atoms form, CMB emitted
    • (c) Nuclei form
    • (d) Universe inflates

79. What does a pseudo-force do in a non-inertial frame for cosmology calculations?

    • (a) Affects perceived expansion rate
    • (b) Affects particle charge
    • (c) Creates decays
    • (d) Reduces forces

80. What is the dimension of $\mathrm{\Delta }T/T$ in the CMB Doppler effect?

    • (a) Dimensionless
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

81. What is the role of the CMB in spacecraft navigation?

    • (a) Increases radiation
    • (b) Provides a reference for velocity via Doppler shifts
    • (c) Reduces efficiency
    • (d) Increases mass

82. What happens to the universe’s temperature after the Big Bang?

    • (a) Increases
    • (b) Decreases due to expansion
    • (c) Remains constant
    • (d) Becomes zero

83. Why are neutrinos difficult to detect?

    • (a) Due to high charge
    • (b) Due to weak interactions and no charge
    • (c) Due to strong force
    • (d) Due to high mass

84. What is the significance of ${10}^{-36}$ s in the Big Bang timeline?

    • (a) Nucleosynthesis begins
    • (b) Inflation occurs
    • (c) Recombination happens
    • (d) CMB is emitted

85. What is the unit of particle mass in the form $m{c}^2$ (e.g., MeV/$c^2$)?

    • (a) MeV/$c^2$
    • (b) Joule
    • (c) Hertz
    • (d) Watt

86. What does a high Lorentz factor $\gamma$ indicate for a cosmic ray?

    • (a) Low energy
    • (b) High energy, relativistic speed
    • (c) No motion
    • (d) Constant mass

87. What is the physical significance of $H_{0}d$?

    • (a) Temperature shift
    • (b) Recession velocity of a galaxy
    • (c) Baryon number
    • (d) Lepton number

88. Why is the CMB uniform across the sky?

    • (a) Due to nucleosynthesis
    • (b) Due to early thermal equilibrium
    • (c) Due to dark matter
    • (d) Due to inflation only

89. What is the dimension of $v/c$ in the CMB Doppler effect?

    • (a) Dimensionless
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

90. How does dark energy affect the universe?

    • (a) Slows expansion
    • (b) Accelerates expansion
    • (c) Stops expansion
    • (d) Reduces density

91. What is the role of gluons in the Standard Model?

    • (a) Mediate the weak force
    • (b) Mediate the strong force
    • (c) Give particles mass
    • (d) Mediate gravity

92. What does the abundance of light elements (H:He ~3:1) indicate?

    • (a) Nucleosynthesis in the early universe
    • (b) Recombination
    • (c) Inflation
    • (d) Dark energy

93. What is the physical significance of $E=\gamma m{c}^2$?

    • (a) Rest energy
    • (b) Total energy of a relativistic particle
    • (c) Baryon number
    • (d) Lepton number

94. What is the dimension of $T_{\text{CMB}}$?

    • (a) $[\text{K}]$
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

95. Why are top quarks short-lived?

    • (a) Due to low mass
    • (b) Due to high mass and rapid decay via weak force
    • (c) Due to strong force
    • (d) Due to electromagnetic force

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NEET-style Numerical Problems

96. Calculate the electric charge of a neutron (udd). Given: $q_u=+2/3$, $q_d=-1/3$.

    • (a) -0.01
    • (b) 0.00
    • (c) 0.01
    • (d) 0.02

97. A galaxy is at 25 Mpc. Using $H_0=67.4\text{km/s/Mpc}$, calculate its recession velocity in km/s.

    • (a) 1684 km/s
    • (b) 1685 km/s
    • (c) 1686 km/s
    • (d) 1687 km/s

98. Calculate the baryon number of a proton (uud). Given: $B_q=1/3$.

    • (a) 0.99
    • (b) 1.00
    • (c) 1.01
    • (d) 1.02

99. A spacecraft moves at $v=800\text{km/s}$ relative to the CMB frame ($T_{\text{CMB}}=2.7\text{K}$). Calculate $\mathrm{\Delta }T$ in K.

    • (a) $7.19\times {10}^{-3}$ K
    • (b) $7.20\times {10}^{-3}$ K
    • (c) $7.21\times {10}^{-3}$ K
    • (d) $7.22\times {10}^{-3}$ K

100. Calculate the electric charge of a $K^{-}$ meson (s$\overline{\text{u}}$). Given: $q_s=-1/3$, $q_{\overline{\text{u}}}=-2/3$.
     - (a) -0.99
     - (b) -1.00
     - (c) -1.01
     - (d) -1.02

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