Quantum Mechanics

The de Broglie wavelength of a particle is given by: where:
is Planck’s constant,
is the mass of the particle,
is the velocity of the particle.
Since all three particles (electron, proton, and deuteron) have the same speed, the wavelength is inversely proportional to their masses: The masses of the particles are:
(electron) is the smallest,
(proton) is larger,
(deuteron) is the largest.
Thus, their wavelengths follow the relation: Hence, the correct answer is (2).

Concept: This refers to Rutherford’s alpha-particle scattering experiment. The graph shows how the number of scattered alpha particles varies with scattering angle.

Graph description: Plot:

X-axis → Scattering angle ( )
Y-axis → Number of scattered particles
Shape of graph:

Very large number of particles at small angles (near )
Rapid decrease as angle increases
Very few particles scattered at large angles
Extremely small number scattered backward (near )
So, the curve starts high at small angles and falls sharply with increasing angle. Conclusion 1: Atom is mostly empty space. Since most alpha particles pass through with little or no deflection:

Positive charge and mass are not uniformly spread.
Most of the atom is empty.
Conclusion 2: Presence of a small, dense nucleus. A very small fraction of particles are deflected through large angles:

Indicates strong repulsive force.
Positive charge is concentrated in a tiny central region (nucleus).
Additional inference (optional):

Nucleus is positively charged and very small compared to atom size.