Mass defect \(\Delta m = (2 m_p + 2 m_n) - m_{\text{He}} = [2(1.0073) + 2(1.0087)] - 4.0015 = 0.0305\text{ u}\). Binding energy is \(E_b = \Delta m \times 931.5\text{ MeV} \approx 0.0305 \times 931.5 \approx 28.4\text{ MeV}\).

In \( \beta^+ \) decay, a proton inside the nucleus converts into a neutron, emitting a positron (\( e^+ \)) and a neutrino (\( \nu_e \)). The nuclear reaction is \( p \rightarrow n + e^+ + \nu_e \).

The energy of an electron in a hydrogen-like atom is given by \( E_n \propto Z^2 \). For hydrogen, \( Z = 1 \), and for helium ion \( \text{He}^+ \), \( Z = 2 \). Therefore, \( E_{\text{He}^+} = Z^2 E = 4E \).

The radius of a nucleus is given by \(R \propto A^{1/3}\). Therefore, \(frac{R'}{R} = \left(\frac{16}{2}\right)^{1/3} = 8^{1/3} = 2\), so \(R' = 2R\).

Nuclides with the same atomic number \(Z\) but different neutron number \(N\) are called isotopes. Isotones are nuclides with the same number of neutrons.

Nuclear radius is given by \(R \propto A^{1/3}\), so the surface area is \(S \propto R^2 \propto A^{2/3}\). Thus, \(\frac{S_2}{S_1} = \left(\frac{64}{8}\right)^{2/3} = 8^{2/3} = 4\), meaning the surface area of the nucleus \(Y^{64}\) is \(4A_0\).

Nuclides with the same atomic number \( Z \) but different neutron numbers \( N \) are called isotopes, not isotones. Thus, option C is incorrect.

Electron-positron annihilation converts their mass-energy into radiant energy, typically two gamma-ray photons to conserve momentum. Photons indeed possess non-zero energy and momentum. However, the reason describes a property of the resulting photons rather than the fundamental cause or mechanism of the annihilation process itself.

In Rutherford's alpha-scattering experiment, large-angle deflections are caused by the strong electrostatic repulsion from the positively charged, massive nucleus. Electrons, being much lighter, cause negligible deflection. The nucleus's significant mass (R) ensures it remains largely stationary during collision, allowing for large deflections of \(\alpha\)-particles (A).

Assertion (A) is true; charged particle therapy (e.g., proton therapy) is used for cancer. Reason (R) is true; charged particles lose energy mainly through ionization, creating a Bragg peak that can precisely damage tumor cells. (R) is the correct explanation for (A).