The question asks us to evaluate the correctness of four statements regarding the properties of an atomic nucleus, specifically its radius, volume, and density in relation to its mass number (A).

Statement Analysis

• Statement (A): A nucleus of mass number A has a radius R given by the expression $R = R_0A^{1/3}$

  This is a well-established empirical formula in nuclear physics. It suggests that the radius of a nucleus is approximately proportional to the cube root of its mass number. Here, $R_0$ is a constant, approximately $1.2 \times 10^{-15}$ meters (or 1.2 fm). This statement is correct.

• Statement (B): Volume of nucleus is proportional to mass number A

  The volume ($V$) of a nucleus can be approximated as the volume of a sphere with radius $R$. Using the formula from statement (A), $R = R_0A^{1/3}$, the volume is:

  $ V = \frac{4}{3}\pi R^3 $

  Substituting the expression for $R$:

  $ V = \frac{4}{3}\pi (R_0A^{1/3})^3 $

  $ V = \frac{4}{3}\pi R_0^3 A $

  Since $\frac{4}{3}\pi R_0^3$ is a constant, the volume ($V$) is directly proportional to the mass number ($A$). This statement is correct.

• Statement (C): The density of nucleus increases with the radius of nucleus

  Nuclear density ($\rho$) is defined as the mass of the nucleus divided by its volume. The mass ($m$) of a nucleus is approximately the mass number ($A$) multiplied by the average mass of a nucleon (proton or neutron), let's call it $m_{nucleon}$.

  $ m \approx A \times m_{nucleon} $

  Using the volume derived in statement (B), $V \propto A$. Let $V = k A$, where $k = \frac{4}{3}\pi R_0^3$.

  $ \rho = \frac{m}{V} \approx \frac{A \times m_{nucleon}}{k A} $

  $ \rho \approx \frac{m_{nucleon}}{k} $

  This calculation shows that the density is approximately constant and does not depend on the mass number ($A$) or the radius ($R$). Therefore, the density does not increase with the radius. This statement is incorrect.

• Statement (D): Density of nuclear matter does not depend on its mass number A

  As demonstrated in the analysis of statement (C), the nuclear density ($\rho$) is approximately constant and independent of the mass number ($A$). This characteristic property is often referred to as nuclear saturation. This statement is correct.

Summary of Correct Statements

Based on the analysis, statements (A), (B), and (D) are correct, while statement (C) is incorrect.

Conclusion

The correct statements are (A), (B), and (D). Therefore, the option that includes only these statements is the correct answer.