Which of the following statements are correct?
(A) A nucleus of mass number A has a radius R given by the expression R =R₀A^1/3
(B) Volume of nucleus is proportional to mass number A
(C) The density of nucleus increases with the radius of nucleus.
(D) Density of nuclear matter does not depend on its mass number A
Choose the correct answer from the options given below:

The correct answer is (A), (B) and (D) only.

Let us analyze each statement step-by-step to understand why this option is correct:

Statement (A): A nucleus of mass number A has a radius R given by the expression R = R₀A^1/3
Empirical measurements show that the radius of a nucleus is proportional to the cube root of its mass number (which is the total number of protons and neutrons, $A$). This relationship is mathematically expressed as:
$R=R_0$
where $R_0$ is a constant representing the approximate radius of a single nucleon (typically about $1.2\times {10}^{-15}\text{m}$ or $1.2\text{fm}$). Therefore, Statement (A) is correct.

Statement (B): Volume of nucleus is proportional to mass number A
Assuming a nucleus is spherical, its volume ($V$) is given by the formula:
$V=\frac{4}{3}\pi R^3$
Substituting the relation for radius $R$ from Statement (A) into this volume formula, we get:
$V=\frac{4}{3}\pi {\left(R_0{A}^{1/3}\right)}^3=\frac{4}{3}\pi R_0^{3}A$
Since $R_0$ is a constant, the volume of the nucleus $V$ is directly proportional to the mass number $A$ ($V\propto A$). Therefore, Statement (B) is correct.

Statement (C): The density of nucleus increases with the radius of nucleus
The density ($\rho$) of nuclear matter is defined as the ratio of nuclear mass to nuclear volume:
$\rho =\frac{\text{Mass of Nucleus}}{\text{Volume of Nucleus}}$
If $m$ is the average mass of a nucleon (proton or neutron), the total mass of a nucleus containing $A$ nucleons is approximately $mA$. Using the volume from above:
$\rho =\frac{mA}{\frac{4}{3}\pi R_0^{3}A}=\frac{3m}{4\pi R_0^3}$
The expression shows that the density depends only on the average mass of a nucleon $m$ and the constant $R_0$. Hence, the nuclear density is constant and does not increase with the radius. Therefore, Statement (C) is incorrect.

Statement (D): Density of nuclear matter does not depend on its mass number A
As derived in the step above, the mass number $A$ cancels out completely from the numerator and denominator in the density equation. This indicates that the density of nuclear matter is constant and independent of the mass number $A$ for all nuclei. Therefore, Statement (D) is correct.

Consequently, statements (A), (B), and (D) are correct, which matches the option (A), (B) and (D) only.