Nucleus

Step 1: Understanding the Concept:
This question assesses the understanding of the basic properties of the atomic nucleus, specifically its size (radius), volume, and density, and how they relate to the mass number A.

Step 2: Detailed Explanation:
(A) A nucleus of mass number A has a radius R given by the expression R = R A
This is the standard empirical formula for the nuclear radius, where is a constant approximately equal to 1.2 fm. This relationship is derived from experimental data and is a fundamental concept in nuclear physics. This statement is correct.
(B) Volume of nucleus is proportional to mass number A
Assuming the nucleus is spherical, its volume is given by . Substituting the expression for radius from statement (A): Since is a constant, the volume is directly proportional to the mass number A ( ). This statement is correct.
(C) The density of nucleus increases with the radius of nucleus.
Nuclear density ( ) is defined as mass divided by volume ( ). The mass of the nucleus is approximately , where is the mass of a nucleon (proton/neutron). The volume of the nucleus is . So, the density is: As we can see, the mass number cancels out. The density is approximately constant for all nuclei, independent of their radius or mass number. Therefore, the statement that density increases with radius is incorrect.
(D) Density of nuclear matter does not depend on its mass number A
As derived above, the expression for nuclear density is independent of the mass number A. This remarkable property means that all atomic nuclei have roughly the same extremely high density. This statement is correct.

Step 3: Final Answer:
Statements (A), (B), and (D) are correct, while statement (C) is incorrect. The correct option is (C).