A : Nuclear density is almost same for all nuclei .
R: The radius (r ) of a nucleus depends only on the mass number (A) as `r prop A^(1//3)`.

Options

A. if both assertion & reason are true and the reason is correct explanation of the assertion then mark (1)
B. if both assertion & reason are true and the reason is not correct explanation of the assertion then mark (2)
C. if assertion is true statement but reason is false then mark (3)
D. if both assertion & reason are false then mark (4)

Answer

To solve the problem, we need to analyze the assertion (A) and reason (R) provided in the question.

### Step-by-Step Solution:

1. **Understanding Nuclear Density**:
- The assertion states that nuclear density is almost the same for all nuclei.
- Density (\( \rho \)) is defined as mass per unit volume. For a nucleus, the mass can be approximated by the mass number \( A \) (which counts the total number of protons and neutrons).

2. **Calculating Nuclear Density**:
- The formula for density is given by:
\[ \rho = \frac{\text{mass}}{\text{volume}} \]
- For a nucleus, we can express this as:
\[ \rho = \frac{A \cdot m_u}{V} \]
where \( m_u \) is the atomic mass unit.

3. **Volume of the Nucleus**:
- The volume of a nucleus can be approximated assuming it is spherical:
\[ V = \frac{4}{3} \pi r^3 \]
- The radius \( r \) of the nucleus is related to the mass number \( A \) by the relation:
\[ r \propto A^{1/3} \]
- This implies that:
\[ r = r_0 A^{1/3} \]
where \( r_0 \) is a constant (approximately \( 1.2 \) Fermi or \( 1.2 \times 10^{-15} \) meters).

4. **Substituting the Radius into Volume**:
- Substituting \( r \) into the volume formula gives:
\[ V = \frac{4}{3} \pi (r_0 A^{1/3})^3 = \frac{4}{3} \pi r_0^3 A \]

5. **Finding Nuclear Density**:
- Now substituting the volume back into the density formula:
\[ \rho = \frac{A \cdot m_u}{\frac{4}{3} \pi r_0^3 A} \]
- The \( A \) cancels out:
\[ \rho = \frac{3 m_u}{4 \pi r_0^3} \]
- This shows that the nuclear density \( \rho \) is independent of \( A \) and is a constant value for all nuclei.

6. **Conclusion**:
- Since the density does not depend on the mass number \( A \), the assertion (A) is true.
- The reason (R) correctly explains why the density is constant by showing that the radius depends on \( A \) as \( r \propto A^{1/3} \). Therefore, the reason is also true.

### Final Statement:
Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion.