Consider the following two statements A and B and identify the correct answer given below.
A. Nuclear density is same for all nuclei.
B.Radius of the nucleus R and its mass number A are related as `sqrt(A) alpha R^(1/(6))`.

Options

A. A and B are true
B. A and B are false
C. A is true but B is false
D. A is false but B is true

Answer

To solve the problem, we need to evaluate the two statements A and B regarding nuclear density and the relationship between the radius of a nucleus and its mass number.

### Step 1: Analyze Statement A
**Statement A:** "Nuclear density is the same for all nuclei."

- The nuclear density (ρ) can be expressed as:
\[ \rho = \frac{mass}{volume} = \frac{m_p \cdot A}{\frac{4}{3} \pi R^3} \]
where \(m_p\) is the mass of a proton, \(A\) is the mass number, and \(R\) is the radius of the nucleus.

- It is known that the radius \(R\) of a nucleus is proportional to \(A^{1/3}\):
\[ R \propto A^{1/3} \]

- Substituting \(R\) in the density formula, we find that the density remains constant for different nuclei because both the mass and volume change proportionally with \(A\).

**Conclusion for Statement A:** True.

### Step 2: Analyze Statement B
**Statement B:** "Radius of the nucleus \(R\) and its mass number \(A\) are related as \(\sqrt{A} \propto R^{1/6}\)."

- From the known relationship:
\[ R \propto A^{1/3} \]

- If we express this in terms of \(A\):
\[ A \propto R^3 \]

- Taking the square root of \(A\):
\[ \sqrt{A} \propto R^{3/2} \]

- This indicates that the relationship is not as stated in Statement B. The correct relationship is:
\[ \sqrt{A} \propto R^{3/2} \]

**Conclusion for Statement B:** False.

### Final Conclusion
- Statement A is true, and Statement B is false. Therefore, the correct answer is:
**A is true but B is false.**

### Answer: A is true but B is false.

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