The following question of two statements, one labelled as the 'Ass er tion (A)' and the other as 'Reason (R )' You are toe examine these two statements carefully ans select the answer. Assertion (A): If `Z_(1)=3+sqrt(-16), and Z_(2)=3+sqrt(-25),Z_(1)//Z_(2)` is a complex number. Reason (R ):If `Z_(1), Z_(2)` are complex numbers then `Z_(1)//Z_(2)` is always a complex number.

Options

A. Both A and R are individually true, and R is the correct explanation of A.
B. Both A and R are individually true, and R is the correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

Explanation

To solve the given problem, we need to analyze the two statements: the Assertion (A) and the Reason (R).

### Step-by-Step Solution

1. **Identify the Complex Numbers**:
- Given \( Z_1 = 3 + \sqrt{-16} \)
- Given \( Z_2 = 3 + \sqrt{-25} \)

2. **Simplify the Complex Numbers**:
- For \( Z_1 \):
\[ Z_1 = 3 + \sqrt{-16} = 3 + 4i \quad (\text{since } \sqrt{-16} = 4i) \]
- For \( Z_2 \):
\[ Z_2 = 3 + \sqrt{-25} = 3 + 5i \quad (\text{since } \sqrt{-25} = 5i) \]

3. **Calculate \( Z_1 / Z_2 \)**:
- We need to compute \( \frac{Z_1}{Z_2} = \frac{3 + 4i}{3 + 5i} \).
- To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(3 + 4i)(3 - 5i)}{(3 + 5i)(3 - 5i)} \]

4. **Multiply the Numerator**:
- Calculate \( (3 + 4i)(3 - 5i) \):
\[ = 3 \cdot 3 + 3 \cdot (-5i) + 4i \cdot 3 + 4i \cdot (-5i) \]
\[ = 9 - 15i + 12i - 20i^2 \]
\[ = 9 - 3i + 20 \quad (\text{since } i^2 = -1) \]
\[ = 29 - 3i \]

5. **Multiply the Denominator**:
- Calculate \( (3 + 5i)(3 - 5i) \):
\[ = 3^2 - (5i)^2 = 9 - 25(-1) = 9 + 25 = 34 \]

6. **Combine the Results**:
- Now we have:
\[ \frac{Z_1}{Z_2} = \frac{29 - 3i}{34} \]
- This can be expressed as:
\[ = \frac{29}{34} - \frac{3}{34}i \]
- Since this expression contains the imaginary unit \( i \), it is indeed a complex number.

7. **Evaluate the Assertion and Reason**:
- **Assertion (A)**: \( Z_1 / Z_2 \) is a complex number. This is true.
- **Reason (R)**: If \( Z_1, Z_2 \) are complex numbers, then \( Z_1 / Z_2 \) is always a complex number. This is also true.

8. **Conclusion**:
- Both the assertion and reason are true, and the reason correctly explains the assertion.

### Final Answer
Both Assertion (A) and Reason (R) are true, and the reason is the correct explanation of the assertion.