If ` sqrt(( 1 - p^(2)) (1 - q^(2))) = ( sqrt(3))/( 2)` and q=0 then what is the value of `sqrt( 2 p^( 2) + 2 q ^(2) + 2 pq) + sqrt( 2 p ^(2) + 2 q^(2) - 2pq ) `?

Options

A. 2
B. `sqrt(2)`
C. 1
D. none of these

Answer

To solve the problem step by step, we start with the given equation and conditions:

1. **Given Equation**:
\[ \sqrt{(1 - p^2)(1 - q^2)} = \frac{\sqrt{3}}{2} \]
**Condition**: \( q = 0 \)

2. **Substituting \( q = 0 \)**:
\[ \sqrt{(1 - p^2)(1 - 0^2)} = \frac{\sqrt{3}}{2} \]
This simplifies to:
\[ \sqrt{1 - p^2} = \frac{\sqrt{3}}{2} \]

3. **Squaring both sides**:
\[ 1 - p^2 = \left(\frac{\sqrt{3}}{2}\right)^2 \]
\[ 1 - p^2 = \frac{3}{4} \]

4. **Solving for \( p^2 \)**:
\[ p^2 = 1 - \frac{3}{4} = \frac{1}{4} \]

5. **Now, we need to evaluate**:
\[ \sqrt{2p^2 + 2q^2 + 2pq} + \sqrt{2p^2 + 2q^2 - 2pq} \]

6. **Substituting \( p^2 = \frac{1}{4} \) and \( q = 0 \)**:
- First term:
\[ \sqrt{2p^2 + 2q^2 + 2pq} = \sqrt{2 \cdot \frac{1}{4} + 2 \cdot 0 + 2 \cdot \frac{1}{4} \cdot 0} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

- Second term:
\[ \sqrt{2p^2 + 2q^2 - 2pq} = \sqrt{2 \cdot \frac{1}{4} + 2 \cdot 0 - 2 \cdot \frac{1}{4} \cdot 0} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

7. **Adding both terms**:
\[ \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

Thus, the final answer is:
\[ \sqrt{2} \]