Consider the system of equations `cos^(-1)x + (sin^(-1) y)^(2) = (p pi^(2))/(4) and (cos^(-1) x) (sin^(-1) y)^(2) = (pi^(4))/(16), p in Z`
The value of p for which system has a solution is

Options

A. 1
B. 2
C. 0
D. `-1`

Explanation

To solve the given system of equations:

1. **Equations**:
\[ \cos^{-1} x + (\sin^{-1} y)^2 = \frac{p \pi^2}{4} \quad (1) \]
\[ (\cos^{-1} x)(\sin^{-1} y)^2 = \frac{\pi^4}{16} \quad (2) \]

2. **Substitutions**:
Let:
\[ \cos^{-1} x = a \quad \text{and} \quad \sin^{-1} y = b \]
Thus, the equations become:
\[ a + b^2 = \frac{p \pi^2}{4} \quad (3) \]
\[ ab^2 = \frac{\pi^4}{16} \quad (4) \]

3. **Range of Variables**:
- Since \( a = \cos^{-1} x \), we have \( a \in [0, \pi] \).
- Since \( b = \sin^{-1} y \), we have \( b \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

4. **Finding the Range of \( b^2 \)**:
- The minimum value of \( b^2 \) is \( 0 \) (when \( b = 0 \)).
- The maximum value of \( b^2 \) is \( \left(\frac{\pi}{2}\right)^2 = \frac{\pi^2}{4} \).

5. **Substituting \( b^2 \) into Equation (3)**:
- From equation (3):
\[ a + b^2 = \frac{p \pi^2}{4} \]
- The minimum value of \( a + b^2 \) is \( 0 + 0 = 0 \).
- The maximum value of \( a + b^2 \) is \( \pi + \frac{\pi^2}{4} \).

6. **Setting Up Inequalities**:
From the range of \( a + b^2 \):
\[ 0 \leq \frac{p \pi^2}{4} \leq \pi + \frac{\pi^2}{4} \]

7. **Multiplying Through by 4**:
\[ 0 \leq p \pi^2 \leq 4\pi + \pi^2 \]

8. **Dividing by \( \pi^2 \)** (assuming \( \pi^2 > 0 \)):
\[ 0 \leq p \leq \frac{4\pi}{\pi^2} + 1 = \frac{4}{\pi} + 1 \]

9. **Calculating \( \frac{4}{\pi} \)**:
- Approximate \( \pi \approx 3.14 \):
\[ \frac{4}{\pi} \approx \frac{4}{3.14} \approx 1.273 \]
- Therefore:
\[ 0 \leq p \leq 2.273 \]

10. **Finding Integer Values**:
Since \( p \) must be an integer (\( p \in \mathbb{Z} \)):
\[ p = 0, 1, 2 \]

11. **Verifying Solutions**:
- Substitute \( p = 0, 1, 2 \) back into the equations to check for valid solutions.
- For \( p = 2 \), we can find specific values for \( a \) and \( b \) that satisfy both equations.

12. **Conclusion**:
The values of \( p \) for which the system has a solution are:
\[ \boxed{2} \]