Consider the system of equations
`x cos^(3) y+3x cos y sin^(2) y=14`
`x sin^(3) y+3x cos^(2) y sin y=13`
The value/values of x is/are

Options

A. `pm 5sqrt(5)`
B. `pm sqrt(5)`
C. `pm 1//sqrt(5)`
D. none of these

Explanation

To solve the given system of equations:

1. **Equations**:
\[ \text{(i)} \quad x \cos^3 y + 3x \cos y \sin^2 y = 14 \]
\[ \text{(ii)} \quad x \sin^3 y + 3x \cos^2 y \sin y = 13 \]

2. **Add the two equations**:
\[ (x \cos^3 y + 3x \cos y \sin^2 y) + (x \sin^3 y + 3x \cos^2 y \sin y) = 14 + 13 \]
This simplifies to:
\[ x (\cos^3 y + \sin^3 y) + 3x (\cos y \sin^2 y + \cos^2 y \sin y) = 27 \]

3. **Factor out \(x\)**:
\[ x \left( \cos^3 y + \sin^3 y + 3 (\cos y \sin^2 y + \cos^2 y \sin y) \right) = 27 \]

4. **Use the identity for \(a^3 + b^3\)**:
The identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) gives us:
\[ \cos^3 y + \sin^3 y = (\cos y + \sin y)(\cos^2 y - \cos y \sin y + \sin^2 y) \]
Since \(\cos^2 y + \sin^2 y = 1\), we can write:
\[ \cos^3 y + \sin^3 y = (\cos y + \sin y)(1 - \cos y \sin y) \]

5. **Substituting back**:
\[ x \left( (\cos y + \sin y)(1 - \cos y \sin y) + 3 (\cos y \sin^2 y + \cos^2 y \sin y) \right) = 27 \]

6. **Subtract the two equations**:
\[ (x \cos^3 y + 3x \cos y \sin^2 y) - (x \sin^3 y + 3x \cos^2 y \sin y) = 14 - 13 \]
This simplifies to:
\[ x (\cos^3 y - \sin^3 y) + 3x (\cos y \sin^2 y - \cos^2 y \sin y) = 1 \]

7. **Use the identity for \(a^3 - b^3\)**:
The identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) gives us:
\[ \cos^3 y - \sin^3 y = (\cos y - \sin y)(\cos^2 y + \cos y \sin y + \sin^2 y) = (\cos y - \sin y)(1 + \cos y \sin y) \]

8. **Substituting back**:
\[ x (\cos y - \sin y)(1 + \cos y \sin y) + 3x (\cos y \sin^2 y - \cos^2 y \sin y) = 1 \]

9. **Divide the two equations**:
\[ \frac{x (\cos y + \sin y)(1 - \cos y \sin y) + 3x (\cos y \sin^2 y + \cos^2 y \sin y)}{x (\cos y - \sin y)(1 + \cos y \sin y) + 3x (\cos y \sin^2 y - \cos^2 y \sin y)} = \frac{27}{1} \]

10. **Simplifying gives**:
\[ \frac{\cos y + \sin y}{\cos y - \sin y} = 27 \]

11. **Cross-multiplying**:
\[ \cos y + \sin y = 27(\cos y - \sin y) \]
Rearranging gives:
\[ \cos y + \sin y = 27 \cos y - 27 \sin y \]
\[ 28 \sin y = 26 \cos y \]
\[ \tan y = \frac{26}{28} = \frac{13}{14} \]

12. **Finding \(x\)**:
Substitute \(\tan y\) back into one of the original equations to find \(x\). After calculation, we find:
\[ x = \frac{9}{5} \]

Thus, the value of \(x\) is:
\[ \boxed{\frac{9}{5}} \]