Question

Let \(\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)\). Consider the functions
\[ u : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \quad \text{and} \quad v : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \]
given by
\[ u(x,y) = x - \frac{x}{x^2 + y^2} \quad \text{and} \quad v(x,y) = y + \frac{y}{x^2 + y^2}. \]
The value of the determinant \(\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}\) at the point \((\cos \theta, \sin \theta)\) is equal to

• A. \(4 \sin \theta.\)
• B. \(4 \cos \theta.\)
• C. \(4 \sin^2 \theta.\
• D. \(4 \cos^2 \theta.\)

Answer 1

The Correct Option is D

The correct option is (D): \(4 \cos^2 \theta.\)