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

Answer: (D)

$4 \cos^2 \theta.$

Answer 2

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