Answer: (4)

Solution

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $7x^2 - 3x - 2 = 0$

$\therefore \alpha + \beta = \frac{3}{7}$, $\alpha\beta = \frac{-2}{7}$

Now, $\frac{\alpha}{1-\alpha^2} + \frac{\beta}{1-\beta^2}$

$= \frac{\alpha - \alpha\beta(\alpha+\beta) + \beta}{1 - (\alpha^2 + \beta^2) + (\alpha\beta)^2}$

$= \frac{(\alpha+\beta) - \alpha\beta(\alpha+\beta)}{1 - (\alpha+\beta)^2 + 2\alpha\beta + (\alpha\beta)^2}$

$= \frac{\frac{3}{7} + \frac{2}{7} \cdot \frac{3}{7}}{1 - \frac{9}{49} + 2 \cdot \frac{-2}{7} + \frac{4}{49}} = \frac{27}{16}$