Correct Answer: $\frac{1}{4}$

Solution : Given that $\alpha + \beta = 90^\circ$ and $\alpha = 2\beta$,
Substitute $\alpha$ in terms of $\beta$ into the first equation,
$⇒2\beta + \beta = 90^\circ$
$⇒\beta = 30^\circ$
$\therefore \alpha = 60^\circ$
Substitute these values into the expression $3\cos^2\alpha - 2\sin^2\beta$:
$3\cos^2(60^\circ) - 2\sin^2(30^\circ) = 3\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right)^2 = 3\left(\frac{1}{4}\right) - 2\left(\frac{1}{4}\right) = \frac{1}{4}$
Hence, the correct answer is $ \frac{1}{4}$.