$P(\alpha \max (X, Y) < XY)$
$= P(\alpha X < XY \text{ AND } \alpha Y < XY) $
$= P(\alpha < Y \text{ AND } \alpha < X )$
$= P(\alpha < Y) \times P(\alpha < X )$ (Since, $X$ and $Y$ are independent)
$= (P(Y) - P(\alpha)) \times (P(X) - P(\alpha)) $
$= (1 - \alpha) \times (1 - \alpha)$ (Uniform distribution in the interval $0$ to $1$)
$= (1 - \alpha)^2$
Option D.