The area (in sq. units) of the region

A = {(x, y) : x² $$ \le $$ y $$ \le $$ x + 2} is

Let $$\alpha $$ and $$\beta $$ be the roots of the equation x² + x + 1 = 0. Then for y $$ \ne $$ 0 in R,
$$$\left| {\matrix{ {y + 1} & \alpha & \beta \cr \alpha & {y + \beta } & 1 \cr \beta & 1 & {y + \alpha } \cr } } \right|$$$ is equal to

Four persons can hit a target correctly with probabilities $${1 \over 2}$$, $${1 \over 3}$$, $${1 \over 4}$$ and $${1 \over 8}$$ respectively. if all hit at the target independently, then the probability that the target would be hit, is :

If the line, $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 2} \over 4}$$ meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is :