Let $$\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)} $$ where the function ƒ satisfies
ƒ(x + y) = ƒ(x)ƒ(y) for all natural numbers x, y and ƒ(1) = 2. then the natural number 'a' is

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 :

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

The area (in sq. units) of the region

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