If    $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},$$ then 'a' is equal to :

If the function

f(x) = $$\left\{ {\matrix{ { - x} & {x < 1} \cr {a + {{\cos }^{ - 1}}\left( {x + b} \right),} & {1 \le x \le 2} \cr } } \right.$$

is differentiable at x = 1, then $${a \over b}$$ is equal to :

If the tangent at a point P, with parameter t, on the curve x = 4t² + 3, y = 8t³−1, t $$ \in $$ R, meets the curve again at a point Q, then the coordinates of Q are :

The minimum distance of a point on the curve y = x²−4 from the origin is :