Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $$\Delta $$S'BS is a right angled triangle with right angle at B and area ($$\Delta $$S'BS) = 8 sq. units, then the length of a latus rectum of the ellipse is :

Let S be the set of all real values of $$\lambda $$ such that a plane passing through the points (–$$\lambda $$², 1, 1), (1, –$$\lambda $$², 1) and (1, 1, – $$\lambda $$²) also passes through the point (–1, –1, 1). Then S is equal to :

If the function f given by f(x) = x³ – 3(a – 2)x² + 3ax + 7, for some a$$ \in $$R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, $${{f\left( x \right) - 14} \over {{{\left( {x - 1} \right)}^2}}} = 0\left( {x \ne 1} \right)$$ is :

Let Z be the set of integers.
If A = {x $$ \in $$ Z : 2^{(x + 2) (x2 $$-$$ 5x + 6)} = 1} and
B = {x $$ \in $$ Z : $$-$$ 3 < 2x $$-$$ 1 < 9},
then the number of subsets of the set A $$ \times $$ B, is