Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system of linear equations

2x – y + 2z = 2
x – 2y + $$\lambda $$z = –4
x + $$\lambda $$y + z = 4

has no solution. Then the set S :

If a function f(x) defined by

$$f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < 1} \cr {c{x^2},} & {1 \le x \le 3} \cr {a{x^2} + 2cx,} & {3 < x \le 4} \cr } } \right.$$

be continuous for some $$a$$, b, c $$ \in $$ R and f'(0) + f'(2) = e, then the value of of $$a$$ is :

Area (in sq. units) of the region outside

$${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$$ is :

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :