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 :

Let $$\alpha $$ > 0, $$\beta $$ > 0 be such that
$$\alpha $$³ + $$\beta $$² = 4. If the maximum value of the term independent of x in
the binomial expansion of $${\left( {\alpha {x^{{1 \over 9}}} + \beta {x^{ - {1 \over 6}}}} \right)^{10}}$$ is 10k,
then k is equal to :

Let A be a 2 $$ \times $$ 2 real matrix with entries from {0, 1} and |A| $$ \ne $$ 0. Consider the following two statements :

(P) If A $$ \ne $$ I₂ , then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,

where I₂ denotes 2 $$ \times $$ 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :