A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be ($$\alpha$$, $$\beta$$). Then, the value of 7$$\alpha$$ + 3$$\beta$$ is equal to ____________.

Let A = {1, a₁, a₂ ....... a₁₈, 77} be a set of integers with 1 < a₁ < a₂ < ....... < a₁₈ < 77.

Let the set A + A = {x + y : x, y $$\in$$ A} contain exactly 39 elements. Then, the value of a₁ + a₂ + ...... + a₁₈ is equal to _____________.

The number of positive integers k such that the constant term in the binomial expansion of $${\left( {2{x^3} + {3 \over {{x^k}}}} \right)^{12}}$$, x $$\ne$$ 0 is 2⁸ . l, where l is an odd integer, is ______________.

The number of elements in the set {z = a + ib $$\in$$ C : a, b $$\in$$ Z and 1 < | z $$-$$ 3 + 2i | < 4} is __________.