A student representative council consists of 8 girls and 6 boys. If an editorial board consisting of 5 persons is to be formed, what is the probability that the board consists of

(a) 3 girls and 2 boys ;

(b) either all girls or all boys.

The coordinates of the vertices of triangle ABC are A(-2, 1), B(4, -2) and C(1, 8) respectively. If D(x, y) is the foot perpendicular from A to BC, find 

(a) an equation connecting x and y ; 

(b) the unit vector in the direction of BC.

The position vector of a body, with respect to the origin, is given by \(r = 4ti + (12 - 3t)j\) at any time t seconds. 

(a) Find the velocity of the body ;

(b) Calculate the magnitude of the displacement between t = 0 and t = 5.

(a) The 3rd and 6th terms of a Geometric Progression (G.P) are 2 and 54 respectively. Find the : (i) common ratio ; (ii) first term ; (iii) sum of the first 10 terms, correct to the nearest whole number. 

(b) The ratio of the coefficient of \(x^{4}\) to that of \(x^{3}\) in the binomial expansion of \((1 + 2x)^{n}\) is \(3 : 1\). Find the value of n.

(a) Using the same axes, sketch the curves \(y = 6 - x - x^{2}\) and \(y = 3x^{2} - 2x + 3\).

(b) Find the x- coordinates of the points of intersection of the two curves in (a).

(c) Calculatethe area of the finite region bounded by the two curves in (a).