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).

(a) Evaluate \(\int_{1} ^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\)

(b)(i) Evaluate: \(\begin{vmatrix} 2 & -3 & 1 \\ 0 & 1 & -2 \\ 1 & 2 & -3 \end{vmatrix}\)

(ii) Using your answer in b(i), solve the simultaneous equations :

\(2x - 3y + z = 10\)

\(y - 2z = -7\)

\(x + 2y - 3z = -9\)