(a) A body P of mass q kg is suspended by two light inextensible strings AB and DB attached to a horizontal table. The strings are inclined at 30° and 60° respectively to the horizontal and the tension in AB is 48N. If the system is in equilibrium :

(i) sketch a diagram to represent the information ; (ii) calculate the tension in DB ; 

(a) Given that \(m = (6i + 8j)\) and \(n = (-8i + \frac{7}{3}j)\), find the :

(i) magnitudes and direction of m and n ; (ii) angle between m and n.

(b) The position vectors of points P, Q, R and S are \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}, \begin{pmatrix} 10 \\ 4 \end{pmatrix}, \begin{pmatrix} 3 \\ 12 \end{pmatrix}\) and \(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\) respectively. Show that \(\overrightarrow{PQ}\) is perpendicular to \(\overrightarrow{RS}\).

A binary operation \(*\) is defined on the set, R, of real numbers by \(m * n = m + n + 2\). Find the :

(a) identity element under the operation ;

(b) inverse of n under the operation .

Given that (5, 2), (-4, k) and (2, 1) lie on a straight line, find the value of k.

(a) If \(f(x + 2) = 6x^{2} + 5x - 8\), find \(f(5)\).

(b) Express \(\frac{7\sqrt{2} + 3\sqrt{3}}{4\sqrt{2} - 2\sqrt{3}}\) in the form \(p + q\sqrt{r}\), where p, q and r are rational numbers.