(a) If \(f(x) = \int (4x - x^{2}) \mathrm {d} x\) and f(3) = 21, find f(x).

(b) The second, fourth and eigth terms of an Arithmetic Progression (A.P) form the first three consecutive terms of a Geometric Progression (G.P). The sum of the third and fifth terms of the A.P is 20, find the :

(i) first four terms of the A.P

(ii) sum of the first ten terms of the A.P

(a) In a school, the ratio of those who passed to those who failed in a History test is 4 : 1. If 7 students are selected at random from the school, find, correct to two decimal places, the probability that :

(i) at least 3 passed the test ; (ii) between 3 and 6 students failed the test.

(b) A fair die is thrown five times; find the probability of obtaining a six three times.

The table shows the frequency distribution of the ages of patients in a clinic.

(a) Draw a histogram for the distribution

(b) Find, correct to two decimal places, the mean age of the patients.

(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 \(\overrightarrow{RS}\).