(a) Simplify : \(\frac{1}{1 - \cos \theta} + \frac{1}{1 + \cos \theta}\) and leave your answer in terms of \(\sin \theta\).

(b) Find the equation of the line joining the stationary points of \(y = x^{2} (x - 3)\) and the distance between them.

(a) If \(f(x) = \frac{2x - 3}{(x^{2} - 1)(x + 2)}\)

(i) find the values of x for which f(x) is undefined.

(ii) express f(x) in partial fractions.

(b) A circle with centre (-3, 1) passes through the point (3, 1). Find its equation.

(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.