(a) The functions \(f : x \to x^{2} + 1\) and \(g : x \to 5 - 3x\) are defined on the set of the real numbers, R.

(i) State the domain of \(f^{-1}\), the inverse of f ; (ii) find \(g^{-1} (2)\).

(b) Evaluate : \(\int \frac{(x + 3)}{x^{2} + 6x + 9} \mathrm {d} x\)

(a) If \(\frac{\sqrt{5} + 4}{3 - 2\sqrt{5}} - \frac{2 + \sqrt{5}}{4 - 2\sqrt{5}} = a + b\sqrt{5}\), find the values of a and b.

(b)(i) Evaluate : \(\begin{vmatrix} 2 & -1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{vmatrix}\)

(ii) Using the result in b(i), find, correct to two decimal places, the value of x in the system of equations.

\(2x - y + 2z + 5 = 0\)

\(x + 3y + 4z - 1 = 0\)

\(x + 2y + z + 2 = 0\)

(a)(i) Write down the binomial expansion of \((1 + x)^{4}\).

(ii) Use the result in (a)(i) to evaluate, correct to three decimal places \((\frac{5}{4})^{4}\).

(b) The first, second and fifth terms of a linear sequence (A.P) are three consecutive terms of an exponential sequence (G.P). If the first term of the linear sequence is 7, find the common difference.

The table shows the distribution of the heights of a group of people.

(a) Draw a histogram to illustrate the distribution.

(b) Using an assumed mean of 1.1m, find, correct to one decimal place, the mean height of the group.

(a) Edem and his wife were invited to a dinner by a family of 5. They all sat in such a way in such a way that Edem sat next to his wife. Find the number of ways of seating them in a row.

(b) A bag contains 4 red and 5 black identical balls. If 5 balls are selected at random, one after the other with replacement, find the probability that :

(i) a red ball was picked 3 times ; (ii) a black ball was picked at most 2 times.