(a) Differentiate \((x - 3)(x^{2} + 5)\) with respect to x.

(b) If \((x + 1)^{2}\) is a factor of \(f(x) = x^{3} + ax^{2} + bx + 3\), where a and b are constants, find the :

(i) values of a and b ; (ii) zeros of f(x).

(a) Express \(\frac{5 + \sqrt{2}}{3 - \sqrt{2}} - \frac{5 - \sqrt{2}}{3 + \sqrt{2}}\) in the form \(a + b\sqrt{2}\).

(b) Solve the following equations simultaneously using the determinant method.

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

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

\(2x + 3y + z = 0\)

(a) If \(f(x) = \frac{x - 3}{2x - 1} , x \neq \frac{1}{2}\) and \(g(x) = \frac{x - 1}{x + 1}, x \neq -1\), fing \(g \circ f\).

(b)(i) Sketch the curve \(y = 9x - x^{3}\) ; (ii) Calculate the total area bounded by the x- axis and the curve \(y = 9x - x^{3}\).

The histogram above represents the scores of some candidates in an examination. 

(a) Using the histogram, construct a frequency distribution table indicating clearly the class intervals ;

(b) Draw a cumulative frequency curve of the distribution and use it to estimate the :

(i) median ; (ii) quartile deviation.

The probabilities that Kofi, Kwasi and Ama will pass a certain examination are \(\frac{9}{10}, \frac{4}{5}\) and x respectively. If the probability that only one of them will pass the examination is \(\frac{9}{50}\), find the :

(a) value of x ; 

(b) probability that at least one of them will pass the examination.