(a) Solve the equation : \(\sqrt{4x - 3} - \sqrt{2x - 5} = 2\).

(b) Find the finite area enclosed by the curve \(y^{2} = 4x\) and the line \(y + x = 0\).

(a) Given that \(\begin{vmatrix} 5 & 2 & -3 \\ -1 & k & 6 \\ 3 & 9 & (k + 2) \end{vmatrix} = -207\), find the values of the constant k.

(b) The equation of a curve is \(x(y^{2} + 1) - y(x^{2} + 1) + 4 = 0\). Find the:

(i) gradient of the curve at any point (x, y).

(ii) equation of the tangent to the curve at the point (-1, -3).

(a) Using the trapezium rule with seven ordinates, evaluate \(\int_{0}^{3} \frac{\mathrm d x}{x^{2} + 1}\), correct to two decimal places.

(b) Using matrix method, solve \(-2x + y = 3; - x + 4y = 1\).

(a) If \(^{18}C_{r} = ^{18}C_{r + 2}\), find \(^{r}C_{5}\).

(b) In a community, 10% of the people tested positive to the HIV virus. If 6 persons from the community are selected at random, one after the other with replacement, calculate, correct to four decimal places, the probability that : (i) exactly 5 (ii) none (iii) at most 2; tested positive to the virus.

The table below shows the distribution of ages of workers in a company.

(a) Using an assumed mean of 39, calculate the (i) mean (ii) standard deviation; of the distribution.

(b) If a worker is selected at random from the company for an award, what is the probability that he is at most 36 years old?