(a) Differentiate \(\frac{x^{2} + 1}{(x + 1)^{2}}\) with respect to x.

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

(ii) Using the answer in (b)(i), solve the system of equations.

\(x + 2y - z = 4\)

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

\(-x + y + 3z = -1\).

(a) Using a scale of 2 cm to 30° on the x- axis, 2 cm to 0.2 units on the y- axis, on the same graph sheet, draw the graphs of \(y = \sin 2x\) and \(y = \cos x\) for \(0° \leq x \leq 210°\) at intervals of 30°.

(b) Using the graphs in (a), find the truth set of :

(i) \(\sin 2x = 0\) ; (ii) \(\sin 2x - \cos x = 0\).

(a) The sum of the first three terms of a decreasing exponential sequence (G.P) is equal to 7 and the product of these three is equal to 8. Find the :

(i) common ratio ; (ii) first three terms of the sequence.

(b) Using the trapezium rule with the ordinates at x = 1, 2, 3, 4 and 5, calculate, correct to two decimal places, the value of \(\int_{1} ^{5} (x + \frac{2}{x^{2}}) \mathrm {d} x\).

(a) Find the maximum and minimum points of the curve \(y = 2x^{3} - 3x^{2} - 12x + 4\).

(b) Sketch the curve in (a) above.

The table gives the relationship between the height, in metres, of a plant and the number of days it is left to grow.

(a) Using a scale of 2 cm to represent 0.5 units on the y- axis and 2cm to 10 units on the x- axis, draw a scatter diagram for the information.

(b) Find \(\bar{x}\), the mean of x, and \(\bar{y}\), the mean of y, and plot \((\bar{x}, \bar{y})\) on the diagram.

(c) Draw the line of best fit to pass through \((\bar{x}, \bar{y})\) and \((10, 1)\).

(d) From graph, find the :

(i) equation of the line of best fit ; (ii) height of plant in 75 days.