Using a scale of 2cm to 1 unit on the x- axis and 1cm to 1 unit on the y- axis, draw on the same axes the graphs of \(y = 3 + 2x - x^{2}; y = 2x - 3\) for \(-3 \leq x \leq 4\). Using your graph:

(i) solve the equation \(6 - x^{2} = 0\);

(ii) find the maximum value of \(3 + 2x - x^{2}\);

(iii) find the range of x for which \(3 + 2x - x^{2} \leq 1\), expressing all your answers correct to one decimal place.

(a) Prove that the angle which an arc of a circle subtends at the centre is twice that which it subtends at any point on the remaining part of the circumference.

(b) 

In the diagram, O is the centre of the circle, < OQR = 32° and < MPQ = 15°. Calculate (i) < QPR ; (ii) < MQO.

The table below shows the distribution of the waiting times for some customers in a certain petrol station.

(a) Write down the class boundaries of the distribution.

(b) Construct a cumulative frequency curve for the data;

(c) Using your graph, estimate: (i) the interquartile range of the distribution ; (ii) the proportion of customers who could have waited for more than 3 minutes.

(a) Using a ruler and a pair of compasses only, construct a parallelogram PQRS with diagonals |PR| = 9cm and |QS| = 6cm, intersecting at K and < QKR = 60°.

(b) Construct a rectangle PABS which is equal in area to PQRS in (a) above and on the same side of PS as PQRS. Measure |PA|.

From a horizontal distance of 8.5 km, a pilot observes that the angles of depression of the top and the base of a control tower are 30° and 33° respectively. Calculate, correct to three significant figures :

(a) the shortest distance between the pilot and the base of the control tower;

(b) the height of the control tower.