The table shows the scores of 2000 candidates in an entrance examination into a private secondary school.

(a) Prepare a cumulative frequency table and draw the cumulative frequency curve for the distribution.

(b) Use your curve to estimate the : (i) cut off mark, if 300 candidates are to be offered admission ; (ii) probability that a candidate picked at random, scored at least 45%.

A box contains 5 blue balls, 3 black balls and 2 red balls of the same size. A ball is selected at random from the box and then replaced. A second ball is then selected. Find the probability of obtaining

(a) two red balls ;

(b) two blue balls or two black balls ;

(c) one black and one red ball in any order.

(a) Given that \(3 \times 9^{1 + x} = 27^{-x}\), find x.

(b) Evaluate \(\log_{10} \sqrt{35} + \log_{10} \sqrt{2} - \log_{10} \sqrt{7}\)

(a) Evaluate, without using mathematical tables, \(17.57^{2} - 12.43^{2}\).

(b) Prove that angles in the same segment of a circle are equal.

(a) A tower and a building stand on the same horizontal level. From the point P at the bottom of the building, the angle of elevation of the top, T of the tower is 65°. From the top Q of the building, the angle of elevation of the point T is 25°. If the building is 20m high, calculate the distance PT.

(b) Hence or otherwise, calculate the height of the tower. [Give your answers correct to 3 significant figures].