(a) A man travels from a village X on a bearing of 060° to a village Y which is 20km away. From Y, he travels to a village Z, on a bearing of 195°. If Z is directly east of X, calculate, correct to three significant figures, the distance of :

(i) Y from Z ; (ii) Z from X .

(b) An aircraft flies due South from an airfield on latitude 36°N, longitude 138°E to an airfield on latitude 36°S, longitude 138°E. 

(i) Calculate the distance travelled, correct to three significant figures ; (ii) if the speed of the aircraft is 800km per hour, calculate the time taken, correct to the nearest hour.

[Take \(\pi = \frac{22}{7}\), R = 6400km].

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.