The table below shows the marks obtained by forty pupils in a Mathematics test.

(a) Draw a histogram for the mark distribution ;

(b) Use your histogram to estimate the mode ;

(c) Calculate the median of the distribution.

(a) If \(\log_{10} 2 = 0.3010\) and \(\log_{10} 3 = 0.4771\), calculate without using tables, the value of \(\log_{10} 0.72\).

(b) A hawk on top of a tree, 20 metres high views a chick on the ground at an angle of depression of 39°. Find, correct to 2 significant figures, the distance of the chick from the bottom of the tree.

(a) ABCD is a trapezium with AB parallel to DC and /AD/ = /AB/. If  < BAD = 106°, find < BDC.

(b) The table below shows the distribution of 20 cards labelled A - E.

(i) If a card is selected at random from the pack, what is the probability that the card is E? (ii) If two cards are selected at random one after the other without replacement from the pack, what is the probability that one of the two cards is B?

(a) Without using tables, find the value of \(\frac{0.45 \times 0.91}{0.0117}\)

(b) Find the number which is exactly halfway between \(1\frac{6}{7}\) and \(2\frac{11}{28}\).

(c) If each interior angle of a regular polygon is five times the exterior angle, how many sides has the polygon?

(d) Calculate the volume of the material used in making a pipe 20cm long, with an internal diameter 6cm and external diameter 8cm. [Take \(pi = \frac{22}{7}\)].

(a) The universal set U is the set of integers, P, Q and R are subsets of U defined as follows:

\(P = x : x \leq 2 \) ; \(Q = x : -7 < x < 15\) ; \(R = x : -2 \leq x < 19\).

Find (i) \(P \cap Q\) ; (ii) \(P \cap (Q \cup R')\), where R' is the complement of R with respect to U.

(b) The following data shows the marks of 40 students in a History examination.

41 52 37 56 63 48 65 46 54 32 51 66 74 23 35 61 58 44 49 53 45 57 56 38 59 28 50 49 67 56 36 45 79 68 43 56 26 47 55 71.

(i) Form a grouped frequency table with the class intervals 20 - 29, 30 - 39, 40 - 49 etc; (ii) Find the mean of the distribution.