(a) If \(y = (2x + 3)^{7} + \frac{x + 1}{2x - 1}\), find the value of \(\frac{\mathrm d y}{\mathrm d x}\) at x = -1.

(b) Using the substitution, \(u = x + 2\), evaluate \(\int_{1} ^{2} \frac{x - 1}{(x + 2)^{4}} \mathrm d x\).

The images of (3, 2) and (-1, 4) under a linear transformation T are (-1, 4) and (7, 11) respectively. P is another transformation where \(P : (x, y) \to (x + y, x + 2y)\).

(a) Find the matrices T and P of the linear transformations T and P;

(b) Calculate TP.

(c) Find the image of the point X(4, 3) under TP.

The table gives the distribution of marks of 60 candidates in a test.

(a) Draw a cumulative frequency curve of the distribution.

(b) From your curve, estimate the : (i) 80th percentile ; (ii) median ; (iii) semi-interquartile range.

(a) Eight coins are tossed at once. Find, correct to three decimal places, the probability of obtaining :

(i) exactly 8 heads ; (ii) at least 5 heads ; (iii) at most 1 head.

(b) In how many ways can four letters from the word SHEEP be arranged (i) without any restriction ; (ii) with only one E.

(a) A manufacturer produces light bulbs which are tested in the following way. A batch is accepted in either of the following cases:

(i) a first sample of 5 bulbs contains no faulty bulbs ; (ii) a first sample of 5 bulbs contains at least one faulty bulb but a second sample of size 5 has no faulty bulb. If 10% of the bulbs are faulty, what is the probability that the batch is accepted?

(b) A bag contains 15 identical marbles of which 3 are black, Keshi picks a marble at random from the bag and replaces it. If this is repeated 10 times; what is the probability that he :

(i) did not pick a black ball? (ii) picked a black ball at most three times?