(a) The sum of the first three terms of a decreasing exponential sequence (G.P) is equal to 7 and the product of these three is equal to 8. Find the :

(i) common ratio ; (ii) first three terms of the sequence.

(b) Using the trapezium rule with the ordinates at x = 1, 2, 3, 4 and 5, calculate, correct to two decimal places, the value of \(\int_{1} ^{5} (x + \frac{2}{x^{2}}) \mathrm {d} x\).

(a) Find the maximum and minimum points of the curve \(y = 2x^{3} - 3x^{2} - 12x + 4\).

(b) Sketch the curve in (a) above.

The table gives the relationship between the height, in metres, of a plant and the number of days it is left to grow.

(a) Using a scale of 2 cm to represent 0.5 units on the y- axis and 2cm to 10 units on the x- axis, draw a scatter diagram for the information.

(b) Find \(\bar{x}\), the mean of x, and \(\bar{y}\), the mean of y, and plot \((\bar{x}, \bar{y})\) on the diagram.

(c) Draw the line of best fit to pass through \((\bar{x}, \bar{y})\) and \((10, 1)\).

(d) From graph, find the :

(i) equation of the line of best fit ; (ii) height of plant in 75 days.

(a) A bag contains 5 blue, 4 green and 3 yellow balls. All the balls are identical except for colour. Three balls are drawn at random without replacement. Find the probability that : (i) all three balls have the same colour ; (ii) two balls have the same colour.

(b) The table shows the ranks of the marks scored by 7 candidates in Physics and Chemistry tests.

Calculate the Spearman's rank correlation coefficient.

(a) The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins : (i) all races ; (ii) no race ; (iii) at most 3 races ?

(b) A class consists of 5 girls and 10 boys. If a committee of 5 is chosen at random from the class, find the probability that :

(i) 3 boys are selected ; (ii) at least one girl is selected.