A circle is drawn through the points (3, 2), (-1, -2) and (5, -4). Find the :

(a) coordinates of the centre of the circle ;

(b) radius of the circle ;

(c) equation of the circle.

(a) Solve : \(2^{3y + 2} - 7(2^{2y + 2}) - 31(2^{y}) - 8 = 0, y \in R\).

(b) Find \(\int (\sqrt{x^{2} + 1}) xdx\).

(a)(i) Write down the binomial expansion of \((2 - \frac{1}{2}x)^{5}\) in ascending powers of x.

(ii) Using the expansion in (a)(i), find, correct to two decimal places, the value of \((1.99)^{5}\).

(b) The polynomial \(x^{3} + qx^{2} + rx + 9\), where q and r are constants, has (x + 1) as a factor and has a remainder -17 when divided by (x + 2). Find the values of q and r.

Ten coins were tossed together a number of times. The distribution of the number of heads obtained is given in the following table :

Calculate, correct to three decimal places, the :

(a) mean number of heads ;

(b) probability of getting an even head ;

(c) probability of getting an odd number.

The probabilities that Ali, Baba and Katty will gain admission to college are \(\frac{2}{3}, \frac{3}{4}\) and \(\frac{4}{5}\) respectively. Find the probability that:

(a) only Katty and Baba will gain admission ;

(b) none of them will gain admission ;

(c) at most two of them will gain admission.