Further Mathematics Past Questions and Answers

(a) The polynomial \(f(x) = x^{3} + px^{2} - 10x + q\) is exactly divisible by \(x^{2} + x - 6\). Find the :
(i) values of p and q ; (ii) third factor.
(b) The volume of a cube is increasing at the rate of \(2\frac{1}{2} cm^{3} s^{-1}\). Find the rate of change of the side of the base when its length is 2cm.

(a) Write down the matrix A of the linear transformation \(A(x, y) \to (2x -y, -5x + 3y)\).
(b) If \(B = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\), find :
(i) \(A^{2} - B^{2}\) ; (ii) matrix \(C = B^{2} A\) ; (iii) the point \(M(x, y)\) whose image under the linear transformation \(C\) is \(M' (10, 18)\).
(c) What is the relationship between matrix A and matrix C?

(a) Evaluate : \(\int_{1} ^{4} \frac{x(3x - 2)}{2\sqrt{x}} \mathrm {d} x\)
(b) The equation of a circle is given by \(2x^{2} + 2y^{2} - 8x + 5y - 10 = 0\). Find the :
(i) coordinates of the centre ; (ii) radius of the circle .

(a)(i) Find the sum of the series \(A(1 + r) + A(1 + r)^{2} + ... + A(1 + r)^{n}\).
(ii) Given that r = 8% and A = GH 40.00, find the sum of the 6th to 10th terms of the series in (i).
(b) Find the equation of the tangent to the curve \(y = \frac{1}{x}\) at the point on the curve when x = 2.

(a) A fair die with six faces is thrown six times. Calculate, correct to three decimal places, the probability of obtaining :
(i) exactly three sixes ; (ii) at most three sixes.
(b) Eight percent of screws produced by a machine are defective. From a random sample of 10 screws produced by the machine, find the probability that :
(i) exactly two will be defective ; (ii) not more than two will be defective.