Further Mathematics Past Questions and Answers

An object is projected vertically upwards. Its height, h m, at time t seconds is given by \(h = 20t - \frac{3}{2}t^{2} - \frac{2}{3}t^{3}\). Find
(a) the time at which it is momentarily at rest (b) correct to two decimal places, the maximum height reached by the object.

(a) The roots of the equation \(x^{2} + mx + 11 = 0\) are \(\alpha\) and \(\beta\), where m is a constant. If \(\alpha^{2} + \beta^{2} = 27\), find the values of m.
(b) The line \(2x + 3y = 1\) intersects the circle \(2x^{2} + 2y^{2} + 4x + 9y - 9 = 0\) at points P and Q where Q lies in the fourth quadrant. Find the coordinates of P and Q.

(a) Solve the equation : \(\sqrt{4x - 3} - \sqrt{2x - 5} = 2\).
(b) Find the finite area enclosed by the curve \(y^{2} = 4x\) and the line \(y + x = 0\).

(a) Given that \(\begin{vmatrix} 5 & 2 & -3 \\ -1 & k & 6 \\ 3 & 9 & (k + 2) \end{vmatrix} = -207\), find the values of the constant k.
(b) The equation of a curve is \(x(y^{2} + 1) - y(x^{2} + 1) + 4 = 0\). Find the:
(i) gradient of the curve at any point (x, y).
(ii) equation of the tangent to the curve at the point (-1, -3).

(a) Using the trapezium rule with seven ordinates, evaluate \(\int_{0}^{3} \frac{\mathrm d x}{x^{2} + 1}\), correct to two decimal places.
(b) Using matrix method, solve \(-2x + y = 3; - x + 4y = 1\).