Further Mathematics Past Questions

Evaluate \(\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x\).

An object is thrown vertically upwards from the top of a cliff with a velocity of \(25ms^{-1}\). Find the time, in seconds, when it is 20 metres above the cliff. \([g = 10ms^{-2}]\).

Given that \(P = \begin{pmatrix} y - 2 & y - 1 \\ y - 4 & y + 2 \end{pmatrix}\) and |P| = -23, find the value of y.

Given that \(\frac{\mathrm d y}{\mathrm d x} = \sqrt{x}\), find y.

Find the range of values of x for which \(x^{2} + 4x + 5\) is less than \(3x^{2} - x + 2\)

The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.

Differentiate \(x^{2} + xy - 5 = 0\).

Find the equation of the line that is perpendicular to \(2y + 5x - 6 = 0\) and bisects the line joining the points P(4, 3) and Q(-6, 1).

Given that \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\) and \(f(3) = 0\), factorize f(x).

If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 6x + 5 = 0\), evaluate \(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\).

If \(\sqrt{x} + \sqrt{x + 1} = \sqrt{2x + 1}\), find the possible values of x.

Find the third term in the expansion of \((a - b)^{6}\) in ascending powers of b.

If \(f(x) = x^{2}\)  and \(g(x) = \sin x\), find g o f.

Express \(\log \frac{1}{8} + \log \frac{1}{2}\) in terms of \(\log 2\).

Given that \(a^{\frac{5}{6}} \times a^{\frac{-1}{n}} = 1\), solve for n.

Solve: \(\sin \theta = \tan \theta\)

A binary operation * is defined on the set of real numbers, R, by \(x * y = x + y - xy\). If the identity element under the operation * is 0, find the inverse of \(x \in R\).

Simplify \(\frac{x^{3n + 1}}{x^{2n + \frac{5}{2}}(x^{2n - 3})^{\frac{1}{2}}}\)

The diagram above is a velocity- time graph of a moving object. Calculate the distance travelled when the acceleration is zero.