GCE Further Mathematics 2019
Past Questions

If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {2, 4, 6, 8}, find A' (the complement of A).

A binary operation * is defined on the set of real numbers by $a * b = a + b - 2ab$. Find the value of $3 * 2$.

A binary operation Δ is defined on the set $S = \{0, 1, 2, 3, 4\}$ by $a \Delta b = (a + b) \mod 5$. Find the inverse of 3 under this operation, given that the identity element is 0.

If log₂ 8 + log₂ 4 = log₂ x, find x.

Rationalize the denominator of $\frac{5}{3 - \sqrt{2}}$ and simplify.

The polynomial $g(x) = x^3 + px^2 - 7x + q$ has $(x - 2)$ as a factor and leaves a remainder of 24 when divided by $(x + 3)$. Find $p + q$.

The sum of the first $n$ terms of a series is $S_n = 3n^2 + n$. Find the $n$th term.

Find the value of x if $\begin{vmatrix} x & 2 \\ 5 & x+1 \end{vmatrix} = 6$.

Find the $2 \times 2$ matrix X such that $X\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

A vector has magnitude 10 and makes an angle of 60° with the positive x-axis. Find its y-component.

Forces F₁ = (3i + 2j) N and F₂ = (i - 4j) N act on a particle. Find the magnitude of the resultant force.

The angle between vectors a = 2i + j and b = i + 2j is

A chord of a circle of radius 10 cm subtends an angle of $\frac{2\pi}{3}$ radians at the centre. Find the length of the chord.

The gradient of the curve $y = 2x^3 - 9x^2 + 12x - 4$ at the point where $x = 1$ is

The displacement of a particle at time $t$ seconds is given by $s = t^3 - 6t^2 + 9t + 4$ metres. The particle is momentarily at rest when $t$ equals

Using the substitution $u = x^2 + 1$, evaluate $\int_{0}^{2} \frac{x}{x^2+1}\, dx$

A student is to answer 8 out of 12 questions in an examination. In how many ways can the student make the selection if questions 1 and 2 must both be answered?

(a) A binary operation * is defined on the set of real numbers R by: x * y = x + y + 3xy, for all x, y ∈ R. (i) Show that the operation * is commutative. (ii) Show that the operation * is associative…

1. (a) Given that f(x) = 2x³ + 3x² - 11x - 6, show that (x - 2) is a factor of f(x). [2 marks] (b) Hence, factorize f(x) completely. [4 marks] (c) Express (7x² - 11x - 6) / [(2x + 1)(x - 2)(x + 3)] …

2. (a) The polynomial p(x) = x⁴ - 3x³ + ax² + bx - 10 has (x - 1) and (x + 2) as factors. Find the values of the constants a and b. [6 marks] (b) Using the values of a and b found in (a), factorize p…