Simple algebraic division; use of the Factor Theorem and the Remainder Theorem.

Only division by \((x + a)\) or \((x – a)\) will be required.

Students should know that if \(f(x) = 0\) when \(x = a\), then \((x – a)\) is a factor of \(f(x)\).

Students may be required to factorise cubic expressions such as \(x^3 + 3x^2 – 4\) and \(6x^3 + 11x^2 – x – 6\).

Students should be familiar with the terms 'quotient' and 'remainder' and be able to determine the remainder when the polynomial \(f(x)\) is divided by \((ax + b)\).

Coordinate geometry of the circle using the equation of a circle in the form \((x – a)^2 + (y – b)^2 = r^2\) and including use of the following circle properties:

(i) the angle in a semicircle is a right angle;

(ii) the perpendicular from the centre to a chord bisects the chord;

(iii) the perpendicularity of radius and tangent.

Students should be able to find the radius and the coordinates of the centre of the circle given the equation of the circle, and vice versa.

The sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1.

The general term and the sum to n terms are required.

The proof of the sum formula should be known.

Binomial expansion of \((1 + x)^n\) for positive integer n.

The notations n! and \(\displaystyle\binom{n}{r}\).

Expansion of \((a + bx)^n\) may be required.

The sine and cosine rules, and the area of a triangle in the form \(\dfrac{1}{2} ab \sin C\).

Radian measure, including use for arc length and area of sector.

Use of the formulae \(s = r\theta\) and \(A = \dfrac{1}{2} r^2 \theta\) for a circle.

Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.

Knowledge of graphs of curves with equations such as

\(y = 3 \sin x, y = \sin \left( x + \dfrac{\pi}{6} \right), y = \sin 2x\) is expected.

Knowledge and use of \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}\), and \(\sin^2 \theta + \cos^2 \theta = 1\).

Solution of simple trigonometric equations in a given interval.

Students should be able to solve equations such as

\(\sin \left( x + \dfrac{\pi}{2} \right) = \dfrac{3}{4}\) for \(0 < x < 2\pi\),

\(\cos (x + 30^{\circ}) = \dfrac{1}{2}\) for \(-180^{\circ} < x < 180^{\circ}\),

\(\tan 2x = 1\) for \(90^{\circ} < x < 270^{\circ}\),

\(6 \cos^2 x + \sin x - 5 = 0, 0^{\circ} \leq x < 360\),

\(\sin^2 \left( x + \dfrac{\pi}{6} \right) = \dfrac{1}{2}\) for \(-\pi \leq x < \pi\).

\(y = a^x\) and its graph.

Laws of logarithms

To include

\(log_a  xy = log_a  x + log_a  y\) ,

\(log_a  \dfrac{x}{y} = log_a  x − log_a  y\) ,

\(log_a  x^k = k  log_a  x\) ,

\(log_a  \dfrac{1}{x} = − log_a  x\) ,

\(log_a  a = 1\)

The solution of equations of the form \(a^x = b\).

Students may use the change of base formula.

Applications of differentiation to maxima and minima and stationary points, increasing and decreasing functions.

The notation f′′(x) may be used for the second order derivative.

To include applications to curve sketching. Maxima and minima problems may be set in the context of a practical problem.

Evaluation of definite integrals.

Interpretation of the definite integral as the area under a curve.

Students will be expected to be able to evaluate the area of a region bounded by a curve and given straight lines.

Eg find the finite area bounded by the curve \(y  =  6x  -  x^2\) and the line \(y  =  2x\).

\(\displaystyle\int x  dy\) will not be required.

Approximation of area under a curve using the trapezium rule.

For example,

evaluate \(\displaystyle\int_{0}^{1} \surd (2 x + 1) dx\)

using the values of \(\surd (2x + 1)\) at x = 0, 0.25, 0.5, 0.75 and 1.