Pure 1
1. Algebra and functions
2. Coordinate geometry
3. Sequences and series
4. Exponentials and logarithms
5. Trigonometry
6. Differentiation
7. Integration
1. Algebra and functions
Laws of indices for all rational exponents.
Use and manipulation of surds.
Quadratic functions and their graphs.
The discriminant of a quadratic function.
Completing the square. Solution of quadratic equations.
Solution of quadratic equations by factorisation, use of the formula and completing the square.
Simultaneous equations: analytical solution by substitution.
For example, where one equation is linear and one equation is quadratic.
Solution of linear and quadratic inequalities.
For example, ax + b ≥ cx + d, px² + qx + r ≥ 0, px² + qx + r < ax + b.
Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation.
Factorisation of polynomials of degree n, n ≤ 3, e.g. x³ 1 4x² + 3x. The notation f(x) may be used.
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 x³ + 3x² – 4 and 6x³ + 11x² – 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).
Graphs of functions; sketching curves defined by simple equations.
Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations.
Functions to include simple cubic functions and the reciprocal function y = k/x with x ≠ 0.
Knowledge of the term asymptote is expected.
Also y = ax and its graph and trigonometric graphs – see sections 4 and 5.
Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).
Students should be able to apply one of these transformations to any of the above functions (quadratics, cubics, reciprocal, and sketch the resulting graph.
Given the graph of any function y = f(x) students should be able to sketch the graph resulting from one of these transformations.
2. Coordinate geometry in the (x, y) plane
Equation of a straight line, including the forms y – y1 + m(x 2 x1) and ax + by + c = 0.
To include:
(i) the equation of a line through two given points
(ii) the equation of a line parallel (or perpendicular) to agiven line through a given point. For example, the line perpendicular to the line 3x + 4y = 18 through the point(2, 3) has equation y – 3 = 4/3 (x – 2).
Conditions for two straight lines to be parallel or perpendicular to each other.
Coordinate geometry of the circle using the equation of a circle in the form(x 2 a)2 1 ( y 2 b)2 5 r2 and including use of the following circle properties:
(i) the angle in a semicircle is a rightangle;
(ii) the perpendicular from the centre toa chord bisects the chord;
(iii) the perpendicularity of radius and tangent.
Students should be able to find the radius and the coordinatesof the centre of the circle given the equation of the circle, and vice versa.
3. Sequences and series
Sequences, including those given bya formula for the nth term and those generated by a simple relation of the form xn+1 = f(xn).
Arithmetic series, including the formula for the sum of the first n natural numbers.
The general term and the sum to n terms of the series are required. The proof of the sum formula should be known.
Understanding of Σ notation will be expected.
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 (n r ).
Expansion of (a + bx)n may be required.
4. Exponentials and logarithms
y = ax and its graph.
Laws of logarithms.
To include
loga xy ≡ loga x + loga y,
loga x/y ≡ loga x – loga y,
loga xk ≡ k loga x,
loga 1/x ≡ –loga x,
loga a = 1
The solution of equations of the form ax = b
Students may use the change of base formula.
5. Trigonometry
The sine and cosine rules, and the area of a triangle in the form 1/2 ab sin C.
Radian measure, including use for arc length and area of sector.
Use of the formulae s = rθ and A = 1/2 r2θ 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 (x + π/6), y = sin 2x is expected.
Knowledge and use of tan θ = sin θ/cos θ,and sin2 θ + cos2 θ = 1.
Solution of simple trigonometric equations in a given interval.
Students should be able to solve equations such as
sin ( x + π/2 ) = 3/4 for 0 ‹ x ‹ 2p,
cos (x + 30°) = 1/2 for – 180° ‹ x ‹ 180°,
tan 2x = 1 for 90° ‹ x ‹ 270°,
6 cos2 x + sin x – 5 = 0 for 0° ‹ x ‹ 360°,
sin2 ( x + π/6 ) = 1/2 for –π ‹ x ‹ π.
6. Differentiation
The derivative of f(x) as the gradient of the tangent to the graph of y 5 f(x) at a point; the gradient of the tangent as a imit; interpretation as a rate of change; second order derivatives.
For example, knowledge that dy/dx is the rate of change of y with respect to x. Knowledge of the chain rule is not required.
The notation f '(x) and f"(x) may be used.
Differentiation of xn, and related sums and differences.
The ability to differentiate expressions such as (2x + 5)(x – 1) and (x2 + 5x – 3)/3x1/2 is expected.
Applications of differentiation to gradients, tangents and normals.
Use of differentiation to find equations of tangents and normals at specific points on a curve.
Applications of differentiation to maxima and minima and stationary points, increasing and decreasing functions.
To include applications to curve sketching. Maxima and minima problems may be set in the context of a practical problem.
7. Integration
Indefinite integration as the reverse of differentiation.
Students should know that a constant of integration is required.
Integration of xn.
For example, the ability to integrate expressions such as (1/2) x2 – 3x – 1/2 and (x + 2)2 /x1/2 is expected.
Given f'(x) and a point on the curve, students should be able to find an equation of the curve in the form y = f(x).
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.
E.g. find the finite area bounded by the curve y = 6x – x2 and the line y = 2x.
∫x dy will not be required.
Approximation of area under a curve using the trapezium rule.
For example, evaluate ∫ 0 1 √(2x + 1) dx using the values of √((2x + 1) at x = 0, 0.25, 0.5, 0.75 and 1.