📚 Subject Content
The subject content is organised by topic. It is not presented in a teaching order. Candidates are expected to use techniques listed in the content and apply them to solve problems with or without a calculator, as appropriate.
1. Functions
Candidates should be able to:
- Understand the terms: function, domain, range (image set), one-one function, many-one function, inverse function and composition of functions.
- Find the domain and range of functions (including inverse functions and composite functions).
- Recognise and use function notation.
2. Quadratic functions
Candidates should be able to:
- Find the maximum or minimum value of the quadratic function \(f(x) = ax^2 + bx + c\) by any method.
- Use the maximum or minimum value of \(f(x)\) to sketch the graph or determine the range for a given domain.
- Know the conditions for \(f(x) = 0\) to have: (i) two real roots, (ii) two equal roots, (iii) no real roots and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve.
- Solve quadratic equations for real roots and find the solution set for quadratic inequalities.
3. Factors of polynomials
Candidates should be able to:
- Know and use the remainder and factor theorems.
- Find factors of polynomials.
- Solve cubic equations.
4. Equations, inequalities and graphs
Candidates should be able to:
- Solve graphically or algebraically equations of the type \(|ax+b|=c\) (\(c \ge 0\)) and \(|ax+b| = |cx+d|\).
- Solve graphically or algebraically inequalities of the type \(|ax+b| > c\) (\(c \ge 0\)), \(|ax+b| \le c\) (\(c > 0\)) and \(|ax+b| \le |cx+d|\).
- Use substitution to form and solve a quadratic equation in order to solve a related equation.
- Sketch the graphs of cubic polynomials and their moduli, when given in factorised form \(y=k(x-a)(x-b)(x-c)\).
- Solve cubic inequalities in the form \(k(x-a)(x-b)(x-c) \le d\) graphically.
5. Simultaneous equations
Candidates should be able to:
- Solve simple simultaneous equations in two unknowns by elimination or substitution.
6. Logarithmic and exponential functions
Candidates should be able to:
- Know simple properties and graphs of the logarithmic and exponential functions including \(\ln x\) and \(e^x\) (series expansions are not required) and graphs of \(ke^{nx}+a\) and \(k\ln(ax+b)\) where \(n, k, a\) and \(b\) are integers.
- Know and use the laws of logarithms (including change of base of logarithms).
- Solve equations of the form \(a^x = b\).
7. Straight-line graphs
Candidates should be able to:
- Interpret the equation of a straight line graph in the form \(y=mx+c\).
- Transform given relationships, including \(y = ax^n\) and \(y = Ab^x\), to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph.
- Solve questions involving mid-point and length of a line.
- Know and use the condition for two lines to be parallel or perpendicular, including finding the equation of perpendicular bisectors.
8. Coordinate geometry of the circle
Candidates should be able to:
- Understand and use the equation of a circle in the form \((x-a)^2 + (y-b)^2 = r^2\).
- Find the centre and radius of a circle from its equation.
- Determine whether a point lies inside, on, or outside a circle.
- Find the intersection of a line and a circle using the discriminant.
- Find the equation of a tangent to a circle at a given point.
9. Circular measure
Candidates should be able to:
- Solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure.
10. Trigonometry
Candidates should be able to:
- Know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent).
- Understand amplitude and periodicity and the relationship between graphs of related trigonometric functions.
- Draw and use the graphs of \(y = a\sin(bx)+c\), \(y = a\cos(bx)+c\), \(y = a\tan(bx)+c\) where \(a\) is a positive integer, \(b\) is a simple fraction or integer, and \(c\) is an integer.
- Use the relationships \(\sin^2 A + \cos^2 A = 1\), \(\sec^2 A = 1 + \tan^2 A\), \(\cosec^2 A = 1 + \cot^2 A\), \(\frac{\sin A}{\cos A} = \tan A\), \(\frac{\cos A}{\sin A} = \cot A\).
- Solve simple trigonometric equations involving the six trigonometric functions and the above relationships.
- Prove simple trigonometric identities.
11. Permutations and combinations
Candidates should be able to:
- Recognise and distinguish between a permutation case and a combination case.
- Know and use the notation \(n!\) (with \(0! = 1\)), and the expressions for permutations and combinations of \(n\) items taken \(r\) at a time.
- Answer simple problems on arrangement and selection.
12. Series
Candidates should be able to:
- Use the Binomial Theorem for expansion of \((a+b)^n\) for positive integer \(n\).
- Use the general term \(\binom{n}{r} a^{n-r} b^r\), \(0 \le r \le n\).
- Recognise arithmetic and geometric progressions.
- Use the formulae for the \(n\)th term and for the sum of the first \(n\) terms to solve problems involving arithmetic or geometric progressions.
- Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.
13. Vectors in two dimensions
Candidates should be able to:
- Use vectors in any form, e.g. \(\vec{a}\), \(\overrightarrow{AB}\), \(\mathbf{p}\), \(a\mathbf{i} - b\mathbf{j}\).
- Know and use position vectors and unit vectors.
- Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars.
- Compose and resolve velocities.
14. Calculus
Candidates should be able to:
- Understand the idea of a derived function.
- Use the notations \(f'(x)\), \(f''(x)\), \(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\).
- Use the derivatives of the standard functions \(x^n\) (for any rational \(n\)), \(\sin x\), \(\cos x\), \(\tan x\), \(e^x\), \(\ln x\), together with constant multiples, sums and composite functions of these.
- Differentiate products and quotients of functions.
- Apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems.
- Use the first and second derivative tests to discriminate between maxima and minima.
- Understand integration as the reverse process of differentiation.
- Integrate sums of terms in powers of \(x\) including \(\frac{1}{x}\) and \(\frac{1}{ax+b}\).
- Integrate functions of the form \((ax+b)^n\) for any rational \(n\), \(\sin(ax+b)\), \(\cos(ax+b)\), \(e^{ax+b}\).
- Evaluate definite integrals and apply integration to the evaluation of plane areas.
- Apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of \(x\)-\(t\) and \(v\)-\(t\) graphs.
📋 Formula List Provided in the Exam
A list of formulas is provided on page 2 of the examination papers for candidates to refer to. However, note that not all required formulas are given; candidates must memorise many key formulas.
Formulas that are provided in the exam:
- Curved surface area of a cone
- Surface area of a sphere
- Volume of a pyramid or a cone
- Volume of a sphere
- Sine rule
- Cosine rule
- Area of a non-right-angled triangle