IGCSE Additional Mathematics tutor: live classes, unit by unit

What the exam asks for

• Distinguishing functions from relations and identifying one-to-one and many-to-one mappings.
• Finding domains and ranges, including restrictions introduced by inverse and composite functions.
• Forming and evaluating composite functions and inverses, and explaining their relationships.

Common slips to watch for

• Treating a relation as a function when an input has more than one output.
• Giving an inverse domain or range without swapping the original function’s domain and range.
• Composing functions in the wrong order or overlooking restrictions on allowable inputs.

What the exam asks for

• Finding quadratic maxima or minima by completing the square or differentiating.
• Using the extremum to sketch a graph or determine a range on a specified domain.
• Connecting the discriminant to the number of real roots and the graph's intersections with the axis.
• Choosing a suitable algebraic or graphical representation to support conclusions about roots and range.

Common slips to watch for

• Misreading the sign of the squared term after completing the square and naming a minimum instead of a maximum, or vice versa.
• Using the vertex value as the range boundary without checking the specified domain's endpoints.
• Confusing a zero discriminant with two distinct real roots.
• Sketching a graph with intercepts or opening direction inconsistent with the quadratic.

What the exam asks for

• Solving equations with absolute values by separating cases or interpreting intersections on a graph.
• Solving absolute-value inequalities and expressing the complete solution set accurately.
• Selecting an algebraic or graphical method suited to the expression and showing how boundaries determine the answer.
• Using accurate graphs where needed to distinguish intervals that satisfy the equation or inequality.

Common slips to watch for

• Removing absolute-value bars without considering both positive and negative cases.
• Including boundary values that do not satisfy a strict inequality, or excluding valid equality boundaries.
• Keeping solutions introduced by squaring or other transformations without checking them in the original equation.
• Reading a graph's intersection points correctly but shading the wrong intervals for an inequality.

What the exam asks for

• Apply index rules accurately when simplifying expressions embedded in questions on other topics.
• Use surd techniques as supporting algebra in multistep solutions.
• Choose and carry out appropriate index or surd manipulation within a wider problem.
• Practise the same supporting skills in both calculator and non-calculator settings.

Common slips to watch for

• Applying an index rule to only part of a term or expression.
• Changing a surd expression incorrectly while simplifying or combining terms.
• Relying on a calculator for a result that requires exact index or surd form.

What the exam asks for

• Use the remainder and factor theorems to connect substitution with remainders and factors.
• Factor cubic polynomials, including finding linear and quadratic factors.
• Solve cubic equations by factoring and identifying the resulting roots.

Common slips to watch for

• Confusing the remainder from substitution with a factor, which requires a zero remainder.
• Making sign errors when substituting a value into a polynomial or when dividing by a linear factor.
• Stopping after finding one factor without completing the factorisation or listing all roots.

What the exam asks for

• Choosing elimination or substitution to solve pairs of linear equations efficiently.
• Using substitution to handle pairs that include a nonlinear relationship.
• Checking candidate solutions in both original equations and interpreting the resulting pair of values.

Common slips to watch for

• Making a sign error while eliminating a variable or rearranging an equation.
• Accepting a value pair that satisfies only one equation.
• Losing a valid solution when substituting into a nonlinear equation or failing to check all resulting candidates.

What the exam asks for

• Using logarithm laws, including change of base, to rewrite and evaluate expressions.
• Relating logarithmic and exponential functions as inverses and reading their graphs and asymptotes.
• Selecting suitable logarithmic or exponential methods to solve equations and checking solutions against the original conditions.

Common slips to watch for

• Applying logarithm laws to sums or differences as though they were products or quotients.
• Confusing the inverse relationship between logarithmic and exponential functions with a reciprocal relationship.
• Accepting a logarithmic solution that makes an argument non-positive.

What the exam asks for

• Form and interpret straight-line equations from relationships between coordinates and gradients.
• Use gradient conditions to establish whether lines are parallel or perpendicular.
• Calculate midpoints and lengths, and determine or apply equations of perpendicular bisectors.

Common slips to watch for

• Interchanging coordinate differences when calculating a gradient or length.
• Using equal gradients for perpendicular lines instead of the negative reciprocal relationship where defined.
• Finding a midpoint correctly but using it as the segment length or failing to substitute it into the required line equation.

What the exam asks for

• Use radian measure to calculate arc lengths and sector areas.
• Select the relevant formula and identify the angle and radius in a diagram.
• Combine arc and sector calculations when solving compound-shape problems.

Common slips to watch for

• Using a degree-based formula or treating a radian angle as degrees.
• Confusing arc length with sector area, or omitting a squared radius in an area calculation.
• Using only part of a compound shape or misidentifying which boundary is an arc.

What the exam asks for

• Apply the six trigonometric functions and fundamental identities, including for angles beyond the acute range.
• Solve trigonometric equations by finding all solutions in the stated domain.
• Interpret and sketch sine, cosine and tangent graphs using amplitude, period factor and vertical shift.
• Relate graph features to the function and use the specified domain when reading or drawing graphs.

Common slips to watch for

• Finding only one solution when the domain contains others, or including solutions outside the domain.
• Applying an identity with a missing sign or confusing reciprocal and quotient functions.
• Treating the period factor as the period itself, or applying a vertical shift to the amplitude.
• Drawing tangent branches or graph features incorrectly across the stated domain.

What the exam asks for

• Deciding whether a counting situation requires ordered arrangements or unordered selections.
• Applying permutations or combinations to counts involving a choice of a specified number from a larger set.
• Using factorial notation correctly, including the value of zero factorial, in numerical and algebraic counting.

Common slips to watch for

• Using permutations when order does not matter, or combinations when it does.
• Using the wrong number of selected items in the counting formula.
• mishandling factorial cancellation or treating zero factorial as zero.

What the exam asks for

• Expand positive-integer powers with the binomial theorem and simplify coefficients accurately.
• Use the general term to identify a particular term or the term independent of a variable.
• Recognise arithmetic and geometric progressions and choose the appropriate nth-term or sum formula.
• Use the convergence condition and sum-to-infinity formula for geometric progressions, stating why the condition holds.

Common slips to watch for

• Using the wrong binomial term or exponent when identifying a specified term.
• Missing powers of the variable when deciding which term is independent of it.
• Confusing the common difference with the common ratio, or substituting the wrong index into a progression formula.
• Applying the sum-to-infinity formula without checking convergence.

What the exam asks for

• Represent points and directions with position vectors and unit vectors in the notation required by the problem.
• Calculate vector magnitudes and use vector addition, subtraction and scalar multiplication.
• Translate between a geometric description and an algebraic vector expression.
• Keep track of direction when forming displacement vectors between points.

Common slips to watch for

• Reversing the order of endpoints when calculating a displacement vector.
• Treating a magnitude as a vector, or omitting the magnitude’s square root.
• Applying scalar multiplication to only one component of a vector.
• Using inconsistent vector notation for points, directions and position vectors.

What the exam asks for

• Differentiate power, trigonometric, exponential and logarithmic functions, including composites, products and quotients.
• Use derivatives to find gradients, tangent and normal lines, and stationary points; interpret the result in context.
• Integrate the specified forms and include an arbitrary constant in indefinite integrals.
• Evaluate definite integrals to find areas between lines or curves, including regions made from more than one area.
• Connect displacement, velocity and acceleration using differentiation and integration, and interpret motion graphs.

Common slips to watch for

• Applying a differentiation rule incorrectly to a composite, product or quotient, or confusing the derivative notation.
• Giving a normal gradient equal to the tangent gradient, or failing to use the point on the curve in a line equation.
• Omitting the arbitrary constant, mishandling a specified reciprocal-power case, or using incorrect limits in a definite integral.
• Ignoring sign when interpreting a definite integral as area, or confusing distance with displacement and speed with velocity.

What the exam asks for

• Read the centre and radius from circle equations in different forms, completing the square when needed.
• Find intersections of a line and circle, then use the results to classify the relationship.
• Form tangent equations using the circle’s centre, radius and the point of contact.
• Combine coordinate methods with circle properties to solve problems involving tangents and intersecting circles.

Common slips to watch for

• Reading the radius directly from a squared-radius term without taking its square root.
• Making a sign error when converting a circle equation to centre-radius form.
• Classifying a line-circle relationship without checking the intersection results or discriminant.
• Using a tangent direction that is not perpendicular to the radius at the contact point.