Mathematics: Algebra, Calculus and Problem Strategy
Core concepts to master
Algebra underpins almost every question: quadratic equations and the discriminant, factorisation, inequalities, indices and logarithms, sequences and series, and the binomial theorem. Coordinate geometry covers straight lines, distance, midpoint and circles. Trigonometry requires identities, the sine and cosine rules, and radian measure. Calculus tests limits, differentiation rules, maxima and minima, and basic integration. Probability and statistics appear as mean, median, mode, standard deviation and simple combinatorics.
Traps examiners set
- Dividing an inequality by a negative number without flipping the sign.
- Dropping the ± when solving x² = k.
- Applying the product rule but forgetting the chain rule inside it.
- Confusing permutations with combinations when order does not matter.
- Using degrees in a calculus derivative of a trigonometric function.
Rule of thumb
For any optimisation or "find the value" problem, express everything in one variable, differentiate, set to zero, then verify with the second derivative. Always check whether the domain restricts the answer.
Worked example
Find the maximum of f(x) = 12x − x².
f′(x) = 12 − 2x = 0 gives x = 6. Since f″(x) = −2 < 0, this is a maximum. f(6) = 72 − 36 = 36.
Practice habit
Substitute your answer back into the original equation; it takes seconds and catches sign and algebra slips.