Mathematics: Algebra, Calculus, Probability and Statistics
Scope
Mathematics questions test manipulation and interpretation: algebra, functions, logarithms and exponentials, limits and derivatives, elementary integration, probability, and descriptive statistics. Speed comes from recognising standard forms.
Core rules
- Algebra: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac))/2a; the sign of the discriminant gives the number of real roots.
- Logarithms: log(ab) = log a + log b; log(aⁿ) = n log a; ln and log₁₀ are different bases, so check which one a question uses.
- Calculus: the derivative of xⁿ is nxⁿ⁻¹; f′(x) = 0 locates stationary points; the sign of f″ classifies them.
- Probability: P(A∪B) = P(A) + P(B) − P(A∩B); multiply for independent events; "at least one" is usually 1 − P(none).
- Statistics: the mean is sensitive to outliers, the median is not; in a normal distribution about 68% of values lie within one standard deviation of the mean.
Worked example with reasoning
For f(x) = x³ − 3x, find and classify the stationary points. f′(x) = 3x² − 3 = 0 gives x = ±1. Since f″(x) = 6x, f″(1) = 6 > 0, a local minimum with f(1) = −2, and f″(−1) = −6 < 0, a local maximum with f(−1) = 2. The second-derivative test is fast but inconclusive when f″ = 0; in that case examine the sign change of f′.
Traps that cost marks
- Dropping the ± when solving x² = k.
- Applying log rules to sums: log(a + b) ≠ log a + log b.
- Forgetting restricted values when simplifying rational expressions.
- Treating dependent events as independent.
- Rounding intermediate values and drifting away from the answer options.
Transfer tip
Rewrite the question in symbols before touching the calculator and keep exact fractions until the final step.