Mathematics (Algebra, Trigonometry, Calculus, Statistics)
What this domain tests
Mathematics covers algebra, trigonometry, differential and integral calculus, and statistics. Questions test manipulation, identities, limits, derivatives, integration and interpretation of data.
Core rules
- Quadratic: roots are x = [−b ± √(b² − 4ac)]/2a; the discriminant b² − 4ac decides the nature of roots.
- AP: aₙ = a + (n − 1)d; Sₙ = n/2[2a + (n − 1)d]. GP: aₙ = ar^(n−1).
- Trigonometry: sin²θ + cos²θ = 1; sin 2θ = 2 sin θ cos θ; cos 2θ = 1 − 2 sin²θ.
- Calculus: d/dx(xⁿ) = nx^(n−1); product rule (uv)′ = u′v + uv′; ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1.
- Statistics: mean = Σx/n; median is the middle value after ordering; mode is the most frequent value.
Worked example
Find and classify the stationary points of f(x) = x³ − 3x. f′(x) = 3x² − 3 = 0 gives x = ±1. f″(x) = 6x. At x = 1, f″ = 6 > 0, so it is a local minimum. At x = −1, f″ = −6 < 0, so it is a local maximum. The values are f(1) = −2 and f(−1) = 2.
Traps
Do not ignore domain restrictions, especially in logarithms and inverse trigonometric functions. Distinguish permutations (order matters) from combinations (order does not). In statistics, order the data before finding the median, and remember that adding a constant shifts the mean but not the standard deviation.
Exam focus
Memorise standard derivatives, integrals and identities, and show the substitution step in integration by parts.