Mathematics
Mathematics
The Mathematics section of ACET checks the quantitative machinery behind actuarial work: algebra, calculus, matrices, coordinate geometry and basic trigonometry. Questions are short and formula-driven, so recognising the intended method and avoiding arithmetic slips matter more than long proofs.
Core rules
- Indices and logarithms: a^m × a^n = a^(m+n); log(ab) = log a + log b; log(a/b) = log a − log b; log(a^n) = n log a. State domains: log x needs x > 0.
- Quadratics and inequalities: the discriminant D = b² − 4ac fixes the nature of the roots. For an inequality, find the roots, mark them on a number line, then test one interval.
- Calculus: stationary points come from f′(x) = 0; f″(x) > 0 gives a minimum, f″(x) < 0 a maximum. For limits, factorise or rationalise before substituting.
- Matrices: confirm conformability before multiplying; det(AB) = det A × det B; a square system has a unique solution only when the coefficient matrix is non-singular.
Worked example
Solve 2^(x+1) = 8^(x−1).
Write 8 as 2³, so the right-hand side becomes 2^(3x−3). Equate exponents: x + 1 = 3x − 3, giving 4 = 2x, so x = 2. Check: 2³ = 8 and 8¹ = 8. The mark-winning step is rewriting both sides with the same base before equating powers.
Traps
Substituting into a limit before simplifying and getting 0/0; discarding the negative root of a quadratic; forgetting that a logarithm argument must be positive; sign errors when expanding −(x − 3); and answering with the x-coordinate when the question asks for the maximum or minimum value.