Mathematics: algebra, analysis, probability and statistics
Mathematics tests fluent manipulation and interpretation rather than memorized formulas. Expect function domains, logarithmic and exponential equations, limits and derivatives, definite integrals, probability trees and descriptive statistics.
Core rules
- Algebra and functions: state the domain before simplifying; log(ab) = log a + log b; exponential and logarithm are inverse functions. Solve systems by substitution or elimination; matrices help with linear systems.
- Analysis: a limit describes approach, not necessarily value. Derivative rules include product, quotient and chain. Extrema occur where f′(x) = 0 or is undefined, checked against endpoints. The definite integral gives signed area; include +C for indefinite integrals.
- Probability: P(A|B) = P(A∩B)/P(B). For independent events, P(A∩B) = P(A)P(B). Total probability sums over exhaustive cases; Bayes' theorem updates a prior.
- Statistics: mean, median and mode differ in sensitivity to outliers; sample variance uses n − 1; standard error is s/√n.
Worked example
A seed-lot contamination test has prevalence 2%, sensitivity 95% and specificity 90%. If a lot tests positive, what is the probability it is truly contaminated? P(+/contaminated) = 0.95; P(contaminated) = 0.02; P(+/clean) = 0.10; P(clean) = 0.98. P(+) = 0.95×0.02 + 0.10×0.98 = 0.019 + 0.098 = 0.117. P(contaminated|+) = 0.019/0.117 ≈ 0.162, about 16%. Reasoning: the many clean lots generate more false positives than the rare contaminated lots generate true positives.
Traps that cost marks
- Cancelling expressions before restricting the domain.
- Omitting +C, or reversing the limits of integration.
- Confusing P(A|B) with P(B|A).
- Using n instead of n − 1 for sample variance.
- Reporting the mean when the data are skewed and the median is requested.