Mathematics - Analysis, Algebra and Probability
Core concepts
The GS mathematics paper rewards fluency across four blocks: real analysis (limits, continuity, differentiability, the derivative as a rate and as a tangent slope, monotonicity and the variation table), integration (antiderivatives, area between curves, integration by parts, computing a mean value), complex numbers (algebraic and trigonometric form, modulus-argument, roots of unity, geometric interpretation) and probability/statistics (conditional probability, independence, binomial and normal approximations). Sequences and recurrences also appear, often coupled with induction.
Traps
- Confusing continuity with differentiability: continuity is necessary, not sufficient.
- Forgetting the domain restriction when simplifying (dividing by a term that may vanish).
- Sign errors in the variation table: the sign of f′ governs monotonicity, not the sign of f.
- In probability, mixing P(A|B) with P(B|A); always write the conditioning event on the right of the bar.
- Losing the constant of integration, or forgetting to change bounds when substituting in a definite integral.
Worked rule of thumb
For f(x) = x·e^(−x) on [0, +∞): f′(x) = e^(−x)(1 − x). So f′ > 0 on (0,1) and f′ < 0 on (1,∞): f rises then falls, maximum at x = 1 with value 1/e. For an integral like ∫ x·e^(−x) dx, integrate by parts with u = x, dv = e^(−x)dx, giving −x·e^(−x) − e^(−x) + C. Always check by differentiating the answer - the fastest self-correction available in the exam.