Mathematics
Mathematics tests algebraic fluency and calculus, plus elementary probability and combinatorics. Questions are short and computational, so a single sign or domain error loses the item.
Core rules
- Domain restrictions: denominator ≠ 0, logarithm argument > 0, square root argument ≥ 0.
- Derivatives: (xⁿ)' = nxⁿ⁻¹, (ln x)' = 1/x, (eˣ)' = eˣ, product and quotient rules, and the chain rule for composites.
- Limits: factor and cancel, or use L'Hôpital only for 0/0 and ∞/∞ forms.
- Integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C; a definite integral equals F(b) − F(a).
- Logarithms: log(ab) = log a + log b; ln(eˣ) = x; aˣ = e^(x ln a).
- Probability: P(Aᶜ) = 1 − P(A); P(A∩B) = P(A)P(B|A); combinations nCr = n!/(r!(n−r)!).
Worked example
Differentiate f(x) = x² ln x for x > 0. Apply the product rule with u = x², v = ln x: u' = 2x, v' = 1/x. Then f'(x) = 2x·ln x + x²·(1/x) = 2x ln x + x = x(2 ln x + 1). Note the domain restriction: ln x is undefined at x ≤ 0, so the derivative exists only for x > 0.
Traps
- Omitting the + C in indefinite integrals while including it in definite ones.
- Writing (ln x)' = 1/x for negative x.
- Confusing permutations (order matters) with combinations (order does not).
- Working in degree mode on a calculator for calculus and series questions.
- Multiplying inequalities by a negative without flipping the sign.