Mathematics: Course 1 and Course 2
What the domain tests
Mathematics is a 0–200, 80-minute paper with two courses. Course 1 assumes Mathematics I and A and suits humanities and social science applicants. Course 2 adds Mathematics II, III, and B and is required for many science and engineering programs. Both courses test fluent manipulation, not just answer recognition: you must choose the expression that follows from the conditions. Answers are machine-scored, so a small algebra slip can lead you to a listed distractor.
Core rules
- Write the domain before simplifying: denominators ≠ 0, arguments of logs > 0, and angles in the specified unit.
- In quadratics, use the discriminant to classify roots and check endpoints for inequalities.
- In calculus, differentiate to find critical points, then use the second derivative or a sign chart to classify them.
- In probability, define the sample space and count without double-counting; use complementary events when “at least one” appears.
Worked example
For f(x) = x^3 − 3x, find the local extrema. f′(x) = 3x^2 − 3 = 3(x − 1)(x + 1), so critical points are x = −1 and x = 1. f″(x) = 6x. At x = −1, f″ < 0, so f(−1) = 2 is a local maximum. At x = 1, f″ > 0, so f(1) = −2 is a local minimum. Reasoning: the derivative locates candidates; the second derivative classifies them.
Traps
- Dropping the negative sign when substituting −1 into an even or odd power.
- Forgetting that dividing by a variable can lose the solution x = 0.
- Using degrees in trigonometric calculus where radians are required.
- Choosing an answer before checking whether the question asks for a value, a range, or a count.