Mathematics (Mathematics I, II, III, A, B)
What the exam tests
Mathematics at this entrance level rewards precise manipulation and clear justification across Mathematics I, II, III, A, and B: calculus, sequences, vectors, complex numbers, probability, and integers. Questions often combine topics, so a derivative may appear inside a probability setting or a vector proof.
Core rules
- Differentiate and integrate polynomials, exponentials, logs, and trigonometric functions; track domains and absolute values.
- For definite integrals, area is the integral of the absolute difference, not automatically the integral of f minus g.
- Vectors: use components; perpendicular means dot product 0; parallel means one is a scalar multiple of the other.
- Probability: with replacement vs without replacement changes the denominator; conditional probability uses P(A and B)/P(B).
- Complex numbers: represent z=a+bi, use modulus and argument; rotation comes from multiplying by cos theta + i sin theta.
Worked example
Find the area between y=x^2 and y=2x. Intersections: x^2=2x gives x=0 and x=2. On (0,2), 2x is above x^2. Area = integral from 0 to 2 of (2x-x^2) dx = [x^2-x^3/3] from 0 to 2 = 4-8/3 = 4/3. The reasoning is to subtract the lower curve from the upper curve and check which is above on the interval.
Traps
Forgetting +C in indefinite integrals; dropping absolute values in integrals of 1/x; confusing permutations with combinations; losing a vector sign; not checking endpoints in inequalities; and stating a result without showing the key identity.