Mathematics: Methods, Traps and a Worked Rule
What the examiner tests
UB entrance mathematics rewards clean technique more than cleverness. You must be fluent in algebra (factorisation, quadratic equations, inequalities with sign tables), functions (domain, limits, derivatives, monotonicity, curve sketching), sequences (arithmetic, geometric, recurrence), basic integration, and elementary probability and counting. Geometry and trigonometry appear regularly: Pythagoras, similar triangles, the sine and cosine rules, radian measure, and standard identities.
Traps that cost marks
- Inequalities: multiplying or dividing by a negative number reverses the sign. Solve by moving everything to one side and using a sign table rather than cross-multiplying.
- Domain errors: when simplifying, state restrictions. Dividing by (x−2) is invalid at x=2.
- Degrees versus radians: trigonometric derivatives and limits only work in radians.
- Counting: distinguish arrangements (order matters) from selections (order does not).
- Units and rounding: keep exact values until the final line.
Worked example
Solve (x−1)/(x+3) ≥ 2. Move everything left: (x−1)/(x+3) − 2 ≥ 0, giving (x−1−2x−6)/(x+3) ≥ 0, so (−x−7)/(x+3) ≥ 0, equivalently (x+7)/(x+3) ≤ 0. The critical values are x = −7 and x = −3, where the denominator is undefined. A sign table shows the expression is negative between the roots and positive outside. Because we need ≤ 0, the solution is −7 ≤ x < −3. Note the bracket at −7 and the open end at −3.
Rule of thumb
Whenever a question involves a quotient, a root, or a logarithm, write the domain first. Half the errors in this paper are domain errors in disguise.