Mathematics Proficiency: Algebra, Counting, and Functions
What Mathematics Proficiency tests
This section moves beyond arithmetic into school-level algebra, functions, sequences, coordinate geometry, trigonometry, permutations and combinations, probability, and inequalities. It rewards recognising structure - an identity, a symmetry, a standard count - instead of brute-force calculation.
Core rules
- Identity: a³ + b³ = (a + b)³ − 3ab(a + b). Use it whenever a sum and a product are given.
- For ax² + bx + c = 0, roots sum to −b/a and multiply to c/a; the discriminant b² − 4ac decides the nature of the roots.
- Counting: arrangements of n distinct objects = n!; selections = C(n, r). For 'together' problems, bind the group and multiply by its internal arrangements.
- Probability = favourable / total only when outcomes are equally likely; otherwise use the complement or split into cases.
- Check domain restrictions: denominators non-zero, logarithm arguments positive, square-root results non-negative.
Worked example
If x + 1/x = 3, find x³ + 1/x³.
Set a = x and b = 1/x. Then ab = 1 and a + b = 3. Applying a³ + b³ = (a + b)³ − 3ab(a + b) gives 27 − 3(1)(3) = 27 − 9 = 18.
Traps
Expanding everything and running out of time. Forgetting that squaring can introduce extraneous roots. Double counting in permutations when objects repeat. Handling 'at least one' without using the complement. Ignoring that x + 1/x = 3 gives two real values of x, so the target expression must be symmetric in x and 1/x. Confirm whether the question asks for a count, a probability, or a value.