Algebra
Algebra in the BIEAP Intermediate Public Examination (IPE) Mathematics papers covers functions, mathematical induction, matrices and rank, complex numbers, quadratic expressions, theory of equations, permutations and combinations, the binomial theorem and partial fractions. The exam rewards clean, justified steps: a 7-mark question wants the method shown, not a bare answer.
Core rules and concepts
- A function is fixed by domain, codomain and rule; the range is a subset of the codomain.
- Mathematical induction: prove P(1), assume P(k), then prove P(k+1) using that assumption.
- Rank of a square matrix is the order of the largest non-vanishing minor. The system AX = B is consistent if and only if rank A = rank [A|B]; the solution is unique when this common rank equals the number of unknowns.
- de Moivre: (cos θ + i sin θ)^n = cos nθ + i sin nθ. The cube roots of unity satisfy 1 + ω + ω² = 0 and ω³ = 1.
- For ax² + bx + c, the discriminant and the sign of a decide where the expression is positive or negative and its extreme value.
Worked example
Solve 2x³ − 9x² + 13x − 6 = 0, given that x = 1 is a root. Synthetic division by (x − 1) on coefficients 2, −9, 13, −6 gives 2, −7, 6 with remainder 0, so the cubic equals (x − 1)(2x² − 7x + 6). Factorising the quadratic: 2x² − 7x + 6 = (2x − 3)(x − 2). Roots: x = 1, 3/2, 2.
Traps that cost marks
- Induction proofs that never invoke P(k).
- Declaring a system inconsistent after one zero row without comparing ranks.
- Writing ω² = ω instead of ω² = −1 − ω.
- Mixing nPr with nCr in binomial coefficient questions.
- Forgetting to test whether a root is repeated before listing solutions.