BCECE Mathematics: Domains, Limits and Valid Inverses
What BCECE Mathematics tests
The Mathematics paper tests algebra, trigonometry, coordinate geometry, calculus, vectors, three-dimensional geometry, probability, matrices and determinants, and sequences and series. It is speed-sensitive, so recognising standard forms matters more than deriving everything from scratch.
Core rules
- Before dividing by an expression, check whether it can be zero; otherwise you lose roots.
- For logarithms, state the domain: the argument must be positive, and the base must be positive and not 1.
- In definite integrals, change the limits when you substitute; do not mix old limits with a new variable.
- For a tangent to a curve, slope = dy/dx at the point; the normal slope is the negative reciprocal.
- In probability, decide whether order matters before choosing nPr or nCr.
- For an inverse trigonometric function, restrict the principal range before simplifying.
Worked example
Evaluate ∫₀¹ x e^x dx. Integrate by parts with u = x and dv = e^x dx: ∫ x e^x dx = x e^x − ∫ e^x dx = e^x(x − 1). Apply limits: [e^x(x − 1)]₀¹ = e(0) − (1)(−1) = 1. So the value is 1.
Traps
- Omitting the constant of integration in indefinite integrals.
- Forgetting to check extraneous solutions after squaring both sides.
- Using degrees when a calculus limit or trigonometric derivative assumes radians.
- Sign errors in the section formula and in vector dot and cross products.
- Confusing permutation and combination in arrangement questions.
- Applying a formula outside its domain, such as a logarithm of a negative number.