MECEE-BL Mathematics: exactness in algebra, calculus and probability
What the mathematics paper tests
Algebra, trigonometry, coordinate geometry, calculus, vectors, matrices, sequences and series, permutations and combinations, probability and statistics. MECEE-BL items are usually short but exact; a single domain restriction or missed solution can decide the answer.
Core rules
- Quadratic ax² + bx + c = 0: discriminant D = b² − 4ac; D > 0 two real roots, D = 0 one repeated root, D < 0 complex roots.
- Logarithms require positive arguments; squaring can introduce extraneous roots, so check.
- Derivatives: d/dx(x^n) = nx^(n−1); stationary points where f′(x) = 0; f″ decides maximum or minimum.
- Trigonometry: use radians in calculus; sin²θ + cos²θ = 1; check the quadrant for signs.
- Probability: P(A∪B) = P(A) + P(B) − P(A∩B); combinations count selections, permutations count arrangements.
- Vectors: dot product gives cosθ; cross product gives a perpendicular vector and area.
Worked example
For f(x) = x³ − 3x, f′(x) = 3x² − 3 = 3(x − 1)(x + 1). Critical points are x = 1 and x = −1. f″(x) = 6x, so f″(1) = 6 > 0: local minimum at x = 1, f(1) = −2. f″(−1) = −6 < 0: local maximum at x = −1, f(−1) = 2. A sketch confirms the cubic rises, turns down, then rises.
Traps
Do not cancel across an inequality without checking sign. In inverse trigonometry, give the principal value. In matrices, AB ≠ BA in general. In probability, decide with or without replacement before multiplying. In coordinate geometry, a zero denominator means a vertical line, not an undefined slope to ignore.