Mathematics - Mie Entrance Science
The Mathematics section of Mie Entrance Science samples the Japanese senior-high sequence - Mathematics I–III and A/B - with an emphasis on exact manipulation and multi-step justification. Expect algebra, trigonometry, sequences, vectors, probability, and calculus; answers are often exact values or short derivations rather than guesses.
Core rules
- Domains and extraneous roots: after squaring or clearing denominators, check every candidate in the original equation.
- Calculus: differentiate composites with the chain rule. For area, split a definite integral at zeros, because an integral gives signed area.
- Vectors: use a·b = |a||b|cos θ. Perpendicular means a·b = 0; parallel means one is a scalar multiple of the other (in 2D, determinant 0).
- Sequences: identify arithmetic or geometric structure; recurrences often telescope or reduce to a characteristic equation.
- Probability: without replacement changes denominators; use complementary counting for “at least one”.
Worked example
Evaluate ∫₀¹ x e^(−x) dx. Integrate by parts with u = x and dv = e^(−x)dx, so du = dx and v = −e^(−x). Then ∫x e^(−x)dx = −x e^(−x) + ∫e^(−x)dx = −(x+1)e^(−x). Evaluate from 0 to 1: [−2e^(−1)] − [−1] = 1 − 2/e. The negative sign in v is the usual slip.
Traps
- Using degrees in calculus; trigonometric derivatives and integrals assume radians.
- Forgetting ± after x² = a or |x| = a.
- Reporting an indefinite integral without +C, or a definite integral with +C.
- Mixing permutations and combinations when order is irrelevant.
- Ignoring stated domains, such as log x requiring x > 0.