Mathematics
Mathematics
Hiroshima's science entrance paper tests fluent manipulation rather than recall: algebra, functions and graphs, differentiation and integration, vectors, sequences, and probability. Questions often combine two topics, so identify the constraint before calculating.
Core rules
- A function's domain is part of the answer; restrict before simplifying.
- For extrema on a closed interval, compare critical points with endpoints.
- Derivative sign gives monotonicity; the second derivative is a convenience, not a proof by itself.
- Vectors: a·b = |a||b|cosθ; perpendicular means a·b = 0.
- Probability: P(A∪B)=P(A)+P(B)−P(A∩B); check independence rather than assuming it.
Worked example
Find the global maximum and minimum of f(x)=x³−3x on [−2,2]. Differentiate: f′(x)=3x²−3=3(x−1)(x+1). Critical points x=−1 and x=1 lie in the interval. Evaluate: f(−2)=−8+6=−2, f(−1)=−1+3=2, f(1)=1−3=−2, f(2)=8−6=2. Compare all four values: maximum 2 at x=−1 and x=2; minimum −2 at x=−2 and x=1. The derivative alone would miss the endpoint value at x=2, so the interval check is mandatory.
Traps that cost marks
- Reporting only stationary points; endpoints and domain restrictions change the answer.
- Cancelling a factor that can be zero without stating the excluded case.
- Mixing degrees and radians in trigonometric calculus.
- Giving a vector magnitude when a direction or component was requested.