Mathematics: algebra, geometry, calculus and sequences
The mathematics section of the Iwate entrance science paper tests fluent algebraic manipulation, plane and coordinate geometry, introductory differential and integral calculus, and sequences and series. Questions are short but chained: an error in line one removes every later mark, so show each step.
Core rules
- Algebra: factorise before dividing; for ax²+bx+c the roots are (−b±√(b²−4ac))/2a and the discriminant b²−4ac fixes how many real roots exist.
- Geometry: perpendicular gradients multiply to −1; distance between (x₁,y₁) and (x₂,y₂) is √((Δx)²+(Δy)²); a circle with centre (a,b) and radius r satisfies (x−a)²+(y−b)²=r².
- Calculus: d/dx(xⁿ)=nxⁿ⁻¹; ∫xⁿdx=xⁿ⁺¹/(n+1)+C for n≠−1; a stationary point requires f′(x)=0 and f″(x) classifies it.
- Sequences: arithmetic aₙ=a₁+(n−1)d with Sₙ=n/2·(2a₁+(n−1)d); geometric aₙ=a₁rⁿ⁻¹ with Sₙ=a₁(1−rⁿ)/(1−r).
Worked example
Find the area enclosed by y=x and y=x² between x=0 and x=1. Intersections: x=x² gives x=0, 1. On (0,1), x exceeds x², so area = ∫₀¹(x−x²)dx = [x²/2−x³/3]₀¹ = 1/2−1/3 = 1/6. The segment's maximum height is 0.25, so an area below 0.25 is plausible - a useful sanity check.
Traps that cost marks
Dropping +C in an indefinite integral; discarding the negative root of a square; differentiating trigonometric functions in degrees instead of radians; giving the nth term when the sum was asked; rounding intermediate values too early; applying the geometric sum formula when r=1.
Key skills to drill
Rewrite expressions into standard forms, sketch curves before integrating, and state the domain whenever you divide or take a root.