- Without a calculator, find the exact value of sin²1° + sin²2° + sin²3° + ... + sin²88° + sin²89°.
- Find the smallest positive integer n for which (1 + i√3)ⁿ is a positive real number, and the value it then takes.
- The number of tūī seen at a garden feeder in an hour follows a Poisson distribution. The chance of seeing exactly 1 tūī equals the chance of seeing exactly 2. What is the probability of seeing exactly 3?
- Kiwifruit weights are normally distributed with mean 100 g. 20% of kiwifruit weigh under 94 g. A kiwifruit is chosen at random and is known to weigh more than 94 g. What is the probability it weighs more than 106 g?
- A rectangle is drawn inside the ellipse x²/9 + y²/4 = 1 with its sides parallel to the axes and all four corners on the ellipse. What is the largest possible area of the rectangle?
- At a whānau Secret Santa, 10 people each put their name in a hat, then each draws one name at random without replacement. What is the expected number of people who draw their own name?
- Find the exact value of ∫ from 0 to π/2 of √(sin x) ÷ (√(sin x) + √(cos x)) dx.
- Water is poured at 8π cm³ per second into a cone-shaped funnel held point down. The funnel is 12 cm deep and its top has radius 6 cm. How fast is the water level rising when the water is 4 cm deep?
- z is a complex number with z⁵ = 1 and z ≠ 1. Find the value of (1 + z)(1 + z²)(1 + z³)(1 + z⁴).
- A function satisfies f(x) = x + ∫ from 0 to 1 of t f(t) dt for all x. Find f(x).
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