IB Mathematics: Applications and Interpretation practice questions.

USD 5000 is invested at 4% per annum, compounded quarterly. The value of the investment after 3 years is

• 5624.32
• 5634.13
• 5600.00
• 5665.29

Answer: 5634.13

FV = 5000(1 + 0.04/4)^(4×3) = 5000(1.01)¹² ≈ 5634.13. With compound interest, divide the rate and multiply the periods by the compounding frequency. 5624.32 is the annual-compounding trap.

A population follows the logistic model P(t) = 8000 / (1 + 15e^(-0.3t)). As t becomes very large, P(t) approaches

• 500
• 8000
• 15
• 1200

Answer: 8000

As t → ∞, e^(−0.3t) → 0, so P → 8000/1 = 8000. The numerator L is the carrying capacity; 500 = 8000/16 is the initial value, not the limit.

A facility must be placed inside a region as far as possible from all of the existing sites (the 'toxic waste dump' problem). Using a Voronoi diagram of the sites, the best candidates are

• the midpoints between pairs of adjacent sites
• the sites themselves
• the vertices of the Voronoi diagram
• the centroid of all the sites

Answer: the vertices of the Voronoi diagram

The nearest-site distance is locally maximised at Voronoi vertices, where three or more cells meet — each vertex is the centre of the largest empty circle through its nearby sites. Compare vertex distances (and edge–boundary intersections) and pick the largest.

A hypothesis test at the 5% significance level gives a p-value of 0.03. The correct conclusion is to

• reject H₀, since 0.03 < 0.05
• accept H₀ as proven true
• reject H₀, since 0.03 < 0.5
• fail to reject H₀, since 0.03 < 0.05

Answer: reject H₀, since 0.03 < 0.05

Reject H₀ when p-value < significance level: 0.03 < 0.05, so reject. Note we never 'prove' a hypothesis — we either reject H₀ or fail to reject it, and always compare p with the stated significance level.

A population is modelled by the logistic differential equation dP/dt = 0.2P(1 - P/600). Its non-zero equilibrium (long-term) population is

• 0.2
• 120
• 600
• 3000

Answer: 600

Equilibria occur where dP/dt = 0: P = 0 or 1 − P/600 = 0, giving P = 600 — the carrying capacity. 0.2 is the growth rate, not a population value.