Mathematics

1. 1.

   A random sample of 36 independent observations from a normal population has mean 52. The population standard deviation is known to be 6. Test H0: mu = 50 against H1: mu > 50 at the 5% level using a z-test.

   [4 marks]
2. 2.

   A particle moves on a straight line with velocity v(t) = 3t^2 m s^-1 for 0 <= t <= 2, where t is in seconds. Find its displacement over this interval and acceleration at t = 2.

   [3 marks] · no calculator
3. 3.

   In ten independent trials, nine successes are observed. Test H0: p = 0.5 against H1: p > 0.5 at 5% significance using the exact binomial upper-tail probability.

   [4 marks]
4. 4.

   A fill amount X is modelled as normal with mean 16 and standard deviation 2. Find P(X < 14), then find c such that P(X < c) = 0.05. Give both answers to three significant figures.

   [4 marks]
5. 5.

   A population model is P(t) = 1000/(1 + 9e^(-0.4t)), t >= 0 in years. Find P(0), the initial growth rate, and when it reaches 500. Using P' = 0.4P(1 - P/1000), find the maximum growth rate on t >= 0.

   [5 marks]
6. 6.

   For x > 0 a model cost is C(x) = x^2 + 100/x. Find the exact minimising x and minimum cost, and justify that the minimum is global.

   [5 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.