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CAPE Applied Mathematics Unit 2 · 2010 · Paper 2

32 questions and parts from this paper. Open one to see it in full, then practise it on Quelpr and get it marked against the mark scheme.

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  1. 1(a)(i)8 marksOn the answer sheet provided, graph the feasible region for the programming problem.
  2. 1(a)(ii)4 marksHence, solve the linear programming problem.
  3. 1(b)(i)8 marksUse the Hungarian algorithm to determine the task to which EACH person must be assigned in order to minimise the total time.
  4. 1(b)(ii)2 marksHence, determine the total time taken by the FIVE persons.
  5. 1(c)3 marksFor the diagram provided, list any THREE paths which start and finish at A.
  6. 2(a)4 marksDesign a switching circuit to allow current to pass when and only when at least TWO members vote 'yes'.
  7. 2(b)5 marksConstruct a truth table to show that the proposition (~p ∨ ~q) ⇒ (p ∧ ~q) always takes the value of p.
  8. 2(c)5 marksIf a ⇒ b is equivalent to ~a ∨ b, draw a circuit for a ⇒ b using OR and NOT gates only.
  9. 2(d)(i)9 marksCopy and complete the table for the earliest start time, latest start time, and float time for activities A through G.
  10. 2(d)(ii)2 marksHence, determine the critical path.
  11. 3(a)(i)4 marksCalculate the value of the constant, k, and sketch the graph y = F(x).
  12. 3(a)(ii)3 marksCalculate P(3.5 ≤ X ≤ 5).
  13. 3(a)(iii)3 marksCalculate the median of X.
  14. 3(a)(iv)4 marksCalculate the lower quartile of X.
  15. 3(b)(i)4 marksDetermine the probability density function f(x) and sketch its graph.
  16. 3(b)(ii)3 marksDetermine the expectation of X.
  17. 3(b)(iii)4 marksDetermine the variance of X.
  18. 4(a)(i)6 marksDetermine the standard deviation of the length of wood cut.
  19. 4(a)(ii)5 marksDetermine the proportion of the pieces that are longer than 3.2 metres.
  20. 4(b)(i)2 marksState clearly the null and alternative hypotheses for a chi-square test to determine whether the number of employees who work overtime is independent of the distance of their home from the workplace.
  21. 4(b)(ii)3 marksFind the expected frequencies if the null hypothesis is true.
  22. 4(b)(iii)9 marksUsing a 5% significance level, determine whether the number of employees who work overtime is independent of the distance of their home from the workplace. Clearly state your conclusion of the test.
  23. 5(a)(i)7 marksCalculate the acceleration of the particle.
  24. 5(a)(ii)8 marksCalculate the time taken to travel 105 metres from O.
  25. 5(b)(i)6 marksCalculate the acceleration.
  26. 5(b)(ii)4 marksCalculate the velocity of the particle at a distance of 50 metres from P.
  27. 6(a)(i)9 marksIf in the position of equilibrium P = 6 N, find the magnitude of the tension in OB.
  28. 6(a)(ii)3 marksIf in the position of equilibrium P = 6 N, find the magnitude of the (acute) angle of inclination OB makes with the vertical.
  29. 6(b)(i)3 marksFind the magnitude of the (acute) angle of inclination OB makes with the vertical.
  30. 6(b)(ii)2 marksFind the magnitude of the pull, P.
  31. 6(c)(i)4 marksFind the magnitude of ai + bj.
  32. 6(c)(ii)4 marksFind the angle of inclination of ai + bj to Ox.

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