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CAPE Applied Mathematics Unit 2 · 2018 · Paper 2

38 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)6 marksDetermine the earliest and latest start times for EACH activity, in order for the project to be completed in the minimum time. Write your answers in the relevant spaces in the table.
  2. 1(a)(ii)2 marksIdentify the critical path.
  3. 1(a)(iii)1 markDetermine the minimum time for completion of the project.
  4. 1(a)(iv)4 marksDraw a fully labelled activity network to represent these activities and their precedence.
  5. 1(a)(v)4 marksAssume that the duration of Activity C is increased to 6 days. Determine which activities are now critical, and the new minimum time for completion of the project.
  6. 1(b)(i)3 marksUse a truth table to show that for any propositions, p and q, p \Rightarrow q \Leftrightarrow \sim p \vee q.
  7. 1(b)(ii)5 marksUsing the laws of Boolean algebra, show that for any two propositions, p and q, \sim(p \vee \sim(p \wedge q)) is a contradiction.
  8. 2(a)2 marksIdentify the TWO variables required for this problem.
  9. 2(b)4 marksWrite the FIVE inequalities of the constraints that must be satisfied by the variables.
  10. 2(c)2 marksWrite the objective function for this problem.
  11. 2(d)8 marksOn the grid provided on page 9, draw the graph representing the five inequalities.
  12. 2(e)1 markLabel the feasible region on your graph.
  13. 2(f)3 marksState the coordinates of the vertices of the feasible region.
  14. 2(g)3 marksDetermine the maximum profit.
  15. 2(h)2 marksState the values of the variables that give the maximum profit.
  16. 3(a)(i)2 marksDetermine the number of ways in which this can be done if there are no restrictions.
  17. 3(a)(ii)3 marksDetermine the number of ways in which this can be done if exactly 2 women must be chosen.
  18. 3(a)(iii)2 marksCalculate the probability that the chosen committee will include exactly 2 women.
  19. 3(b)(i)2 marksIn how many ways can 3 jobs be assigned to the 5 teams if any team can be assigned only one job?
  20. 3(b)(ii)2 marksIn how many ways can the jobs be assigned so that Team M is given the first job?
  21. 3(b)(iii)2 marksWhat is the probability that Team M gets the first job?
  22. 3(c)(i)3 marksShow that k = 1/9.
  23. 3(c)(ii)6 marksCalculate E(X) and Var(X).
  24. 3(c)(iii)3 marksCalculate P(X > 2).
  25. 4(a)(i)4 marksWrite a distribution, V, for the total mass of the 10 biscuits.
  26. 4(a)(ii)3 marksWrite a distribution, T, for the total mass of the package of biscuits.
  27. 4(a)(iii)8 marksCalculate the probability that the total mass of the package of biscuits is less than 510 g.
  28. 4(b)(i)2 marksComplete the table to show the expected frequency for each day, assuming that haircuts are the same on each day.
  29. 4(b)(ii)8 marksCarry out a goodness-of-fit analysis to test the hypothesis that the number of haircuts is the same for each day, using a 10% level of significance. Show full details of your method, and state your conclusion clearly.
  30. 5(a)(i)6 marksDraw a clearly labelled diagram showing all the forces (tensions and weights), and the points P, Q, R and S.
  31. 5(a)(ii)10 marksUsing your diagram, calculate the tension in the strings PQ and QR, and the weight, w.
  32. 5(b)(i)2 marksDraw a diagram which illustrates the forces acting on the particle.
  33. 5(b)(ii)4 marksDetermine the coefficient of friction.
  34. 5(b)(iii)3 marksCalculate the least force which, acting along the slope of the plane, will just cause the particle to move upwards.
  35. 6(a)(i)9 marksGiven that the impulse acting on T has magnitude 0.8 N, calculate the speed, v, and the loss of kinetic energy in the collision.
  36. 6(a)(ii)3 marksParticle T subsequently collides and coalesces with a stationary particle of mass 0.3 kg. Calculate the speed, v_c, of the combined particles after this collision.
  37. 6(b)(i)4 marksIgnoring all air resistances, determine the time taken to reach a speed of 30 ms^-1.
  38. 6(b)(ii)9 marksAt the instant when the speed is 30 ms^-1, the engine is disengaged and a constant braking force of magnitude F Newtons is applied. The truck comes to rest 6 seconds later. Calculate the magnitude of F and the work done…

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