CAPE Applied Mathematics Unit 2 · 2017 · Paper 2

34 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.

1. 1(a)(i)5 marksUsing x for the number of acres of alfalfa and y for the number of acres of corn, formulate a linear programming problem and state clearly the maximizing function.
2. 1(a)(ii)8 marksOn the grid provided, draw all lines defined by the constraints and shade the feasible region satisfying all constraints.
3. 1(a)(iii)6 marksDetermine the number of acres of alfalfa and the number of acres of corn that the farmer must plant to maximize income.
4. 1(b)2 marksWrite the contrapositive of p \rightarrow \sim q.
5. 1(c)(i)2 marksList the vertices that have a degree greater than 2.
6. 1(c)(ii)2 marksList all the paths from C to E.
7. 2(a)(i)6 marksConstruct the activity network for the project.
8. 2(a)(ii)4 marksComplete the table of earliest start time, latest start time, and float for each activity.
9. 2(a)(iii)2 marksState the critical path of the activity network.
10. 2(a)(iv)2 marksDetermine the minimum time needed to complete the project.
11. 2(b)(i)9 marksUse the Hungarian algorithm to determine the task assignment for each worker that maximizes the total income.
12. 2(b)(ii)2 marksDetermine the total income of the workers for the four tasks.
13. 3(a)5 marksFind the probability that the 3 locals are on the committee.
14. 3(b)(i)4 marksCalculate the value of the constant k and sketch the graph of f(x).
15. 3(b)(ii)6 marksDetermine E(X) and \operatorname{Var}(X).
16. 3(b)(iii)3 marksCalculate P(3 < X < 4.5).
17. 3(b)(iv)4 marksDetermine the cumulative distribution function, F(x) = P(X \le x), and sketch its graph.
18. 3(b)(v)3 marksCalculate the median, m, of X.
19. 4(a)4 marksDetermine, to 3 significant figures, the probability that a piece of metal 15\text{ m}^2 will contain at most three blisters.
20. 4(b)5 marksJustifying the use of a suitable approximation, calculate the probability that at least 2 customers intend to pay their arrears.
21. 4(c)(i)1 markState clearly the distribution which can be used to approximate the Poisson distribution.
22. 4(c)(ii)4 marksCalculate the probability that in one hour less than 32 calls are received.
23. 4(c)(iii)4 marksCalculate the probability that in one hour between 31 and 39 calls are received.
24. 4(d)(i)1 markComplete the table by giving the expected number of persons preferring each brand, assuming that each brand is equally likely to be preferred.
25. 4(d)(ii)6 marksPerform a \chi^2 goodness-of-fit test at the 1% significance level to determine whether a uniform distribution fits the tea-drinking preference of the 250 persons.
26. 5(a)(i)4 marksDraw a diagram to illustrate this information, showing the forces acting on the system.
27. 5(a)(ii)6 marksDetermine the masses of the particles, A and B.
28. 5(a)(iii)2 marksCalculate the tension, T, in the string.
29. 5(b)(i)9 marksIf air resistance is ignored, determine the time taken for the object to hit the ground.
30. 5(b)(ii)4 marksShow that the distance from the foot of the cliff to the point where the object strikes the ground is x = 5(\sqrt{120} - \sqrt{3})\text{ m}.
31. 6(a)(i)7 marksDetermine the horizontal force P.
32. 6(a)(ii)3 marksDetermine the frictional force.
33. 6(b)10 marksCalculate the velocities V_1 and V_2.
34. 6(c)5 marksIf the ropes are 60^\circ apart, determine the tensions in the ropes.