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CAPE Applied Mathematics Unit 2 · 2008 (second paper) · Paper 2

35 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)12 marksFormulate a linear programming model to determine the number of each product that maximizes the profit.
  2. 1(b)9 marksUsing the answer graph sheet provided, graph the inequalities of the linear programming model and identify the feasible region.
  3. 1(c)4 marksObtain the maximum profit and state the number of each product manufactured that gives this profit.
  4. 2(a)3 marksExpress in words: ~p ^ ~q.
  5. 2(b)3 marksDetermine the truth value of the statement: 'London is in England or 2 x 3 = 5'.
  6. 2(c)5 marksBy constructing the truth table for the proposition (p ^ q) => (p v q), determine whether this proposition is a tautology or a contradiction.
  7. 2(d)(i)4 marksRepresent the Boolean expression (A ^ ~B) v [(~A v C) ^ B] as a switching circuit.
  8. 2(d)(ii)5 marksRepresent ~a ^ (a v b) as a logic circuit using only AND, OR and NOT gates.
  9. 2(e)5 marksUse the laws of Boolean algebra to show that the Boolean expression (A ^ B) v (A ^ ~B) v (~A ^ ~B) is equivalent to A v ~B.
  10. 3(a)4 marksDetermine how many odd numbers greater than 500 000 can be made from the digits 4, 5, 6, 7, 8 and 9 if repetitions are allowed.
  11. 3(b)(i)2 marksDetermine the number of ways of choosing the team if there are no restrictions.
  12. 3(b)(ii)3 marksDetermine the number of ways of choosing the team if it contains exactly two girls.
  13. 3(b)(iii)4 marksDetermine the number of ways of choosing the team if it contains more girls than boys.
  14. 3(c)(i)2 marksFind P(A).
  15. 3(c)(ii)3 marksFind P(B).
  16. 3(c)(iii)5 marksFind P(A' ∩ B').
  17. 3(d)2 marksState, with reason, whether A and B are mutually exclusive events.
  18. 4(a)(i)2 marksShow that k = 1/10.
  19. 4(a)(ii)3 marksCalculate E(X).
  20. 4(a)(iii)3 marksShow that P(Y = 4) = 1/10.
  21. 4(a)(iv)6 marksTabulate all the possible values of Y with their corresponding probabilities.
  22. 4(a)(v)5 marksHence, or otherwise, calculate E(Y) and Var(Y).
  23. 4(a)(vi)2 marksDetermine E(3X + 2Y).
  24. 4(b)4 marksThe random variable X has a Poisson distribution with mean 1.5. Calculate the probability that X is at least 2.
  25. 5(a)(i)3 marksState the principle of conservation of energy.
  26. 5(a)(ii)3 marksState the principle of conservation of linear momentum.
  27. 5(b)(i)8 marksCalculate the kinetic energy after impact.
  28. 5(b)(ii)11 marksCalculate the distance and the time required to bring the vehicle to rest by a braking force of 1000 N applied immediately after the impact.
  29. 6(a)3 marksCalculate the greatest height reached by the first particle.
  30. 6(b)(i)2 marksFor the first particle, calculate the distance travelled at the end of 5 seconds horizontally.
  31. 6(b)(ii)3 marksFor the first particle, calculate the distance travelled at the end of 5 seconds vertically.
  32. 6(b)(iii)4 marksHence, determine the value of u cos α.
  33. 6(b)(iv)4 marksHence, determine the value of u sin α.
  34. 6(b)(v)4 marksHence, determine the value of u.
  35. 6(b)(vi)5 marksHence, determine the value of α.

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