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CAPE Applied Mathematics Unit 2 · 2016 · Paper 2

71 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.

Practise this paper
  1. 1(a)(i)1 markState whether the 40 buses owned by "I'll take you there tours" constitute a sample or a population.
  2. 1(a)(i)4 marksShow that (p ∧ ~q) ∨ (~p ∧ q) is equivalent to (p ∨ q) ∧ (~p ∨ ~q) using truth tables.
  3. 1(a)(ii)1 markState whether 40 of the persons who attended the health seminar last week constitute a sample or a population.
  4. 1(a)(ii)4 marksShow that (p ∧ ~q) ∨ (~p ∧ q) is equivalent to (p ∨ q) ∧ (~p ∨ ~q) using the laws of Boolean algebra.
  5. 1(b)(i)3 marksDraw the switching circuit for (p ∧ ~q) ∨ (~p ∧ q).
  6. 1(b)(i)a)1 markState ONE reason why visiting beauty shops on Saturday morning to interview a selection of customers buying the product may be unsatisfactory.
  7. 1(b)(i)b)1 markState ONE reason why posting a questionnaire on the manufacturer's Facebook page for fans or followers to respond may be unsatisfactory.
  8. 1(b)(ii)3 marksDraw the switching circuit for (p ∨ q) ∧ (~p ∨ ~q).
  9. 1(b)(ii)a)1 markDescribe the sampling method used when all names are put into a box at the end of the exhibition and a random sample of 50 names is drawn.
  10. 1(b)(ii)b)1 markDescribe the sampling method used when every fifth name is selected from the list of names collected over the three days.
  11. 1(c)2 marksUsing stratified random sampling to select a sample of 15 students from the group, determine how many boys will be in the sample.
  12. 1(c)3 marksDetermine the Boolean expression for the output s in the given logic circuit.
  13. 1(d)1 markState the advantage of using a stem-and-leaf diagram rather than grouping data into a frequency distribution.
  14. 1(d)(i)2 marksWrite an expression in logical notation for: "If a person eats red meat then that person may have a high cholesterol reading or suffer a heart attack."
  15. 1(d)(ii)2 marksWrite an expression in logical notation for: "If a person does not suffer a heart attack then that person has a normal cholesterol reading and does not eat red meat."
  16. 1(d)(iii)4 marksConstruct a truth table for the statement in (d)(i) and state with a reason whether it is a tautology.
  17. 1(e)(i)1 markRelative to the stem-and-leaf diagram, state what 4|6 represents.
  18. 1(e)(ii)1 markDetermine how many patients were in the sample.
  19. 1(e)(iii)1 markDetermine how many patients waited more than 35 minutes.
  20. 1(e)(iv)3 marksDetermine the range of waiting times.
  21. 1(e)(v)2 marksDetermine the median waiting time for the sample.
  22. 1(e)(vi)3 marksCalculate the interquartile range for the waiting time data.
  23. 2(a)(i)2 marksCalculate the probability that all three markers are red.
  24. 2(a)(i)9 marksOn the provided grid, draw the system of inequalities representing the constraints.
  25. 2(a)(ii)4 marksCalculate the probability that exactly two of the markers are black.
  26. 2(a)(ii)2 marksUsing the same grid, shade the feasible region.
  27. 2(a)(iii)3 marksUse your diagram to find the maximum value of P.
  28. 2(b)(i)4 marksFind the probability that a randomly chosen customer spends more than 8 minutes completing a transaction.
  29. 2(b)(i)9 marksUse the Hungarian algorithm to determine the optimal assignment of each driver to a town to minimize total travel time.
  30. 2(b)(ii)4 marksFind the time below which 80% of customers spend completing a transaction.
  31. 2(b)(ii)2 marksDetermine the total minimum time taken by the four drivers.
  32. 2(c)(i)4 marksIn a random sample of 10 households, calculate the probability that exactly 6 have internet access.
  33. 2(c)(ii)2 marksIn a random sample of 80 households, calculate the expected number that have internet access.
  34. 3(a)3 marksA random sample of 49 items with a sample mean of 13 is taken from a normal distribution with standard deviation 4. Calculate a 96% confidence interval for the population mean µ.
  35. 3(a)(i)3 marksCalculate the total number of distinct portfolios of 12 paintings that can be created.
  36. 3(a)(ii)4 marksDetermine the number of portfolios that contain exactly 8 watercolours and 4 oil paintings.
  37. 3(a)(iii)2 marksHence, calculate the probability that a randomly chosen portfolio contains 8 watercolours and 4 oil paintings.
  38. 3(b)(i)5 marksDetermine the estimated linear regression equation y = a + bx for this data.
  39. 3(b)(i)4 marksCalculate the probability that a player starts the game on the third toss of the die.
  40. 3(b)(ii)2 marksEstimate the width of a stem when the stem density is 8 using your regression equation.
  41. 3(b)(ii)3 marksCalculate the probability that at most 5 tosses are necessary to start the game.
  42. 3(b)(iii)2 marksThe correlation coefficient is r = -0.63. Interpret this value in the context of the data.
  43. 3(b)(iii)2 marksDetermine the expected number of tosses required to start the game.
  44. 3(c)(i)2 marksState appropriate null and alternative hypotheses to test for association between gender and rating of restroom facilities using a chi-squared test.
  45. 3(c)(i)3 marksCalculate the probability that exactly 4 faulty reports are received on Monday.
  46. 3(c)(ii)2 marksDetermine the critical region for the chi-squared test at the 5% significance level.
  47. 3(c)(ii)4 marksCalculate the probability that exactly 5 faulty reports are received over a five-day period.
  48. 3(c)(iii)2 marksCalculate the expected frequency corresponding to cell (row 2, column 2), which has an observed value of 24.
  49. 3(c)(iv)a)1 markState whether you reject or fail to reject the null hypothesis.
  50. 3(c)(iv)b)1 markInterpret the test decision in the context of the problem.
  51. 4(a)(i)2 marksState appropriate null and alternative hypotheses to test whether grades are uniformly distributed.
  52. 4(a)(ii)3 marksDetermine the critical region for the chi-squared goodness-of-fit test at the 5% significance level.
  53. 4(a)(iii)4 marksCalculate the value of the chi-squared test statistic.
  54. 4(a)(iv)2 marksState a valid conclusion for the test, giving a reason.
  55. 4(b)(i)3 marksDetermine the value of k.
  56. 4(b)(ii)3 marksCalculate P(X > 1).
  57. 4(b)(iii)3 marksCalculate the expected value E(X).
  58. 4(b)(iv)3 marksState fully the cumulative distribution function F(x).
  59. 4(b)(v)2 marksHence, or otherwise, find P(1.5 < X < 2).
  60. 5(a)(i)9 marksUse Lami's theorem, or otherwise, to show that the weight of the fixture is 50((cos θ)/√3 + sin θ).
  61. 5(a)(ii)3 marksShow that the tension in string B is TB = (100/√3) cos θ.
  62. 5(b)(i)3 marksDraw a clearly labelled diagram showing all the forces acting on the system after the engine is shut off.
  63. 5(b)(ii)4 marksDetermine the decelerating force acting on the system.
  64. 5(b)(iii)6 marksDetermine the time taken to reduce speed from 80 km h⁻¹ to 44 km h⁻¹.
  65. 6(a)(i)5 marksCalculate the tractive force acting parallel to the plane.
  66. 6(a)(ii)3 marksCalculate the work done, to the nearest whole number, after one second to move the body.
  67. 6(a)(iii)1 markCalculate the power developed during that one second.
  68. 6(b)6 marksIf the particle is just about to slip, find the coefficient of friction between the particle and the surface.
  69. 6(c)(i)4 marksCalculate the speed of vehicle B immediately after the collision.
  70. 6(c)(ii)1 markState the direction of motion of vehicle B after the collision.
  71. 6(c)(iii)5 marksCalculate the time taken for vehicle A to come to rest after the collision.

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