CAPE Applied Mathematics Unit 2 · 2013 · Paper 2

40 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)5 marksConstruct the truth table for (p \wedge q) \wedge \sim(p \vee q).
2. 1(b)2 marksState, with reason, whether (p \wedge q) \wedge \sim(p \vee q) is a tautology or a contradiction.
3. 1(c)2 marksWrite a statement to show the proposition that is logically equivalent to \sim(p \vee q).
4. 1(d)3 marksUsing de Morgan's Law, or otherwise, write an equivalent statement to 'It is not true that it is hot and sunny.'
5. 1(e)(i)5 marksDraw the circuit represented by this expression.
6. 1(e)(ii)5 marksSimplify this Boolean expression to obtain an equivalent expression and draw the corresponding circuit.
7. 1(f)3 marksUse logic gates to represent the expression \sim[(p \wedge q) \vee r].
8. 2(a)(i)3 marksDetermine the EARLIEST start times of the activities S and X.
9. 2(a)(ii)2 marksDetermine the MINIMUM completion time of the project.
10. 2(a)(iii)3 marksDetermine the LATEST start times of the activities X and Q.
11. 2(a)(iv)2 marksDetermine a critical path of the activity network.
12. 2(a)(v)3 marksDetermine the float time for activity T.
13. 2(b)2 marksState the degree of the vertices A and C.
14. 2(c)(i)8 marksUse the Hungarian algorithm to determine the supermarket to which EACH warehouse must be assigned in order to MINIMIZE the cost of delivery.
15. 2(c)(ii)2 marksHence, determine the total cost for EACH item at the four warehouses.
16. 3(a)(i)2 marksDetermine E(X + Y).
17. 3(a)(ii)2 marksDetermine E(2X - 3Y).
18. 3(a)(iii)4 marksDetermine \text{Var}(2X - 3Y).
19. 3(b)(i)3 marksCalculate P(X \ge 3).
20. 3(b)(ii)2 marksCalculate E(X).
21. 3(b)(iii)3 marksCalculate P(X = 5).
22. 3(c)(i)4 marksCalculate, to 3 decimal places, P(X \ge 65).
23. 3(c)(ii)5 marksCalculate, to 3 decimal places, P(40 \le X \le 80).
24. 4(a)(i)3 marksDetermine the probability of exactly 2 accidents in any one-week period.
25. 4(a)(ii)4 marksGiven that on average 4 accidents occur in a 4-week period, show that the probability of at least 4 accidents in a 4-week period is 0.567 to 3 decimal places.
26. 4(a)(iii)5 marksA year consists of approximately thirteen 4-week periods. Determine the probability that in a particular year there are exactly eleven periods during which AT LEAST 4 accidents occur.
27. 4(b)6 marksWith the use of another justified probability distribution function as an approximation, determine the probability that a box contains AT MOST two defective nails.
28. 4(c)(i)3 marksDetermine the value of the constant k.
29. 4(c)(ii)4 marksDetermine the value of t such that P(X > t) = \frac{1}{4}.
30. 5(a)(i)5 marksDraw a diagram to illustrate this information, and hence draw the coplanar force diagram.
31. 5(a)(ii)2 marksCalculate the tension in the string.
32. 5(b)8 marksCalculate the frictional resistance of the railway.
33. 5(c)(i)7 marksShow that t = \frac{1}{k}\ln\left(\frac{10}{10 - kx}\right).
34. 5(c)(ii)3 marksOn reaching the top of the 75 m long hill, his velocity has dropped to 4\text{ ms}^{-1}. Find the value of k.
35. 6(a)(i)3 marksDetermine the acceleration, a\text{ ms}^{-2}, of the system.
36. 6(a)(ii)2 marksDetermine the tension, T\text{ N}, in the string.
37. 6(a)(iii)3 marksDetermine the distance travelled by EACH particle during the first 6 seconds.
38. 6(b)4 marksA ball, of mass 1.5 kg, strikes a smooth vertical wall horizontally with a speed of 7\text{ ms}^{-1} and bounces off it at 5\text{ ms}^{-1}. Calculate the impulse of the ball.
39. 6(c)5 marksA particle is projected with a velocity v\text{ ms}^{-1} at an angle of \alpha to the horizontal. If it passes through a point P(x, y), determine the equation of the particle's trajectory.
40. 6(d)8 marksA bullet fired from a point O with velocity 30\text{ ms}^{-1} at an angle of \alpha to the horizontal passes through a point P(25, 40). Calculate, to the nearest degree, the two possible values of \alpha.