CAPE Integrated Mathematics · 2016 · Paper 2

50 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)4 marksFind the complex solutions of the equation x^2 - 2x + 5 = 0, giving your answer in the form x ± iy.
2. 1(b)5 marksSolve the pair of simultaneous equations: 3y = x^2 - x and y + 4 = 2x.
3. 1(c)3 marksFind the equation of the straight line, l, that passes through the point P(0, 2) and is parallel to the line AB with coordinates A(5, -4) and B(-3, 2).
4. 1(d)4 marksObtain the binomial expansion of (2x - 1)^3.
5. 1(e)4 marksDetermine the range of values of x for which 2x^2 + 5x - 3 > 0.
6. 1(f)5 marksA function, f, is such that f(x) = x^3 + x^2 - 4x - 4 where f(-1) = 0. Factorize f(x) completely.
7. 2(a)(i)3 marksGiven that f(x) = 2 ln x^2 - ln x, show that f(x) can be rewritten as f(x) = ln x^3.
8. 2(a)(ii)5 marksHence, solve the equation 2 ln x^2 = 2 + ln x, giving your answer correct to 2 decimal places.
9. 2(b)(i)5 marksSolve the equation 2 sin(theta + pi/3) - 1 = 0 for 0 <= theta <= pi.
10. 2(b)(ii)3 marksDetermine the minimum value of f(theta) = 2 sin(theta + pi/3) - 1.
11. 2(c)3 marksA length of string is cut into 10 pieces such that the shortest and longest pieces measure 0.2 metres and 1 metre, respectively. When the 10 pieces are arranged in order of length, there is a constant difference, d,…
12. 2(d)6 marksA system of linear equations models the cost of computers from three manufacturers: 2x + y + z = 1200 x + 4y + z = 1900 3x + 2z = 1600 Using Cramer's rule or otherwise, where the determinant of the coefficient matrix of…
13. 3(a)(i)1 markIdentify the type of sample if the psychologist assigns each student a number from 1 to 3960, randomly chooses one of the first 132 numbers, and selects every 132nd number thereafter.
14. 3(a)(ii)1 markIdentify the type of sample if the psychologist assigns each student a number from 0001 to 3960 and uses a computer to randomly generate a list of 30 numbers from 0001 to 3960.
15. 3(a)(iii)1 markIdentify the type of sample if students are listed by neighbourhood, 6 neighbourhoods are randomly selected, and all students from those neighbourhoods form the sample.
16. 3(a)(iv)1 markIdentify the type of sample if an equal proportion of students is randomly selected from each of the 5 departments at the college.
17. 3(b)(i)4 marksCalculate the mean height for the 51 children.
18. 3(b)(ii)5 marksCalculate the standard deviation of the heights for the 51 children.
19. 3(c)(i)3 marksA team of 3 girls and 3 boys is to be chosen from a group of 6 girls and 5 boys to represent its class at a mathematics competition. Find the number of ways in which the team may be chosen.
20. 3(c)(ii)3 marksSarah and Hannah are in the group of 6 girls. Find the number of ways in which the team may be chosen if neither Sarah nor Hannah is on the team.
21. 3(d)(i)3 marksShow that a = 3.
22. 3(d)(ii)3 marksFind the variance, Var(X).
23. 4(a)(i)2 marksFind, to the nearest whole number, the value of a.
24. 4(a)(ii)2 marksFind, to the nearest whole number, the value of b.
25. 4(b)(i)1 markDetermine the median time.
26. 4(b)(ii)2 marksFind the lower quartile time.
27. 4(b)(iii)3 marksUsing the axis provided, construct a box-and-whisker plot for the time spent completing the job, given that the upper quartile is 25 minutes.
28. 4(c)(i)2 marksGiven that X ~ Bin(n, 0.4) where the mean is 3.2, show that the number of trials, n = 8.
29. 4(c)(ii)4 marksFind P(X >= 6).
30. 4(d)(i)2 marksThe mass of barrels arriving in a specific country follows a normal distribution with mean 140 kg and standard deviation 5 kg. Show that the z-value of a barrel of mass 132 kg is z = -1.6.
31. 4(d)(ii)4 marksFind the probability that a barrel selected at random weighs more than 132 kg.
32. 4(e)(i)2 marksOn the grid provided on page 19, draw a scatter plot to represent the data.
33. 4(e)(ii)1 markBriefly comment on the shape of the scatter plot.
34. 5(a)(i)2 marksEvaluate lim_{x -> 1+} f(x).
35. 5(a)(ii)2 marksDetermine whether the function f(x) is continuous at x = 1. Justify your answer.
36. 5(b)(i)5 marksGiven that f(x) = x^3 ln 2x, find f'(x), the first derivative of f(x).
37. 5(b)(ii)3 marksDetermine the derivative of the function g(x) = e^(2x) + cos 3x with respect to x.
38. 5(c)(i)3 marksFind the second derivative, P''(t).
39. 5(c)(ii)2 marksGiven that the function P(t) has a stationary point when t = 48, show that the stationary point is a maximum point.
40. 5(c)(iii)3 marksFind the population count when t = 48 days.
41. 5(d)(i)1 markObtain an expression for dV/dt.
42. 5(d)(ii)4 marksFind the rate of increase of the volume of the balloon, dV/dt, at the instant when r = 5 cm. Give your answer in terms of pi.
43. 6(a)(i)3 marksDetermine the integral: int (4x + 7)^6 dx.
44. 6(a)(ii)2 marksDetermine the integral: int (5x^3 - 3x^2 + 1) dx.
45. 6(b)4 marksEvaluate the definite integral int_0^(pi/2) (cos x + 2 sin x) dx.
46. 6(c)(i)4 marksShow that P satisfies the equation P(t) = e^(kt + c).
47. 6(c)(ii)2 marksWhen t = 0, P = 4000. Calculate the value of c.
48. 6(c)(iii)3 marksGiven that k = ln 2, determine the population size when t = 3.
49. 6(d)(i)3 marksFind the value of Q.
50. 6(d)(ii)4 marksUsing integration, determine the area reserved for agricultural activities. Give your answer to the nearest whole number.