CAPE Integrated Mathematics · 2017 · Paper 2

36 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)3 marksA complex number is given by Z = \frac{\sqrt{3}}{4} - \frac{1}{4}i. Find the modulus of Z.
2. 1(b)(i)4 marksDetermine the equation of the straight line which passes through the point (1, 5) and is perpendicular to the line 3x + y = 4.
3. 1(b)(ii)5 marksThe line y = x + 6 meets the curve y = x^2 + x + 2 at the points P and Q. Determine the coordinates of points P and Q.
4. 1(c)5 marksObtain the binomial expansion of (1 + 2x)^3.
5. 1(d)4 marksDetermine the range of values of x for which |2x + 5| \le 3.
6. 1(e)4 marksSolve the equation tan(\theta + \frac{\pi}{3}) = 1, for 0 < \theta < \pi.
7. 2(a)(i)6 marksShow that the expansion 2\log 2 - \frac{1}{2}\log 9 + 4\log 3, when expressed as a single logarithm is \log 108.
8. 2(a)(ii)3 marksSolve \log (x + 3) = 1.
9. 2(b)5 marksLet f(x) = x^3 - 6x^2 + ax - 6. If (x - 2) is a factor of f(x), find the value of a.
10. 2(c)5 marksGiven that A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ p & 0 & 0 \end{pmatrix} and \det(A) = -4, find the value of p.
11. 2(d)3 marksThe graph shows f(x) = \sin x and g(x) = a + b\sin x. Determine the values of a and b and write the equation of g(x).
12. 2(e)3 marksThe first term and the second term of a geometric series are 120 and 60 respectively. Find the sum to infinity.
13. 3(a)4 marksIdentify TWO suitable sample methods and explain how ONE of the methods could have been used by the teacher to select the 50 students from the 100 Class 1 students.
14. 3(b)4 marksComplete the histogram to represent the data. The first two columns are already drawn.
15. 3(c)5 marksA student is selected at random from the 50 students in the sample. Using the histogram or otherwise, determine the probability that the student spends 26 minutes or less than 26 minutes reading.
16. 3(d)(i)2 marksDetermine the value of q in the probability distribution table.
17. 3(d)(ii)5 marksComplete the probability distribution table.
18. 3(d)(iii)5 marksCalculate the mean and variance of X.
19. 4(a)5 marksA survey of 200 people shopping Sunday to Thursday is shown in the pie chart. If E = 2A and B = 3A, determine the number of persons who shop on Thursday.
20. 4(b)(i)2 marksEstimate the median weight before and after the diet.
21. 4(b)(ii)2 marksWhat conclusion can be drawn about the special diet programme? Give a reason for your answer.
22. 4(c)5 marksGiven that X ~ Bin(8, 0.65), show that P(X > 1) = 0.99643, correct to five decimal places.
23. 4(d)6 marksA machine produces plastic bottles with capacity having mean 24.3 ml and standard deviation 0.6 ml, normally distributed. Find the probability that a plastic bottle selected at random has a capacity less than 24 ml.
24. 4(e)(i)3 marksUsing the regression equation \hat{y} = 0.186 + 0.64x, where \hat{y} is weight loss in kilograms and x is number of weeks of exercise, determine the weight loss during the 24th week if no changes were made to the…
25. 4(e)(ii)2 marksThe correlation coefficient, r, was found to be 0.981. Comment on the accuracy of the answer obtained in part (e)(i) above.
26. 5(a)4 marksGiven that f(x) = \frac{x^2 - 4}{x + 2}, evaluate \lim_{x \to -2} f(x).
27. 5(b)(i)3 marksDifferentiate y = (2x + 3)^6 with respect to x.
28. 5(b)(ii)4 marksDifferentiate y = e^x \sin x with respect to x.
29. 5(c)7 marksDetermine the stationary points for the function f(x) = x^3 - 2x^2 + x - 1, stating whether EACH point is a maximum or minimum.
30. 5(d)4 marksIf Z(x, y) = x^3 + 4y^2 + \ln y, calculate the partial derivative \frac{\partial Z}{\partial y}.
31. 5(e)3 marksA small business profit function (in dollars) for a year is given by P(x) = 150x - \frac{9}{20}x^2 - 1500, where x represents the number of cakes sold. Estimate the number of cakes that must be sold to get the maximum…
32. 6(a)4 marksDetermine \int \cos(5x - 3) \, dx.
33. 6(b)5 marksShow that \int_{1}^{3} (x^{\frac{5}{2}} - 2) \, dx is approximately 9.08.
34. 6(c)(i)5 marksA radioactive substance decays at a rate of \frac{dR}{dt} = -kR, where t is time and k is a constant. Show that R satisfies the equation R(t) = e^{-kt + c}.
35. 6(c)(ii)4 marksIf at time t = 0 the substance has an original mass of 500 mg, show that R(t) = 500e^{-kt}.
36. 6(d)7 marksFind the area bounded by the function f(x) = \frac{4}{3}x^3 - 7.5x^2 + 10x, the x-axis and the line x = C, where C is the critical value of f(x) which corresponds to the maximum point. State your answer to three decimal…