CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2

33 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 marksFind the first derivative of the function f(x) = \cos^{-1}(\sin^{-1} x).
2. 1(b)(i)5 marksGiven that \frac{\partial w}{\partial x} = -\frac{1}{9} at the point (4, y_0), calculate the value of y_0.
3. 1(b)(ii)5 marksShow that \frac{\partial^2 w}{\partial y \partial x} - 2 \frac{\partial^2 w}{\partial y^2} = 0.
4. 1(c)(i)7 marksFind the complex numbers u = x + \mathrm{i}y such that x and y are real and u^2 = -15 + 8\mathrm{i}.
5. 1(c)(ii)5 marksHence, or otherwise, solve the equation z^2 - (3 + 2\mathrm{i})z + (5 + \mathrm{i}) = 0, for z.
6. 2(a)(i)4 marksUse integration by parts to derive the reduction formula aI_n = x^n e^{ax} - nI_{n-1}, where I_n = \int x^n e^{ax} \, \mathrm{d}x.
7. 2(a)(ii)6 marksHence, or otherwise, determine \int x^3 e^{3x} \, \mathrm{d}x.
8. 2(b)5 marksCalculate \int_0^1 \frac{\sin^{-1} x}{\sqrt{1 - x^2}} \, \mathrm{d}x.
9. 2(c)(i)5 marksUse partial fractions to show that \frac{2x^2 - x + 4}{x^3 + 4x} = \frac{1}{x} + \frac{x}{x^2 + 4} - \frac{1}{x^2 + 4}.
10. 2(c)(ii)5 marksHence, or otherwise, determine \int \frac{2x^2 - x + 4}{x^3 + 4x} \, \mathrm{d}x.
11. 3(a)(i)6 marksDetermine the Taylor series expansion about x = 2 of the function f(x) = \ln(5 + x) up to and including the term in x^3.
12. 3(a)(ii)2 marksHence, obtain an approximation for f(7) - \ln(7).
13. 3(b)(i)9 marksUse mathematical induction to prove that 1^3 + 2^3 + \ldots + n^3 = \frac{1}{4}n^2(n+1)^2, for n \in \mathbb{N}.
14. 3(b)(ii)3 marksHence, or otherwise, show that \sum_{i=1}^{2n+1} i^3 = (2n + 1)^2(n + 1)^2.
15. 3(b)(iii)5 marksUse the results of Parts (b)(i) and (ii) to show that \sum_{i=1}^{n+1} (2i - 1)^3 = (n + 1)^2(2n^2 + 4n + 1).
16. 4(a)5 marksEight boys and two girls are to be seated on a bench. How many seating arrangements are possible if the girls can neither sit together nor sit at the ends?
17. 4(b)(i)4 marksShow that the binomial expansion of \left(1 + \frac{1}{8}x\right)^8 up to and including the term in x^4 is 1 + x + \frac{7}{16}x^2 + \frac{7}{64}x^3 + \frac{35}{2048}x^4.
18. 4(b)(ii)4 marksUse the expansion to approximate the value of (1.0125)^8.
19. 4(c)(i)3 marksUse the intermediate value theorem to show that f(x) = \sqrt{x} - \cos x has a root in the interval [0, 1].
20. 4(c)(ii)4 marksUse two iterations of the interval bisection method to approximate the root of f in the interval [0, 1].
21. 4(d)(i)2 marksShow that x_{n+1} = \sqrt[3]{\frac{9 - 3x_n}{2}} is an appropriate iterative formula for finding the root of f(x) = -2x^3 - 3x + 9.
22. 4(d)(ii)3 marksApply the iterative formula with initial approximation x_1 = 1, to obtain a third approximation, x_3, of the root of the equation.
23. 5(a)(i)3 marksGiven that P(A \cup B) = 0.7, calculate P(A \text{ only}).
24. 5(a)(ii)2 marksHence, determine whether events A and B are independent. Justify your answer.
25. 5(b)(i)4 marksRepresent the outcomes of the draws and their corresponding probabilities on a tree diagram.
26. 5(b)(ii)3 marksDetermine the probability that the second ball drawn is white.
27. 5(c)(i)5 marksShow that AB = 20I.
28. 5(c)(ii)2 marksHence, deduce the inverse, A^{-1}, of the matrix A.
29. 5(c)(iii)6 marksHence, or otherwise, solve the system of linear equations given by: \begin{aligned} x - y + z &= 1 \\ x - 2y + 4z &= 5 \\ x + 3y + 9z &= 25 \end{aligned}
30. 6(a)(i)7 marksFind the general solution of the differential equation (1 + x^2) \frac{\mathrm{d}y}{\mathrm{d}x} + 2xy = \sqrt[3]{x}.
31. 6(a)(ii)3 marksHence, given that y = 2 when x = 0, calculate y(1).
32. 6(b)(i)2 marksUse the substitution u = y' to show that the differential equation y'' + 4y' = 2\cos 3x - 4\sin 3x can be reduced to u' + 4u = 2\cos 3x - 4\sin 3x.
33. 6(b)(ii)13 marksHence, or otherwise, find the general solution of the differential equation.