CAPE Pure Mathematics Unit 2 · May/June 2015 · 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)(i)3 marksExpress the quotient \frac{z_3}{z_2} in the form x + iy where x, y \in \mathbb{R}.
2. 1(a)(ii)6 marksGiven that \arg w = \arg z_3 - [\arg z_1 + \arg z_2], |z_1| = 1 and \arg z_1 = \frac{\pi}{12}, rewrite w = \frac{z_3}{z_1 z_2} in the form r e^{i\theta} where r = |w| and \theta = \arg w.
3. 1(b)7 marksA complex number v = x + iy is such that v^2 = 2 + i. Show that x^2 = \frac{2 + \sqrt{5}}{2}.
4. 1(c)(i)6 marksShow that \frac{dy}{dx} = \frac{e^t (1 - t^2)}{t^2 + t - 1}.
5. 1(c)(ii)3 marksHence, show that f has no stationary value.
6. 2(a)(i)5 marksUse implicit differentiation to show that \frac{dy}{dx} = -\frac{8x + 3y^2 + 7}{3(1 + 2xy)}.
7. 2(a)(ii)5 marksShow that for f(x, y) = 4x^2 + 3xy^2 + 7x + 3y,…
8. 2(b)(i)2 marksExpress f(x) in the form a + \frac{b}{9x^2 + 4} where a, b \in \mathbb{R}.
9. 2(b)(ii)6 marksGiven that f(x) is symmetric about the y-axis, evaluate \int_{-2}^2 f(x)\,dx.
10. 2(c)(i)5 marksShow that \int h^n \ln h\,dh = \frac{h^{n+1}}{(n+1)^2} [-1 + (n+1)\ln h] + C, where -1 \ne n \in \mathbb{Z} and C \in \mathbb{R}.
11. 2(c)(ii)2 marksHence, find \int \sin^2 x \cos x \ln(\sin x)\,dx.
12. 3(a)(i)5 marksDetermine \lim_{n \to \infty} T_n.
13. 3(a)(ii)3 marksShow that T_4 = \frac{9}{4} \left(1 + \frac{1}{16}\right)^{-\frac{1}{2}}.
14. 3(a)(iii)4 marksHence, use the binomial expansion with x = \frac{1}{16} to approximate the value of T_4 for terms up to and including x^3. Give your answer correct to two decimal places.
15. 3(b)(i)2 marksExpress the nth partial sum S_n of the series in sigma notation.
16. 3(b)(ii)4 marksHence, given that \sum_{n=1}^\infty \frac{1}{n^2} converges to \frac{\pi^2}{6}, show that S_n diverges as n \to \infty.
17. 3(c)7 marksUse the method of mathematical induction to prove that \sum_{r=1}^n r(r-1) = \frac{n(n^2-1)}{3}.
18. 4(a)(i)6 marksObtain the Maclaurin series expansion for g(x) up to and including the term in x^4.
19. 4(a)(ii)3 marksHence, estimate g(0.2) correct to three decimal places.
20. 4(b)(i)3 marksUse the intermediate value theorem to show that f has at least one root in the interval [-2, 0].
21. 4(b)(ii)8 marksUse at least three iterations of the method of interval bisection to show that f(-0.538) \approx 0 in the interval [-0.7, -0.3].
22. 4(c)5 marksUse the Newton–Raphson method with initial estimate x_1 = 5.5 to approximate the root of g(x) = \sin 3x in the interval [5, 6], correct to two decimal places.
23. 5(a)(i)1 markDetermine the number of possible ways in which a group of FOUR applicants may be selected if no restrictions are applied.
24. 5(a)(ii)3 marksDetermine the number of possible ways in which a group of FOUR applicants may be selected if at least one of the successful applicants must be female.
25. 5(b)(i)4 marksDetermine the greatest possible amount of numbers that may be formed.
26. 5(b)(ii)3 marksDetermine the probability that a number formed is greater than 100.
27. 5(c)(i)2 marksRewrite the system of equations as an augmented matrix.
28. 5(c)(ii)5 marksUse elementary row operations to reduce the system to echelon form.
29. 5(c)(iii)3 marksHence, solve the system of equations.
30. 5(c)(iv)4 marksShow that the system has no solution if the third equation is changed to 1.5x - 1.5y + 3z = 9.
31. 6(a)(i)3 marksConstruct a tree diagram to show the probabilities that Alicia arrives at school.
32. 6(a)(ii)3 marksWhat is the probability that Alicia is at school on any given school day?
33. 6(a)(iii)4 marksGiven that Alicia is at school today, determine the probability that it is a rainy day.
34. 6(b)(i)5 marksShow that the equation y + xy + x^2 = 0 is a solution of the differential equation \frac{dy}{dx} = \frac{y - x^2}{x(1 + x)}.
35. 6(b)(ii)a)3 marksFind the general solution of the differential equation.
36. 6(b)(ii)b)7 marksHence, show that the solution which satisfies the boundary conditions y(0) = 1 and y'\left(\frac{\sqrt{2}}{2}\right) = 0 is y = \frac{1}{e^2 + 1}\left(e^{\sqrt{2}x} + e^{2 - \sqrt{2}x}\right).