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CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2

32 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.

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  1. 1(a)5 marksCalculate the gradient of the curve \ln(x^2 y) - \sin y = 3x - 2y at the point (1, 0).
  2. 1(b)5 marksLet f(x, y, z) = 3yz^2 - e^{4x}\cos 4z - 3y^2 - 4 = 0. Given that \frac{\partial z}{\partial y} = -\frac{\partial f / \partial y}{\partial f / \partial z}, determine \frac{\partial z}{\partial y} in terms of x,…
  3. 1(c)6 marksUse de Moivre's theorem to prove that \cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.
  4. 1(d)(i)3 marksWrite the complex number z = (-1 + i)^7 in the form r e^{i\theta}, where r = |z| and \theta = \arg z.
  5. 1(d)(ii)6 marksHence, prove that (-1 + i)^7 = -8(1 + i).
  6. 2(a)(i)5 marksDetermine \int \sin x \cos 2x \, dx.
  7. 2(a)(ii)2 marksHence, calculate \int_0^{\frac{\pi}{2}} \sin x \cos 2x \, dx.
  8. 2(b)5 marksLet f(x) = x|x| = \begin{cases} x^2 &; x \ge 0 \\ -x^2 &; x < 0 \end{cases}. Use the trapezium rule with four intervals to calculate the area between f(x) and the x-axis for the domain -0.75 \le x \le 2.25.
  9. 2(c)(i)6 marksShow that \frac{2x^2 + 4}{(x^2 + 4)^2} = \frac{2}{x^2 + 4} - \frac{4}{(x^2 + 4)^2}.
  10. 2(c)(ii)7 marksHence, find \int \frac{2x^2 + 4}{(x^2 + 4)^2} \, dx. Use the substitution x = 2\tan\theta.
  11. 3(a)6 marksThe sequence \{a_n\} is defined by a_1 = 1, a_{n+1} = 4 + 2\sqrt[3]{a_n}. Use mathematical induction to prove that 1 \le a_n \le 8 for all n in the set of positive integers.
  12. 3(b)(i)a)3 marksLet k > 0 and let f(k) = \frac{1}{k^2}. Show that f(k) - f(k+1) = \frac{2k+1}{k^2(k+1)^2}.
  13. 3(b)(i)b)5 marksShow that \sum_{k=1}^n \left( \frac{1}{k^2} - \frac{1}{(k+1)^2} \right) = 1 - \frac{1}{(n+1)^2}.
  14. 3(b)(iii)3 marksHence, or otherwise, prove that \sum_{k=1}^\infty \frac{2k+1}{k^2(k+1)^2} = 1.
  15. 3(c)(i)5 marksObtain the first four non-zero terms of the Taylor Series expansion of \cos x in ascending powers of (x - \frac{\pi}{4}).
  16. 3(c)(ii)3 marksHence, calculate an approximation to \cos(-\frac{\pi}{16}).
  17. 4(a)(i)4 marksObtain the binomial expansion of \sqrt[4]{(1+x)} + \sqrt[4]{(1-x)} up to the term containing x^2.
  18. 4(a)(ii)4 marksHence, by letting x = \frac{1}{16}, compute an approximation of \sqrt[4]{17} + \sqrt[4]{15} to four decimal places.
  19. 4(b)7 marksShow that the coefficient of the x^5 term of the product (x+2)^5(x-2)^4 is 96.
  20. 4(c)(i)3 marksUse the Intermediate Value Theorem to prove that x^3 = 25 has at least one root in the interval [2, 3].
  21. 4(c)(ii)7 marksComplete the table to obtain an approximation of the root of the equation x^3 = 25 correct to 2 decimal places.
  22. 5(a)7 marksThree letters from the word BRIDGE are selected one after the other without replacement. When a letter is selected, it is classified as either a vowel (V) or a consonant (C). Use a tree diagram to show the possible…
  23. 5(b)(i)5 marksThe augmented matrix for a system of three linear equations with variables x, y and z respectively is A = \begin{pmatrix} 1 & 1 & -1 & | & 1 \\ -5 & 1 & 1 & | & 2 \\ 1 & -5 & 3 & | & 3 \end{pmatrix}. By reducing…
  24. 5(b)(ii)5 marksThe augmented matrix for another system is formed by replacing the THIRD row of A in (i) above with (1 \; -5 \; 5 \mid 3). Determine whether the solution of the new system is unique. Give a reason for your answer.
  25. 5(c)(i)5 marksA country, X, has three airports (A, B, C). The percentage of travellers that use each of the airports is 45\%, 30\% and 25\% respectively. Given that a traveller has a weapon in his/her possession, the…
  26. 5(c)(ii)3 marksOn a particular day, a traveller was caught carrying a weapon at an airport in Country X. What is the probability that the traveller used airport C?
  27. 6(a)(i)7 marksObtain the general solution of the differential equation \cos x \frac{dy}{dx} + y\sin x = 2x\cos^2 x.
  28. 6(a)(ii)5 marksHence, given that y = \frac{15\sqrt{2}\pi^2}{32} when x = \frac{\pi}{4}, determine the constant of the integration.
  29. 6(b)(i)a)2 marksThe general solution of the differential equation y'' + 2y' + 5y = 4\sin 2t is y = CF + PI, where CF is the complementary function and PI is a particular integral. Calculate the roots of…
  30. 6(b)(i)b)3 marksHence, obtain the complementary function (CF), the general solution of y'' + 2y' + 5y = 0.
  31. 6(b)(ii)3 marksGiven that the form of the particular integral (PI) is u_p(t) = A\cos 2t + B\sin 2t, show that A = -\frac{16}{17} and B = \frac{4}{17}.
  32. 6(b)(iii)5 marksGiven that y(0) = 0.04 and y'(0) = 0, obtain the general solution of the differential equation.

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