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CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2

30 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.

Practise this paper
  1. 1(a)(i)5 marksShow that f'(x) = x^2 ln x (3 ln x + 2).
  2. 1(a)(ii)5 marksShow that f''(x) = 6x ln^2 x + 10x ln x + 2x.
  3. 1(b)(i)2 marksWhat is the LARGEST number reached by the membership of the club?
  4. 1(b)(ii)6 marksCalculate the EXACT value of k and of r.
  5. 1(b)(iii)2 marksHow many members will there be in the club 3 years after its formation?
  6. 2(a)(i)6 marksExpress (1 + x)/((x - 1)(x^2 + 1)) in partial fractions.
  7. 2(a)(ii)3 marksHence, find int (1 + x)/((x - 1)(x^2 + 1)) dx.
  8. 2(b)(i)4 marksEvaluate I_1.
  9. 2(b)(ii)4 marksShow that I_n = e - n I_(n-1).
  10. 2(b)(iii)3 marksHence, or otherwise, evaluate I_3, writing your answer in terms of e.
  11. 3(a)(i)3 marksShow that the terms of sum_{r=1}^m ln 3^r are in arithmetic progression.
  12. 3(a)(ii)4 marksFind the sum of the first 20 terms of this series.
  13. 3(a)(iii)3 marksHence, show that sum_{r=1}^{2m} ln 3^r = (2m^2 + m) ln 3.
  14. 3(b)(i)7 marksShow, by mathematical induction, or otherwise, that x_n < 1/2 for all positive integers n.
  15. 3(b)(ii)3 marksBy considering x_{n+1} - x_n, or otherwise, show that x_n < x_{n+1}.
  16. 4(a)5 marksSketch the functions y = sin x and y = x^2 on the SAME axes.
  17. 4(b)3 marksDeduce that the function f(x) = sin x - x^2 has EXACTLY two real roots.
  18. 4(c)4 marksFind the interval in which the non-zero root alpha of f(x) lies.
  19. 4(d)8 marksStarting with a first approximation of alpha at x_1 = 0.7, use one iteration of the Newton-Raphson method to obtain a better approximation of alpha to 3 decimal places.
  20. 5(a)(i)5 marksHow many numbers lying between 3 000 and 6 000 can be formed from the digits, 1, 2, 3, 4, 5, 6, if no digit is used more than once in forming the number?
  21. 5(a)(ii)5 marksDetermine the probability that a number in 5(a)(i) above is even.
  22. 5(b)(i)7 marksCalculate the probability that he will hit the target AT LEAST 8 times.
  23. 5(b)(ii)3 marksCalculate the probability that he will hit the target NO MORE than seven times.
  24. 6(a)(i)3 marksFind AB.
  25. 6(a)(ii)3 marksDeduce A^(-1).
  26. 6(b)(i)1 markWrite down an expression in terms of p, q and r, for the number of kilograms of Z-grass in the blend.
  27. 6(b)(ii)3 marksLet c, z and b represent the number of kilograms of C-grass, Z-grass and B-grass respectively in the blend. Write down a set of THREE equations in p, q, r, to represent the number of kilograms of EACH type of grass in…
  28. 6(b)(iii)3 marksRewrite the set of THREE equations in (b)(ii) above in the matrix form MX = D where M is a 3 by 3 matrix, X and D are column matrices.
  29. 6(b)(iv)3 marksGiven that M^(-1) exists, write X in terms of M^(-1) and D.
  30. 6(b)(v)4 marksGiven that M^(-1) = [[-0.2, -0.2, 0.3], [0.35, 0.1, -0.15], [-0.05, 0.2, -0.05]], calculate how many bags of EACH brand, P, Q, and R, are required to produce a blend containing 30 kilograms of C-grass, 30 kilograms of…

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