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CAPE Pure Mathematics Unit 2 · May/June 2009 · 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.

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  1. 1(a)(i)3 marksFind \frac{\mathrm{d}y}{\mathrm{d}x} if y = \sin^2 5x + \sin^2 3x + \cos^2 3x.
  2. 1(a)(ii)4 marksFind \frac{\mathrm{d}y}{\mathrm{d}x} if y = \sqrt{\cos x^2}.
  3. 1(a)(iii)4 marksFind \frac{\mathrm{d}y}{\mathrm{d}x} if y = x^x.
  4. 1(b)(i)7 marksGiven that y = \cos^{-1} x, where 0 \le \cos^{-1} x \le \pi, prove that \frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{1}{\sqrt{1 - x^2}}.
  5. 1(b)(ii)a)4 marksShow that \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\sqrt{1 + t}}{2}.
  6. 1(b)(ii)b)3 marksHence, find \frac{\mathrm{d}^2y}{\mathrm{d}x^2} in terms of t, giving your answer in simplified form.
  7. 2(a)3 marksSketch the region whose area is defined by the integral \int_0^1 \sqrt{1 - x^2}\, \mathrm{d}x.
  8. 2(b)6 marksUsing FIVE vertical strips, apply the trapezium rule to show that \int_0^1 \sqrt{1 - x^2}\, \mathrm{d}x \approx 0.759.
  9. 2(c)(i)9 marksUse integration by parts to show that, if I = \int \sqrt{1 - x^2}\, \mathrm{d}x, then I = x\sqrt{1 - x^2} - I + \int \frac{1}{\sqrt{1 - x^2}}\, \mathrm{d}x.
  10. 2(c)(ii)2 marksDeduce that I = \frac{x\sqrt{1 - x^2} + \sin^{-1} x}{2} + c, where c is an arbitrary constant of integration.
  11. 2(c)(iii)3 marksHence, find \int_0^1 \sqrt{1 - x^2}\, \mathrm{d}x.
  12. 2(c)(iv)2 marksUse the results in Parts (b) and (c)(iii) above to find an approximation to \pi.
  13. 3(a)(i)3 marksDetermine t_2, t_3 and t_4.
  14. 3(a)(ii)5 marksExpress t_n in terms of n.
  15. 3(b)8 marksFind the range of values of x for which the common ratio r of a convergent geometric series is \frac{2x - 3}{x + 4}.
  16. 3(c)(i)3 marksExpress f(r) - f(r + 1) in terms of r.
  17. 3(c)(ii)4 marksHence, or otherwise, find S_n = \sum_{r=1}^n \frac{4}{(r + 1)(r + 2)}.
  18. 3(c)(iii)2 marksDeduce the sum to infinity of the series in (c)(ii) above.
  19. 4(a)(i)5 marksFind n \in \mathbb{N} such that 5\binom{n}{2} = 2\binom{n+2}{2}.
  20. 4(a)(ii)7 marksThe coefficient of x^2 in the expansion of (1 + 2x)^5 (1 + px)^4 is -26. Find the possible values of the real number p.
  21. 4(b)(i)2 marksWrite down the first FOUR non-zero terms of the power series expansion of \ln(1 + 2x), stating the range of values of x for which the series is valid.
  22. 4(b)(ii)7 marksUse Maclaurin's theorem to obtain the first THREE non-zero terms in the power series expansion in x of \sin 2x.
  23. 4(b)(iii)4 marksHence, or otherwise, obtain the first THREE non-zero terms in the power series expansion in x of \ln(1 + \sin 2x).
  24. 5(a)5 marksA committee of 4 persons is to be chosen from 8 persons, including Mr Smith and his wife. Mr Smith will not join the committee without his wife, but his wife will join the committee without him. Calculate the number of…
  25. 5(b)(i)4 marksthe numbers on BOTH balls are even
  26. 5(b)(ii)4 marksthe number on one ball is odd and the number on the other ball is even.
  27. 5(c)(i)7 marksFind complex numbers u = x + \mathrm{i}y such that x and y are real numbers and u^2 = -15 + 8\mathrm{i}.
  28. 5(c)(ii)5 marksHence, or otherwise, solve for z the equation z^2 - (3 + 2\mathrm{i})z + (5 + \mathrm{i}) = 0.
  29. 6(a)10 marksSolve for x the equation \begin{vmatrix} x - 3 & 1 & -1 \\ 1 & x - 5 & 1 \\ -1 & 1 & x - 3 \end{vmatrix} = 0.
  30. 6(b)(i)a)4 marksFind \mathbf{A}\mathbf{B}.
  31. 6(b)(i)b)3 marksHence deduce the inverse \mathbf{A}^{-1} of the matrix \mathbf{A}.
  32. 6(b)(ii)a)3 marksExpress the system in the form \mathbf{A}\mathbf{x} = \mathbf{b}, where \mathbf{A} is a matrix and \mathbf{x} and \mathbf{b} are column vectors.
  33. 6(b)(ii)b)5 marksHence, or otherwise, solve the system of equations.

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