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CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2

37 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)a)5 marksFind \frac{\mathrm{d}y}{\mathrm{d}x} and \frac{\mathrm{d}^2y}{\mathrm{d}x^2}.
  2. 1(a)(i)b)2 marksFind the x-coordinates of the points at which \frac{\mathrm{d}y}{\mathrm{d}x} = 0.
  3. 1(a)(i)c)2 marksFind the x-coordinates of the points at which \frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 0.
  4. 1(a)(ii)7 marksHence, determine if the coordinates identified in (i) b) and c) above are at the maxima, minima or points of inflection of y = x^2 e^x.
  5. 1(b)(i)6 marksFind the gradient of a tangent to the curve at the point with parameter t.
  6. 1(b)(ii)3 marksFind the equation of the tangent at the point where t = \frac{1}{2}.
  7. 2(a)(i)7 marksExpress \frac{x^2 - 3x}{(x - 1)(x^2 + 1)} in partial fractions.
  8. 2(a)(ii)5 marksHence, find \int \frac{x^2 - 3x}{x^3 - x^2 + x - 1} \,\mathrm{d}x.
  9. 2(b)(i)2 marksGiven that \sin A \cos B - \cos A \sin B = \sin(A - B), show that \cos 3x \sin x = \sin 3x \cos x - \sin 2x.
  10. 2(b)(ii)7 marksProve that (m + 3) I_m = m J_{m-1} - \cos^m x \cos 3x.
  11. 2(b)(iii)2 marksHence, by setting m = 1, prove that 4 \int_0^{\frac{\pi}{4}} \cos x \sin 3x \,\mathrm{d}x = \int_0^{\frac{\pi}{4}} \sin 2x \,\mathrm{d}x + \frac{3}{2}.
  12. 2(b)(iv)2 marksEvaluate \int_0^{\frac{\pi}{4}} \sin 2x \,\mathrm{d}x.
  13. 3(a)(i)5 marksCalculate the first term, a, and the common ratio, r.
  14. 3(a)(ii)4 marksHence, calculate n if S_n = 177\,146.
  15. 3(b)(i)2 marksExpress, in terms of r, the r^{\text{th}} term, u_r, of the sequence.
  16. 3(b)(ii)7 marksProve, by mathematical induction, that \sum_{r=1}^n u_r = \frac{1}{6}n(n + 1)(2n + 7), \forall n \in \mathbb{N}.
  17. 3(c)(i)5 marksUse Maclaurin's Theorem to find the first three non-zero terms in the power series expansion of \cos 2x.
  18. 3(c)(ii)2 marksHence, or otherwise, obtain the first two non-zero terms in the power series expansion of \sin^2 x.
  19. 4(a)(i)1 markExpress \binom{n}{r} in terms of factorials.
  20. 4(a)(ii)3 marksHence, show that \binom{n}{r} = \binom{n}{n - r}.
  21. 4(a)(iii)5 marksFind the coefficient of x^4 in \left(x^2 - \frac{3}{x}\right)^8.
  22. 4(a)(iv)8 marksUsing the identity (1 + x)^{2n} = (1 + x)^n (1 + x)^n, show that \binom{2n}{n} = c_0^2 + c_1^2 + c_2^2 + \dots + c_{n-1}^2 + c_n^2, where c_r = \binom{n}{r}.
  23. 4(b)(i)2 marksUse the intermediate value theorem to determine whether the equation f(x) has any roots in the interval [0.2, 2].
  24. 4(b)(ii)6 marksUsing x_1 = 0.6 as a first approximation of a root T of f(x), execute FOUR iterations of the Newton–Raphson method to obtain a second approximation, x_2, of T.
  25. 5(a)(i)4 marksDetermine how many such numbers can be formed if each digit appears at most once.
  26. 5(a)(ii)3 marksDetermine how many such numbers can be formed if there is no restriction on the number of times a digit may appear.
  27. 5(b)(i)3 marksFind the probability that the committee consists entirely of Jamaicans.
  28. 5(b)(ii)6 marksFind the number of ways in which the committee can be formed, given the restriction that there are as many Tobagonians on the committee as there are Guyanese.
  29. 5(c)(i)3 marksFind the matrix \mathbf{B}, where \mathbf{B} = \mathbf{A}^2 - 3\mathbf{A} - \mathbf{I}.
  30. 5(c)(ii)1 markShow that \mathbf{AB} = -9\mathbf{I}.
  31. 5(c)(iii)2 marksHence, find the inverse, \mathbf{A}^{-1}, of \mathbf{A}.
  32. 5(c)(iv)3 marksSolve the system of linear equations \mathbf{B} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}.
  33. 6(a)(i)6 marksDraw the points A and B on an Argand diagram.
  34. 6(a)(ii)5 marksHence, or otherwise, show that the argument of \frac{(1 + \sqrt{2} + i)}{1 - i} is EXACTLY \frac{3\pi}{8}.
  35. 6(b)(i)3 marksFind ALL complex numbers, z, such that z^2 = i.
  36. 6(b)(ii)5 marksHence, find ALL complex roots of the equation z^2 - (3 + 5i)z - (4 - 7i) = 0.
  37. 6(c)6 marksUse de Moivre's theorem to show that \cos 6\theta = \cos^6 \theta - 15\cos^4 \theta \sin^2 \theta + 15\cos^2 \theta \sin^4 \theta - \sin^6 \theta.

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