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CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · 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)4 marksDifferentiate with respect to x: e^{4x} \cos(\pi x)
  2. 1(a)(ii)4 marksDifferentiate with respect to x: \ln\left(\frac{x^2 + 1}{\sqrt{x}}\right)
  3. 1(b)5 marksGiven y = 3^{-x}, show, by using logarithms, that \frac{\mathrm{d}y}{\mathrm{d}x} = -3^{-x} \ln 3.
  4. 1(c)(i)7 marksExpress in partial fractions: \frac{2x^2 - 3x + 4}{(x - 1)(x^2 + 1)}
  5. 1(c)(ii)5 marksHence, find \int \frac{2x^2 - 3x + 4}{(x - 1)(x^2 + 1)} \,\mathrm{d}x.
  6. 2(a)5 marksSolve the differential equation \frac{\mathrm{d}y}{\mathrm{d}x} + y = e^{2x}.
  7. 2(b)5 marksThe gradient at the point (x, y) on a curve is given by \frac{\mathrm{d}y}{\mathrm{d}x} = e^{4x}. Given that the curve passes through the point (0, 1), find its equation.
  8. 2(c)7 marksEvaluate \int_1^e x^2 \ln x \,\mathrm{d}x, writing your answer in terms of e.
  9. 2(d)(i)3 marksUse the substitution v = 1 - u to find \int \frac{\mathrm{d}u}{\sqrt{1 - u}}.
  10. 2(d)(ii)5 marksHence, or otherwise, use the substitution u = \sin x to evaluate \int_0^{\pi/2} \sqrt{1 + \sin x} \,\mathrm{d}x.
  11. 3(a)(i)3 marksA sequence \{u_n\} is defined by the recurrence relation u_{n+1} = u_n + n, u_1 = 3, n \in \mathbb{N}. State the first FOUR terms of the sequence.
  12. 3(a)(ii)8 marksProve by mathematical induction, or otherwise, that u_n = \frac{n^2 - n + 6}{2}.
  13. 3(b)6 marksA GP with first term a and common ratio r has sum to infinity 81 and the sum of the first four terms is 65. Find the values of a and r.
  14. 3(c)(i)3 marksWrite down the first FIVE terms in the power series expansion of \ln(1 + x), stating the range of values of x for which the series is valid.
  15. 3(c)(ii)a)2 marksUsing the result from (c)(i) above, obtain a similar expansion for \ln(1 - x).
  16. 3(c)(ii)b)3 marksHence, prove that \ln\left(\frac{1 + x}{1 - x}\right) = 2\left(x + \frac{1}{3}x^3 + \frac{1}{5}x^5 + \dots\right).
  17. 4(a)(i)3 marksShow that the function f(x) = x^3 - 3x + 1 has a root \alpha in the closed interval [1, 2].
  18. 4(a)(ii)5 marksUse the Newton-Raphson method to show that if x_1 is a first approximation to \alpha in the interval [1, 2], then a second approximation to \alpha in the interval [1, 2] is given by x_2 = \frac{2x_1^3 - 1}{3x_1^2 - 3}.
  19. 4(b)(i)4 marksUse the binomial theorem or Maclaurin's theorem to expand (1 + x)^{-1/2} in ascending powers of x as far as the term in x^3, stating the values of x for which the expansion is valid.
  20. 4(b)(ii)4 marksObtain a similar expansion for (1 - x)^{1/2}.
  21. 4(b)(iii)5 marksProve that if x is so small that x^3 and higher powers of x can be neglected, then \sqrt{\frac{1 - x}{1 + x}} \approx 1 - x + \frac{1}{2}x^2.
  22. 4(b)(iv)4 marksHence, by taking x = \frac{1}{17}, show, without using calculators or tables, that \sqrt{2} is approximately equal to \frac{1635}{1156}.
  23. 5(a)(i)8 marksIn how many ways can this committee be selected so that the committee includes AT LEAST ONE former batsman?
  24. 5(a)(ii)3 marksIn how many ways can this committee be selected so that the committee includes AT LEAST ONE batsman and ONE bowler?
  25. 5(b)(i)a)2 marksDetermine the matrix A - B.
  26. 5(b)(i)b)3 marksDetermine the matrix AM.
  27. 5(b)(ii)3 marksDeduce from (i) b) above the inverse A^{-1} of the matrix A.
  28. 5(b)(iii)6 marksFind the matrix X such that AX + B = A.
  29. 6(a)(i)4 marksExpress the complex number \frac{2 - 3i}{5 - i} in the form \lambda(1 - i).
  30. 6(a)(ii)1 markState the value of \lambda.
  31. 6(a)(iii)5 marksVerify that \left(\frac{2 - 3i}{5 - i}\right)^4 is a real number and state its value.
  32. 6(b)(i)7 marksShow that z + \bar{z} = 6 z\bar{z}.
  33. 6(b)(ii)8 marksShow that as t varies, T lies on a circle, and state the coordinates of the centre of this circle.

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