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CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2

28 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)5 marksDifferentiate, with respect to x, y = \ln(x^2 + 4) - x \tan^{-1}\left(\frac{x}{2}\right).
  2. 1(a)(ii)7 marksA curve is defined parametrically as x = a\cos^3 t, y = a\sin^3 t. Show that the tangent at the point P(x, y) is the line y\cos t + x\sin t = a\sin t\cos t.
  3. 1(b)(i)2 marksDetermine the nature of the roots of the equation.
  4. 1(b)(ii)4 marksExpress \alpha and \beta in the form r e^{i\theta}, where r is the modulus and \theta is the argument, where -\pi < \theta \le \pi.
  5. 1(b)(iii)4 marksUsing de Moivre's theorem, or otherwise, compute \alpha^3 + \beta^3.
  6. 1(b)(iv)3 marksHence, or otherwise, obtain the quadratic equation whose roots are \alpha^3 and \beta^3.
  7. 2(a)(i)3 marksShow that F_n(x) = x(\ln x)^n - n F_{n-1}(x).
  8. 2(a)(ii)7 marksHence, or otherwise, show that F_3(2) - F_3(1) = 2(\ln 2)^3 - 6(\ln 2)^2 + 12\ln 2 - 6.
  9. 2(b)(i)7 marksBy decomposing \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} into partial fractions, show that \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} = \frac{1}{y^2 + 1} + \frac{2y}{(y^2 + 1)^2}.
  10. 2(b)(ii)8 marksHence, find \int_0^1 \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} \, dy.
  11. 3(a)(i)8 marksProve, by mathematical induction, that for n \in \mathbb{N}, S_n = 1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \dots + \frac{1}{2^{n-1}} = 2 - \frac{1}{2^{n-1}}.
  12. 3(a)(ii)3 marksHence, or otherwise, find \lim_{n \to \infty} S_n.
  13. 3(b)14 marksFind the Maclaurin expansion for f(x) = (1 + x)^2 \sin x up to and including the term in x^3.
  14. 4(a)(i)5 marksFor the binomial expansion of (2x + 3)^{20}, show that the ratio of the term in x^6 to the term in x^7 is \frac{3}{4x}.
  15. 4(a)(ii)a)4 marksDetermine the FIRST THREE terms of the binomial expansion of (1 + 2x)^{10}.
  16. 4(a)(ii)b)3 marksHence, obtain an estimate for (1.01)^{10}.
  17. 4(b)6 marksShow that \frac{n!}{(n - r)!r!} + \frac{n!}{(n - r + 1)!(r - 1)!} = \frac{(n + 1)!}{(n - r + 1)!r!}.
  18. 4(c)(i)3 marksShow that the function f(x) = -x^3 + 3x + 4 has a root in the interval [1, 3].
  19. 4(c)(ii)4 marksBy taking x_1 = 2.1 as a first approximation of the root in the interval [1, 3], use the Newton–Raphson method to obtain a second approximation, x_2, in the interval [1, 3].
  20. 5(a)(i)7 marksFive teams are to meet at a round table. Each team consists of two members AND one leader. How many seating arrangements are possible if each team sits together with the leader of the team in the middle?
  21. 5(a)(ii)a)4 marksGiven that 40% of the individuals used red and 50% used blue, calculate the probability that an individual used BOTH colours.
  22. 5(a)(ii)b)2 marksDetermine the TOTAL number of individuals that participated in the experiment.
  23. 5(b)(i)4 marksDetermine the range of values of x for which A^{-1} exists.
  24. 5(b)(ii)4 marksGiven that \det(AB) = -21, show that x = 3.
  25. 5(b)(iii)4 marksHence, obtain A^{-1}.
  26. 6(a)(i)10 marksShow that the general solution of the differential equation y' + y\tan x = \sec x is y = \sin x + C\cos x.
  27. 6(a)(ii)4 marksHence, obtain the particular solution where y = \frac{2}{\sqrt{2}} and x = \frac{\pi}{4}.
  28. 6(b)11 marksA differential equation is given as y'' - 5y' = x e^{5x}. Given that a particular solution is y_p(x) = Ax^2 e^{5x} + Bx e^{5x}, solve the differential equation.

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