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CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2

31 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)3 marksCopy the diagram and on the same axes, sketch the graph of g(x) = ln x.
  2. 1(a)(ii)3 marksDescribe clearly the relationship between f(x) = e^x and g(x) = ln x.
  3. 1(a)(iii)a)1 markUsing a calculator, find the value of r.
  4. 1(a)(iii)b)2 marksUsing a calculator, find the value of p.
  5. 1(b)3 marksGiven that log_a(bc) = x, log_b(ca) = y, log_c(ab) = z and a != b != c, show that a^x b^y c^z = (abc)^2.
  6. 1(c)8 marksFind the values of x in R for which e^x + 3e^(-x) = 4.
  7. 2(a)(i)4 marksFind dy/dx in terms of t.
  8. 2(a)(ii)2 marksFind the gradient of the normal to the curve at the point t = 2.
  9. 2(b)(i)7 marksExpress (2x + 1) / (x^2(x + 1)) in the form A/x + B/x^2 + C/(x + 1), where A, B, and C are constants.
  10. 2(b)(ii)7 marksHence, evaluate the integral from 1 to 2 of (2x + 1) / (x^2(x + 1)) dx.
  11. 3(a)(i)5 marksUse the fact that 1/r - 1/(r + 1) = 1 / (r(r + 1)) to show that S_n = sum_{r=1}^n 1/(r(r + 1)) = 1 - 1/(n + 1).
  12. 3(a)(ii)1 markDeduce that as n -> infinity, S_n -> 1.
  13. 3(b)10 marksThe common ratio, r, of a geometric series is given by r = 5x / (4 + x^2). Find ALL the values of x for which the series converges.
  14. 3(c)(i)2 marksShow that sum_{r=0}^n binom(n, r) = 2^n.
  15. 3(c)(ii)2 marksShow that sum_{r=0}^n binom(n, r) (-1)^r = 0.
  16. 4(a)(i)4 marksShow that f is everywhere strictly decreasing.
  17. 4(a)(ii)4 marksShow that the equation f(x) = 0 has a real root, alpha, in the closed interval [1, 2].
  18. 4(a)(iii)4 marksShow that alpha is the only real root of the equation f(x) = 0.
  19. 4(b)8 marksIf x_n is the nth approximation to alpha, use the Newton-Raphson method to show that the (n + 1)th approximation x_{n+1} is given by x_{n+1} = (2x_n^3 + 6) / (3x_n^2 + 4).
  20. 5(a)(i)3 marksCopy and complete the diagram to represent the event space.
  21. 5(a)(ii)a)2 marksFind the probability that a customer chosen at random who had purchased premium gasoline requested a check for engine oil.
  22. 5(a)(ii)b)2 marksFind the probability that a customer chosen at random who had requested a check of the brake fluid purchased regular gasoline.
  23. 5(a)(ii)c)2 marksFind the probability that a customer chosen at random who had requested a check of the engine oil purchased regular gasoline.
  24. 5(b)(i)3 marksCalculate the total number of ways of choosing the three balls.
  25. 5(b)(ii)3 marksCalculate the probability that ONE ball of EACH colour is drawn.
  26. 5(b)(iii)5 marksCalculate the probability that ALL THREE balls drawn are of the SAME colour.
  27. 6(a)10 marksFind the values of x for which the determinant of the 3x3 matrix with rows [x, 1, 2], [1, x, 2], and [2, 1, x] is equal to 0.
  28. 6(b)(i)2 marksObtain an expression for the number of visitors on the nth day.
  29. 6(b)(ii)3 marksFind the total number of visitors for the first n days.
  30. 6(b)(iii)3 marksThe exhibition closed after 10 days. Determine how many people visited during the period for which it was opened.
  31. 6(b)(iv)2 marksIf the exhibition had been kept opened indefinitely, what would be the maximum number of visitors?

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