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CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2

30 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 marksDetermine the gradient of the curve at the point (1/2, 1/2).
  2. 1(a)(ii)4 marksHence, or otherwise, determine the x and y intercepts of the tangent to the curve at the point (1/2, 1/2).
  3. 1(b)3 marksLet the function f(x, y) = sin(kx) sin(aky). Determine d^2 f(x, y) / (dx dy).
  4. 1(c)5 marksUse De Moivre's theorem to show that sin(5θ) = sin^5(θ) - 10 sin^3(θ) cos^2(θ) + 5 cos^4(θ) sin(θ).
  5. 1(d)(i)3 marksWrite the complex number z = (1 - i) in the form r e^(iθ), where r = |z| and θ = arg(z).
  6. 1(d)(ii)5 marksHence, show that (1 - i)^9 = 16(1 - i).
  7. 2(a)(i)8 marksDetermine the integral of x^5 cos(x^3) dx.
  8. 2(a)(ii)5 marksDetermine the integral of e^(2x) / sqrt(1 - e^(4x)) dx.
  9. 2(b)(i)8 marksUse partial fractions to show that (x^4 + 1) / (x(x^2 + 1)^2) = 1/x - 2x / (x^2 + 1)^2.
  10. 2(b)(ii)4 marksHence, determine the integral of (x^4 + 1) / (x(x^2 + 1)^2) dx.
  11. 3(a)(i)2 marksState the third term, a_3, of the sequence.
  12. 3(a)(ii)8 marksUse mathematical induction to prove that a_n is increasing and bounded above by 3, so a_n < a_(n+1) and a_n <= 3 for all n in N.
  13. 3(b)(i)8 marksLet f(x) = e^(-x^2). By calculating the first three non-zero terms and assuming the pattern continues, show that the Maclaurin series expansion of f(x) may be expressed as sum_(k=0)^infinity ((-1)^k x^(2k)) / k!.
  14. 3(b)(ii)3 marksHence, or otherwise, determine the values of x for which the expansion is valid.
  15. 3(c)4 marksDetermine the sum of the series sum_(n=1)^infinity (sin(1/n) - sin(1/(n+1))).
  16. 4(a)4 marksDetermine the coefficient of the term in x^7 in the expansion of (x^2 - 3/x)^8.
  17. 4(b)6 marksBy expressing ^n C_r and ^n C_(r-1) in terms of factorials, show that ^n C_r + ^n C_(r-1) = ^(n+1) C_r.
  18. 4(c)(i)3 marksUse the intermediate value theorem to show that the equation 4 cos(x) - x^3 + 2 = 0 has a root in the interval (1, 1.5).
  19. 4(c)(ii)8 marksUse linear interpolation to approximate the value of the root of the equation 4 cos(x) - x^3 + 2 = 0 in the interval (1, 1.5), correct to two decimal places.
  20. 4(d)4 marksThe equation 3e^x = 1 - 2 ln(x) has a root in the interval (0, 1). Taking x_1 = 0.2 as the first approximation, use the Newton-Raphson method to find a second approximation, x_2, of the root in the interval (0, 1).
  21. 5(a)(i)3 marksCalculate P(A intersect B).
  22. 5(a)(ii)3 marksDetermine whether events A and B are independent. Justify your response.
  23. 5(b)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…
  24. 5(c)5 marksHow many odd numbers greater than 500 000 can be made from the digits 2, 3, 4, 5, 6, 7 without repetitions?
  25. 5(d)(i)5 marksBy finding AB, deduce that A^(-1) = 1/88 B.
  26. 5(d)(ii)4 marksHence, or otherwise, solve the system of equations given by [[5, -2, 3], [0, 3, -4], [2, 0, 6]] [[x], [y], [z]] = [[7], [11], [-6]].
  27. 6(a)(i)9 marksShow that the general solution of the differential equation is y = (1/2) sec(x) - cos(x) + C sec(x).
  28. 6(a)(ii)2 marksHence, or otherwise, solve the initial value problem y' cos(x) = y sin(x) + sin(2x), y(0) = 0.
  29. 6(b)(i)4 marksDetermine the solution of the complementary equation y'' + 2y' + y = 0.
  30. 6(b)(ii)10 marksGiven that the particular solution has the form y_p = (A x^3 + B x^2) e^(-x), or otherwise, determine the general solution of the differential equation.

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