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CAPE Pure Mathematics Unit 2 · May/June 2021 · 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)6 marksShow that the complex number z₁/z₂ = (√5)/6 e^(i7π/12).
  2. 1(a)(ii)3 marksHence, without using a calculator, determine the value of (z₁/z₂)².
  3. 1(b)4 marksOne root of a quadratic equation is given as 4 – 7i. Determine the quadratic equation with real coefficients which has the root 4 – 7i.
  4. 1(c)6 marksShow that the derivative of sin⁻¹(cos x / (1 + sin x)) with respect to x is -1 / √( (1+sin x)² - cos² x ).
  5. 1(d)6 marksA curve is defined parametrically by x = (3 – 2t)², y = t³ – 2t. Determine the equation of the tangent to the curve at the point where t = 2.
  6. 2(a)8 marksUsing the substitution eˣ = 3 cos θ, or otherwise, determine ∫ eˣ √(9-e²ˣ) dx.
  7. 2(b)(i)12 marksUse partial fractions to prove that (2x + 1) / (2x³ - x² + 8x - 4) = 8 / (17(2x - 1)) - (4x - 15) / (17(x² + 4)).
  8. 2(b)(ii)5 marksHence, find ∫ (2x + 1) / (2x³ - x² + 8x - 4) dx.
  9. 3(a)(i)3 marksDetermine the THIRD partial sum, S₃, of the series.
  10. 3(a)(ii)7 marksShow that Σ (from k=1 to n) 8 / (4k² - 1) = 8n / (2n + 1).
  11. 3(b)(i)8 marksDetermine the Taylor series expansion of e^(cos x) about x = π/2 up to the term in x³.
  12. 3(b)(ii)3 marksUse the series expansion to approximate e^(cos π) correct to 2 decimal places.
  13. 3(c)4 marksThe first and fifth terms of a geometric progression are 16 and 9, respectively. Determine the THIRD term of the progression.
  14. 4(a)4 marksDetermine the first three terms of the expansion of (1 – 8x)^(1/2).
  15. 4(b)6 marksHence, by letting x = 1/100, determine √23.
  16. 4(c)(i)4 marksUse the Intermediate Value Theorem to show that the equation 4 sin 2x + x³ – 3 = 0 has a root in the interval [0, 1].
  17. 4(c)(ii)6 marksUse three iterations of the interval bisection method to obtain an approximation of the root.
  18. 4(d)5 marksUse the iteration x_(n+1) = (sin x_n + 2) / 3 and the initial approximation x = 1 to calculate an approximate value of the root of f(x) = sin x – 3x + 2, correct to 2 decimal places.
  19. 5(a)(i)3 marksRepresent the possible outcomes of a single trial of this experiment on a tree diagram.
  20. 5(a)(ii)4 marksDetermine the probability that a golf ball is drawn on the first trial of the experiment.
  21. 5(b)(i)5 marksIn how many ways can the group be seated in the cars if two particular persons refuse to travel in the same car?
  22. 5(b)(ii)5 marksOn a table, there is space for 10 books out of a total of 16 available books. However, a Bible and a book of ghost stories must go at the ends. In how many ways can the books be arranged on the table?
  23. 5(c)(i)4 marksBy reducing the matrix to row echelon form, show that the system has a finite set of solutions.
  24. 5(c)(ii)4 marksHence, solve the system of linear equations.
  25. 6(a)(i)10 marksShow that the general solution of the differential equation is y = 5/3 + 1/3 cos 2x + C cosec x, where C is a constant.
  26. 6(a)(ii)3 marksHence, determine the particular solution given that y(π/2) = 0.
  27. 6(b)(i)7 marksDetermine the general solution of the differential equation y'' + 2y' + 5y = 0.
  28. 6(b)(ii)5 marksHence, determine the solution of the boundary value problem y'' + 2y' + 5y = 0 with y(0) = 1, y'(π) = 2.

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