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CAPE Pure Mathematics Unit 2 · May/June 2026 · Paper 2

39 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)4 marksExpress the complex number 5-3i / 4+2i in the form x + yi where x and y are real numbers.
  2. 1(b)(i)a)2 marksCalculate the EXACT value of |z|.
  3. 1(b)(i)b)2 marksCalculate the EXACT value of arg z.
  4. 1(b)(ii)4 marksHence, use de Moivre's theorem to determine the value of z^12.
  5. 1(c)6 marksDetermine an expression for dy/dx in terms of x and y.
  6. 1(d)7 marksShow that y = -x + π/2 is the tangent to the curve y = ln(1 + sin 2x) at the point where x = π/2.
  7. 2(a)(i)4 marksApply the trapezium rule with THREE equal intervals to estimate the value of ∫(from 0 to π/3) tan x dx.
  8. 2(a)(ii)4 marksUsing an appropriate trigonometric identity, determine the EXACT value of ∫(from 0 to π/3) tan x dx.
  9. 2(b)7 marksUse the substitution u = cos^(-1)(x/2) to show that ∫(from 0 to 1) (cos^(-1)(x/2))/√(4-x^2) dx = 5π^2/72.
  10. 2(c)(i)6 marksShow that I_n = (n-1)/n * I_(n-2).
  11. 2(c)(ii)a)2 marksVerify that I_0 = π/2.
  12. 2(c)(ii)b)2 marksHence, determine the value of I_4.
  13. 3(a)(i)2 marksState the values of the first FOUR terms of the sequence, a_1, a_2, a_3, and a_4.
  14. 3(a)(ii)2 marksShow that the sequence converges.
  15. 3(b)(i)2 marksExpress the series 4 + 4^2 + 4^3 + ... + 4^n using summation notation.
  16. 3(b)(ii)8 marksProve by mathematical induction that 4 + 4^2 + 4^3 + ... + 4^n = 4/3 (4^n - 1) for all positive integers n.
  17. 3(c)(i)3 marksDetermine the values of A and B such that 4/((2r+1)(2r+3)) = A/(2r+1) + B/(2r+3).
  18. 3(c)(ii)5 marksHence, use the method of differences to show that Σ(from r=1 to n) 4/((2r+1)(2r+3)) = 2(1/3 - 1/(2n+3)).
  19. 3(c)(iii)3 marksHence, calculate Σ(from r=1 to ∞) 4/((2r+1)(2r+3)).
  20. 4(a)3 marksObtain the Maclaurin series expansion of f(x) = e^(2x) up to and including the term in x^4.
  21. 4(b)6 marksObtain the binomial expansion of (8 + x)^(1/3) in ascending powers of x, up to and including the term in x^2. State the values of x for which the expansion is valid.
  22. 4(c)(i)3 marksUsing the intermediate value theorem, show that cos x = xe^(-x) has a root between x = 1 and x = 1.5.
  23. 4(c)(ii)6 marksHence, use the method of interval bisection to determine, correct to 1 decimal place, the approximate root of cos x = xe^(-x) which lies in the interval (1, 1.5).
  24. 4(d)(i)4 marksUse the Newton-Raphson formula to show that for n ≥ 1, with a given initial estimate x_n, x_(n+1) = (3x_n^4 + 13)/(4x_n^3 + 1).
  25. 4(d)(ii)3 marksHence, or otherwise, use the Newton-Raphson method with the initial estimate x_1 = 2 to calculate, to 2 decimal places, a new estimate, x_2, of the root.
  26. 5(a)(i)2 marksDetermine the sample space of the possible outcomes of the experiment.
  27. 5(a)(ii)2 marksCalculate P(H|F).
  28. 5(a)(iii)3 marksState, with reason, whether H and F are independent events.
  29. 5(b)(i)2 marksCalculate the number of ways of arranging 3 letters from the first 6 letters of the alphabet.
  30. 5(b)(ii)3 marksCalculate the number of ways of arranging ALL the letters of the word SUCCESS.
  31. 5(c)4 marksDetermine in how many ways a sub-committee may be formed if 2 Mathematics teachers and 3 English teachers are selected in no particular order.
  32. 5(d)4 marksEvaluate the determinant of A.
  33. 5(e)5 marksDetermine whether the system is consistent.
  34. 6(a)4 marksCalculate the value of k for which the matrix is singular.
  35. 6(b)(i)8 marksSolve the differential equation to obtain the general solution.
  36. 6(b)(ii)2 marksHence, given that y = 1 when x = 0, determine the particular solution.
  37. 6(c)(i)5 marksObtain the complementary function of the differential equation.
  38. 6(c)(ii)5 marksDetermine a particular integral of the differential equation.
  39. 6(c)(iii)1 markHence, state the general solution of the differential equation.

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