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CAPE Pure Mathematics Unit 2 · May/June 2022 · Paper 2

26 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)7 marksUsing DeMoivre's theorem, prove that sin 5θ / sin θ = 5 - 20 sin² θ + 16 sin⁴ θ.
  2. 1(b)7 marksGiven that 3 + 5i is a root of the quadratic equation z² + pz + q = 0, determine the values of p, q ∈ R.
  3. 1(c)5 marksA complex number, z, is such that arg(z-2) = π/2 and arg(z) = π/3. Determine the complex number z.
  4. 1(d)6 marksGiven that (x² + y²)³ = ax²y, use implicit differentiation to determine dy/dx.
  5. 2(a)(i)9 marksUse partial fractions to show that x⁴ / (x⁴ - 1) = 1 + 1/(4(x-1)) - 1/(4(x+1)) - 1/(2(x²+1)).
  6. 2(a)(ii)6 marksHence, or otherwise, determine ∫ x⁴ / (x⁴ - 1) dx.
  7. 2(b)(i)5 marksShow that if I_n = ∫ (1 - x)ⁿeᵃˣ dx for n ≥ 1, then I_n = (1 - x)ⁿeᵃˣ/a + n/a I_n-1.
  8. 2(b)(ii)5 marksHence, determine ∫ (1 - x)²eᵃˣ dx.
  9. 3(a)(i)9 marksUse mathematical induction to prove that 3/4 + 5/36 + ... + (2n-1)/(n²(n-1)²) = 1 - 1/n², n ≥ 1.
  10. 3(a)(ii)4 marksHence, or otherwise, calculate S₃₀ - S₁₀.
  11. 3(b)7 marksCalculate the sum to infinity of the series Σ_{r=2}^{∞} 10/(r² - 1).
  12. 3(c)5 marksDetermine the Taylor series expansion of x sin(x/2) about x = π, up to and including the first THREE non-zero terms.
  13. 4(a)(i)6 marksShow that the binomial expansion of (1 + 5x)^(1/3) up to and including the term in x³ is 1 + x - 2x² + 6x³.
  14. 4(a)(ii)5 marksHence, by letting x = -1/32, compute an estimate of ³√27.
  15. 4(b)(i)3 marksUse the Intermediate Value Theorem to prove that 4eˣ + 2x² - 5 = 0 has a root in the interval [0, 1].
  16. 4(b)(ii)6 marksUse four iterations of the interval bisection method to calculate an approximation of the root of 4eˣ + 2x² - 5 = 0 in the interval [0, 1].
  17. 4(b)(iii)5 marksUse three iterations of the linear interpolation method to approximate the root of 4eˣ + 2x² - 5 = 0 in the interval (-2, -1).
  18. 5(a)(i)4 marksShow that |P| = 183.
  19. 5(a)(ii)5 marksHence, or otherwise, show that the adjoint of P is adj(P) = [[-162, 59, 25], [207, -72, -15], [42, -4, -11]].
  20. 5(a)(iii)8 marksSolve the system of linear equations [[4, 3, 5], [9, 4, 15], [12, 10, -3]] [[x], [y], [z]] = [[11], [13], [4]].
  21. 5(b)4 marksAlex has five blue marbles, four green marbles and three red marbles. In how many ways can he arrange four marbles in a row, if the marbles of any given colour are identical?
  22. 5(c)4 marksLet A and B be two events such that P(A) = 1/2, P(B) = 1/4 and P(A ∩ B) = 1/8. Calculate the value of P(A' ∩ B').
  23. 6(a)(i)2 marksDetermine P(A|B).
  24. 6(a)(ii)2 marksCalculate the probability that a member of the population, selected at random, is colour-blind.
  25. 6(b)9 marksSolve the initial value problem x² dy/dx + 2xy = cos x, where y(π) = 0.
  26. 6(c)12 marksDetermine the general solution of the differential equation y'' - 7y' + 12y = sin x - cos x.

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