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CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 2

32 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)5 marksUse implicit differentiation to show that dy/dx = -(8x + 3y² + 7) / (3(1 + 2xy)).
  2. 1(a)(ii)5 marksShow that 6(∂f(x,y)/∂y) - 10 = (∂²f(x,y)/∂y²) + (∂²f(x,y)/∂y∂x) + (∂²f(x,y)/∂x²).
  3. 1(b)6 marksUse de Moivre's theorem to prove that sin 5x = 16 sin⁵ x - 20 sin³ x + 5 sin x.
  4. 1(c)(i)3 marksWrite the complex number z = (-1 + √3 i)⁷ in the form re^(iθ), where r = |z| and θ = arg z.
  5. 1(c)(ii)6 marksHence, prove that (-1 + √3 i)⁷ = 64 (-1 + √3 i).
  6. 2(a)(i)3 marksShow that F_n(x) = x (ln x)ⁿ - nF_(n-1)(x).
  7. 2(a)(ii)7 marksHence, or otherwise, show that F₂(2) - F₂(1) = 2 (ln 2)³ - 6 (ln 2)² + 12 ln 2 - 6.
  8. 2(b)(i)7 marksBy expressing (y² + 2y + 1) / (y⁴ + 2y² + 1) as partial fractions, show that (y² + 2y + 1) / (y⁴ + 2y² + 1) = 1 / (y² + 1) + 2y / (y² + 1)².
  9. 2(b)(ii)8 marksHence, or otherwise, evaluate ∫(y² + 2y + 1) / (y⁴ + 2y² + 1) dy.
  10. 3(a)3 marksDetermine the coefficient of the term in x³ in the binomial expansion of (3x + 2)⁵.
  11. 3(b)(i)4 marksShow that the binomial expansion of (1 + x)^(1/2) + (1 - x)^(1/2) up to the term in x² is 2 - (3/16)x².
  12. 3(b)(ii)3 marksHence, by letting x = 1/16, compute an approximation of √17 + √15, correct to 4 decimal places.
  13. 3(c)(i)3 marksShow that h(x) = 0 has a root on the interval [0, 1].
  14. 3(c)(ii)6 marksUse the iteration x_(n+1) = (x_n² + 1) / (2x_n + 1) with initial estimate x₁ = 0.7 to estimate the root of h(x) = 0, correct to 2 decimal places.
  15. 3(d)6 marksUse the Newton-Raphson method with initial estimate x₀ = 5.5 to approximate the root of g(x) = sin 3x in the interval [5, 6], correct to 2 decimal places.
  16. 4(a)(i)8 marksObtain the Maclaurin series expansion for g up to the term in x⁴.
  17. 4(a)(ii)2 marksHence, estimate g(2).
  18. 4(b)(i)2 marksExpress the nᵗʰ partial sum S_n of the series using sigma notation.
  19. 4(b)(ii)1 markHence, calculate S₂₀ - S₁₈.
  20. 4(b)(iii)4 marksGiven that Σ(n=1 to ∞) (1/n²) converges, show that S_n diverges.
  21. 4(c)8 marksUse the method of induction to prove that Σ(r=1 to n) r(r-1) = n(n² - 1) / 3.
  22. 5(a)(i)4 marksHow many numbers made up of five digits can be made from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 if each number contains exactly one even digit and no digit is repeated?
  23. 5(a)(ii)4 marksDetermine the probability that the number formed in (a)(i) is less than 30 000.
  24. 5(b)(i)4 marksDetermine the value of x for which A⁻¹ does NOT exist.
  25. 5(b)(ii)4 marksGiven that det(AB) = -10, show that x = 2.
  26. 5(b)(iii)4 marksHence, obtain A⁻¹.
  27. 5(c)(i)3 marksGiven that 40% of the individuals selected green and 50% selected blue, calculate the probability that an individual selected BOTH colours.
  28. 5(c)(ii)2 marksDetermine the total number of individuals who participated in the experiment.
  29. 6(a)(i)5 marksShow that the general solution of the differential equation is y = (c/x) - (2/x) cos x, where c is a constant.
  30. 6(a)(ii)3 marksHence, determine the particular solution of the differential equation that satisfies the condition y = 2 when x = π.
  31. 6(b)7 marksShow that the general solution of the differential equation dy/dx = (xy - y) / (x² - 4) is y = k(x - 2)^(1/4)(x + 2)^(3/4), where k is a constant.
  32. 6(c)10 marksSolve the boundary-value problem y'' - y' - 2y = 0, given that when x = -1, y = 1 and when x = 1, y = 0.

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