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CAPE Pure Mathematics Unit 2 · May/June 2024 · 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)6 marksGiven that 3y² - 4xy + sin xy = 5, determine dy/dx.
  2. 1(b)3 marksDifferentiate cos⁻¹ (3x - 2), expressing your answer in its simplest form.
  3. 1(c)(i)7 marksUse partial fractions to show that (2x+1)/(3x²-x-2) = 3/(5(x-1)) + 1/(5(3x+2)).
  4. 1(c)(ii)4 marksHence, or otherwise, determine ∫ (2x+1)/(3x²-x-2) dx.
  5. 1(d)5 marksGiven that x = 2t - sin t and y = 1 - cos t, determine d²y/dx².
  6. 2(a)8 marksDetermine ∫ e^(2x) sin 3x dx.
  7. 2(b)7 marksUse DeMoivre's theorem to show that (1 + 3i)⁴ = 28 - 96i.
  8. 2(c)(i)6 marksDerive the reduction formula ∫ sinⁿ x dx = -1/n sinⁿ⁻¹ x cos x + (n-1)/n ∫ sinⁿ⁻² x dx.
  9. 2(c)(ii)4 marksHence, or otherwise, show that ∫ sin⁶ x dx = -1/6 sin⁵ x cos x - 5/24 sin³ x cos x + 5/8 ∫ sin² x dx.
  10. 3(a)6 marksUse two iterations of the Newton-Raphson method, with the initial estimate x₀ = 0.3, to calculate a new estimate for the root of f(x) = 3x² - 4x + 1.
  11. 3(b)9 marksDetermine the Taylor series expansion of f(x) = 1/(1-2x) about x = 3 up to and including the term in x².
  12. 3(c)(i)6 marksObtain the binomial expansion of (16-5x)¹/⁴ up to and including the term in x².
  13. 3(c)(ii)2 marksHence, state the values for which the expansion of (16 – 5x)¹/⁴ is valid.
  14. 3(c)(iii)2 marksUse the expansion obtained in (c)(i) to estimate the value of (16 – 5x)¹/⁴ when x = 3.
  15. 4(a)(i)3 marksUse the intermediate value theorem to prove that f(x) = 2x³ + x² + 9x - 5 has at least one root in the interval [0.2, 1].
  16. 4(a)(ii)5 marksHence, use three iterations of the interval bisection method to estimate the root of f in the interval [0.2, 1].
  17. 4(b)(i)4 marksWrite down an expression for the nᵗʰ term of the sequence.
  18. 4(b)(ii)3 marksDetermine whether the sequence converges or diverges.
  19. 4(c)4 marksCalculate ∑(from r=0 to 27) r.
  20. 4(d)6 marksUse the sum of a series to calculate an equivalent fraction for the repeating decimal 0.2727272...
  21. 5(a)6 marksGiven that the matrix A = [[1-p, 3, -2], [2, p, -1], [-3, 2, 1]] is singular, calculate the value(s) of p.
  22. 5(b)9 marksUse row reduction to solve the following system of equations: 3x - 4y + z = 16; 2x + y - 2z = 5; x + 2y - z = -2.
  23. 5(c)(i)7 marksIn how many different ways can the committee be formed?
  24. 5(c)(ii)3 marksWhat is the probability that the committee formed will contain neither of the 2 men who refuse to serve together?
  25. 6(a)5 marksA chemical test kit is 95% accurate in detecting when a solution is acidic. However, the test kit also yields a 'false positive' result for 1% of all solutions tested. If 0.5% of the solutions tested are acidic, what is…
  26. 6(b)5 marksDetermine the general solution of the differential equation dy/dx + (3/x)y = sin(2x)/x².
  27. 6(c)(i)6 marksShow that the general solution of the differential equation is Ae^(3x/4) sin(√19x/4) + Be^(3x/4) cos(√19x/4), where A and B are constants.
  28. 6(c)(ii)9 marksHence, or otherwise, determine the particular solution of the differential equation given that at x = 0, y = 3 and y' = 0.

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