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CAPE Pure Mathematics Unit 2 · May/June 2023 · Paper 2

25 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)3 marksSketch the locus of z on the Argand diagram below.
  2. 1(a)(ii)4 marksDetermine the Cartesian equation of the locus of z.
  3. 1(b)5 marksGiven that x²y – 2xy² = cos(xy), determine dy/dx.
  4. 1(c)7 marksGiven that 3z² - pz + 2q = 0 has root 3 - 5i, determine the values of p and q where p, q ∈ R.
  5. 1(d)6 marksUsing DeMoivre's theorem, determine the square root of 1 - i√3, expressing your answer in radians.
  6. 2(a)6 marksUse integration by parts to evaluate ∫sec² x cosec² x dx.
  7. 2(b)9 marksEvaluate ∫ from 2 to 5 of (3x / (x² - 4x + 5)) dx.
  8. 2(c)10 marksDetermine ∫((x + 1) / (x³ - 9x)) dx.
  9. 3(a)7 marksDetermine the coefficient of x⁴ in the binomial expansion of (1 - 2x) / (1 + 3x)².
  10. 3(b)8 marksShow that if the series 1 + (7 / (3x - 5)) + (7 / (3x - 5))² + (7 / (3x - 5))³ + ... converges, then its sum to infinity is (3x - 8) / (3x - 12).
  11. 3(c)10 marksUse mathematical induction to prove that dⁿ/dxⁿ (eˣ sin x) = 2ⁿ/² eˣ sin(x + nπ/4). You may use the fact that sin x + cos x = √2 sin(x + π/4) and that dᵏ⁺¹/dxᵏ⁺¹ f(x) = d/dx (dᵏ/dxᵏ f(x)).
  12. 4(a)(i)3 marksUse the intermediate value theorem to prove that x⁴ - 2x³ + x² + 2 = 0 has a root in the interval [-2, -1.5].
  13. 4(a)(ii)6 marksHence, use two iterations of the Newton–Raphson method, with initial estimate x₁ = -1.5, to calculate a new estimate of the root of x⁴ - 2x³ + x² + 2 = 0 in the interval [-2, -1.5].
  14. 4(b)(i)4 marksShow that θₙ₊₁ = sin⁻¹(2 / (8 - θₙ²)) is a suitable iteration for the approximation of the roots of the equation cosec θ = 4 - (1/2)θ².
  15. 4(b)(ii)5 marksHence, using θ₁ = 0 as the first approximation, calculate a new estimate for the root of cosec θ = 4 - (1/2)θ², correct to three decimal places.
  16. 4(c)7 marksDetermine the Maclaurin expansion of f(x) = (1 + x²) cos x up to and including the third non-zero term.
  17. 5(a)(i)5 marksCalculate the determinant of M.
  18. 5(a)(ii)4 marksHence, or otherwise, determine the values of k for which the simultaneous equations x + y - z = 1, x + 2y - kz = 0, x - ky - z = 1 have a unique solution.
  19. 5(a)(iii)6 marksUsing k = 2 in the matrix, M, solve the system of linear equations by first reducing it to row echelon form.
  20. 5(b)(i)4 marksDraw a tree diagram to represent the possible events and their respective probabilities.
  21. 5(b)(ii)4 marksDetermine the probability that at least one pencil taken from the bag is blue.
  22. 5(b)(iii)2 marksDetermine whether the result of the second draw is independent of the first draw. Justify your response.
  23. 6(a)9 marksDetermine the general solution of the differential equation y' + y cot x = e²ˣ.
  24. 6(b)(i)11 marksShow that the general solution of the differential equation y'' - y' - 2y = 3e²ˣ is y = Ae²ˣ + Be⁻ˣ + xe²ˣ.
  25. 6(b)(ii)5 marksHence, solve the differential equation given that at x = 0, y = 0 and y' = 7.

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