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CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 2

44 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)1 markConstruct a truth table for the statements
  2. 1(a)(i)1 markp → q
  3. 1(a)(ii)2 marks~(p ∧ q).
  4. 1(b)5 marksSolve the equation 2 ⊕ x = 0.
  5. 1(c)8 marksUse mathematical induction to prove that 5ⁿ + 3 is divisible by 2 for all values of n ∈ N.
  6. 1(d)(i)2 marksGiven that (x + 1) is a factor of f(x), show that p = 6.
  7. 1(d)(ii)4 marksFactorise f(x) completely.
  8. 1(d)(iii)3 marksHence, or otherwise, solve f(x) = 0.
  9. 2(a)7 marksShow that f is one to one.
  10. 2(b)(i)1 markFind
  11. 2(b)(i)a)4 marksf⁻¹(x) and g⁻¹(x)
  12. 2(b)(i)b)1 markf [g(x)] (or f∘g(x)).
  13. 2(b)(ii)5 marksShow that (f∘g)⁻¹ (x) = g⁻¹ (x) ∘ f⁻¹ (x).
  14. 2(c)1 markSolve the following:
  15. 2(c)(i)4 marks3x² + 4x +1 ≤ 5
  16. 2(c)(ii)4 marks|x + 2| = 3x + 5
  17. 3(a)(i)4 marksShow that sin 2θ = (2 tan θ) / (1 + tan² θ)
  18. 3(a)(ii)8 marksHence, or otherwise, solve sin 2θ – tan θ = 0 for 0 ≤ θ ≤ 2π.
  19. 3(b)(i)4 marksExpress f(θ) = 3 cos θ – 4 sin θ in the form r cos (θ+ α) where r > 0 and 0° ≤ α ≤ π/2.
  20. 3(b)(ii)1 markHence, find
  21. 3(b)(ii)a)2 marksthe maximum value of f(θ)
  22. 3(b)(ii)b)2 marksthe minimum value of 1 / (8+f(θ))
  23. 3(b)(iii)1 markshow that
  24. 3(b)(iii)a)3 markssin A = sin (B + C)
  25. 3(b)(iii)b)2 markssin A + sin B + sin C = sin (A + B) + sin (B + C) + sin (A + C).
  26. 4(a)(i)3 marksShow that the centre and the radius of the circle, C, are (3, 2) and 3, respectively.
  27. 4(a)(ii)a)3 marksFind the equation of the normal to the circle C at the point (6, 2).
  28. 4(a)(ii)b)3 marksShow that the tangent to the circle at the point (6, 2) is parallel to the y-axis.
  29. 4(b)4 marksis 4x = y² + 10y + 24.
  30. 4(c)(i)3 marksExpress the vectors AB and BC in the form xi + yj + zk.
  31. 4(c)(ii)5 marksShow that the vector r = − 16j - 8k is perpendicular to the plane through A, B and C.
  32. 4(c)(iii)4 marksHence, find the Cartesian equation of the plane through A, B and C.
  33. 5(a)(i)4 marksFind lim (x→2) f(x).
  34. 5(a)(ii)2 marksDetermine whether f(x) is continuous at x = 2. Give a reason for your answer.
  35. 5(b)5 marksShow that dy/dx = (−4x³ – 10x² – 14x + 4) / (x² + 2)⁴
  36. 5(c)5 marksFind dy/dx in terms of θ.
  37. 5(d)(i)4 marksDetermine the coordinates of the points P and Q at which the curve and the line intersect.
  38. 5(d)(ii)5 marksCalculate the area of the shaded region.
  39. 6(a)(i)5 marksBy using the substitution u = 1 − x, find ∫ x (1-x)² dx.
  40. 6(a)(ii)4 marksshow that ∫[f(t) + g(t)] dt = ∫f(t) dt + ∫g(t) dt.
  41. 6(b)(i)2 marksShow that r = (600-2x) / (2 + π)
  42. 6(b)(ii)6 marksHence, determine the length, x, that maximises the area enclosed by the track.
  43. 6(c)(i)4 marksShow that y'' = x sin x.
  44. 6(c)(ii)4 marksHence, determine the specific solution of the differential equation y'' = x sin x, given that when x = 0, y = 1 and when x = π, y = 6.

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