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CAPE Pure Mathematics Unit 1 · 2021 · 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)4 marksComplete the truth table below.
  2. 1(a)(ii)2 marksHence, state whether the statements ~(p ∨ q) and ~p ∧ ~q are logically equivalent. Justify your response.
  3. 1(b)(i)1 markWrite the converse of the statement "n is an integer ⇒ n² is an integer".
  4. 1(b)(ii)3 marksProve that * is closed in ℝ.
  5. 1(b)(iii)2 marksDetermine whether the operation * is commutative.
  6. 1(c)7 marksCalculate the values of a and b.
  7. 1(d)6 marksSolve the logarithmic equation log₂ x + log₄ x + log₁₆ x = 7.
  8. 2(a)(i)a)4 marksSketch the inverse of f.
  9. 2(a)(i)b)1 markShow that the inverse of f is a function.
  10. 2(a)(ii)4 marksProve that f is one to one.
  11. 2(a)(iii)3 marksDetermine whether f is onto.
  12. 2(b)6 marksSolve the equation |x² - 4| = 3x - 2.
  13. 2(c)8 marksDetermine the equation whose roots are 1/α, 1/β, 1/γ.
  14. 3(a)(i)5 marksExpress 4 sin θ + 3 cos θ in the form of R sin(θ + α).
  15. 3(a)(ii)5 marksHence, solve the equation 4 sin θ + 3 cos θ = 2 for 0 ≤ θ ≤ 2π.
  16. 3(b)7 marksCalculate the value of cos(A - C).
  17. 3(c)8 marksSolve the equation cos θ = sin(θ - π/3) for θ.
  18. 4(a)(i)4 marksDetermine the centre and radius of the circle.
  19. 4(a)(ii)4 marksDetermine the equation of the tangent which touches the circle at (3, 11).
  20. 4(b)5 marksShow that the curve whose parametric equations are x = 3 + 3 sin θ and y = 3 cos θ represents a circle.
  21. 4(c)8 marksDetermine the angle between OA and OB.
  22. 4(d)4 marksDetermine the vector equation of a plane which passes through the point (2, 5, 3) and is perpendicular to the vector 4i + 4j - k.
  23. 5(a)6 marksUse the substitution u = (x³ + 4) to determine ∫ 3x² (x³ + 4)⁴ dx.
  24. 5(b)9 marksCalculate the area of the region enclosed by the parabolas y = x² and y = 6x - 3x².
  25. 5(c)(i)6 marksShow that f(x) has stationary points at x = -1, x = 0 and x = 2.
  26. 5(c)(ii)4 marksDetermine the nature of these stationary points.
  27. 6(a)(i)5 marksDetermine whether or not the lim (x→1) f(x) exists.
  28. 6(a)(ii)2 marksDetermine whether f is continuous at x = 1.
  29. 6(b)4 marksDetermine dy/dx in terms of θ.
  30. 6(c)4 marksCalculate ∫₁² [f(x) + g(x)] dx.
  31. 6(d)(i)6 marksSolve the differential equation dy/dx = sin x / sin y given that when x = 0, y = π/2.
  32. 6(d)(ii)4 marksDetermine the equation of the curve that passes through (1, 5) and for which y = ∫ 6x² dx.

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