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CAPE Pure Mathematics Unit 1 · May/June 2015 · 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.

Practise this paper
  1. 1(a)(i)2 marksState the inverse and the contrapositive of the statement p → q.
  2. 1(a)(ii)4 marksCopy and complete the table below to show the truth table for p → q and ~q → ~p.
  3. 1(a)(iii)2 marksHence, state whether the compound statements p ↔ q and ~q → ~p are logically equivalent. Justify your response.
  4. 1(b)(i)4 marksFind the values of p and q.
  5. 1(b)(ii)5 marksHence, factorize f(x) = x³ + px² – x + q completely.
  6. 1(c)8 marksUse mathematical induction to prove that 4S(n) = 5ⁿ⁺¹ – 5 for n ∈ N.
  7. 2(a)(i)4 marksShow that (g° f) is one-to-one.
  8. 2(a)(ii)4 marksShow that (g° f) is onto.
  9. 2(b)(i)7 marksSolve 3 - 4/(9)ˣ - 4/(81)ˣ = 0.
  10. 2(b)(ii)5 marksSolve |5x - 6| = x + 5.
  11. 2(c)(i)1 markDetermine the number of bacteria present at t = 0.
  12. 2(c)(ii)4 marksDetermine the time required to triple the number of bacteria.
  13. 3(a)(i)6 marksShow that cos 3x = 4 cos³ x – 3 cos x.
  14. 3(a)(ii)9 marksHence, or otherwise, solve cos 6x – cos 2x = 0 for 0 < x < 2π.
  15. 3(b)(i)6 marksExpress f(2θ) = 3 sin 2θ + 4 cos 2θ in the form r sin (2θ + α) where r > 0 and 0 < α < π/2.
  16. 3(b)(ii)4 marksHence, or otherwise, find the maximum and minimum values of 1/(7 - f(2θ)).
  17. 4(a)(i)4 marksDetermine the Cartesian equations of C₁ and C₂ in the form (x – a)² + (y – b)² = r².
  18. 4(a)(ii)9 marksHence or otherwise, find the points of intersection of C₁ and C₂.
  19. 4(b)12 marksShow that the equation of the locus of the point P (x, y) is a circle.
  20. 5(a)4 marksIf f is continuous at x = 0, determine the value of a.
  21. 5(b)6 marksUsing first principles, determine the derivative of f(x) = sin (2x).
  22. 5(c)(i)7 marksShow that x dy/dx = y/(1 + x²).
  23. 5(c)(ii)8 marksShow that d²y/dx² + (3y)/(1 + x²)² = 0.
  24. 6(a)(i)5 marksShow that the coordinates of A, B and C are (4, 5), (3, 2), and (6, 3) respectively.
  25. 6(a)(ii)6 marksHence, use integration to determine the area bounded by the lines.
  26. 6(b)(i)3 marksDetermine the equation of the curve.
  27. 6(b)(ii)8 marksFind the coordinates and nature of the stationary point of the curve in (b) (i) above.
  28. 6(b)(iii)3 marksSketch the curve in (b) (i) by clearly labelling the stationary points.

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