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CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 2

29 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)5 marksCopy and complete the truth table below for the propositions (p ^ q) → r and (p → r) ^ (q → r).
  2. 1(a)(ii)2 marksHence, determine whether or not (p ^ q) → r and (p → r) ^ (q → r) are logically equivalent. Justify your response.
  3. 1(b)8 marksUse mathematical induction to prove that 10ⁿ⁺¹ + 3(10ⁿ) + 5 is divisible by 9 for all natural numbers.
  4. 1(c)10 marksSolve the equation x³ – 6x² – 69x + 154 = 0.
  5. 2(a)(i)5 marksShow that 1/logₐx may be rewritten as ln a / ln x, where a > 0, a ≠ 1.
  6. 2(a)(ii)4 marksHence, or otherwise, solve the equation: 1/log₂x + 1/log₃x + 1/log₄x + 1/log₅x = 1.
  7. 2(b)(i)4 marksOn the same axes, sketch the graphs |f(x)| and |g(x)|.
  8. 2(b)(ii)6 marksHence, or otherwise, solve the equation |x² – 4x + 3| = |2 – 3x|.
  9. 2(c)6 marksDetermine the value of f(f⁻¹[f(1)]).
  10. 3(a)6 marksGiven that tan² x = sin² x / (1 - sin² x), show that sin x = ± tan x / √(1 + tan² x).
  11. 3(b)(i)6 marksShow that f(θ) = √3 sin θ + cos θ may be expressed as f(θ) = 2sin(θ + π/6) where 0 ≤ θ ≤ π/2.
  12. 3(b)(ii)a)5 markssolve the equation f(θ) = √2 for 0 ≤ θ ≤ 2π
  13. 3(b)(ii)b)3 marksdetermine the maximum value of f and the smallest positive value of θ for which it occurs.
  14. 3(c)5 marksWithout the use of a calculator or tables, find the exact value of cos(π/12).
  15. 4(a)8 marksCalculate the distance between the points of intersection of the line 3x – 2y + 6 = 0 and the circle x² + y² = 9.
  16. 4(b)(i)5 marksFind the centre and radius of C.
  17. 4(b)(ii)4 marksFind the equation of the tangent to C at the point (0,0).
  18. 4(c)8 marksFind c given that the angle between the vectors a and b is π/3.
  19. 5(a)5 marksGiven that y = tan⁻¹ (1 − x²), find d²y/dx².
  20. 5(b)(i)4 marksDetermine dy/dx in terms of t.
  21. 5(b)(ii)3 marksDetermine the value of t for which the circle has vertical tangents.
  22. 5(c)(ii)5 marksHence, sketch the graph of f.
  23. 5(c)a)4 marksDetermine whether the function f has turning points.
  24. 5(c)b)4 marksDetermine the vertical and horizontal asymptotes of f.
  25. 6(a)(i)3 marksDetermine the area of the region, A.
  26. 6(a)(ii)5 marksCalculate the volume of the solid that results from revolving the region, A, about the x-axis.
  27. 6(b)7 marksGiven that the curve passes through the point (1,1), determine an expression for f(x).
  28. 6(c)5 marksBy using the substitution, y = x², show that ∫ 1/(√y + √y³) dy = ∫ 2/(1 + x²) dx.
  29. 6(d)5 marksfind ∫sin²x cos²x dx.

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