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CAPE Pure Mathematics Unit 1 · May/June 2008 · Paper 2

36 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)7 marksFind the values of the real constants p, q and r.
  2. 1(b)(i)5 marksWithout using calculators or tables, show that (√6 + √2) / (√6 - √2) = 2 + √3.
  3. 1(b)(ii)5 marksWithout using calculators or tables, show that (√6 + √2) / (√6 - √2) + (√6 - √2) / (√6 + √2) = 4.
  4. 1(c)(i)5 marksShow that Σ (from r=1 to n) r(r + 1) = (1/3)n(n + 1)(n + 2), n ∈ N.
  5. 1(c)(ii)3 marksHence, or otherwise, evaluate Σ (from r=31 to 50) r(r + 1).
  6. 2(a)(i)2 marksWrite down the values of α + β and αβ.
  7. 2(a)(ii)a)2 marksCalculate α² + β².
  8. 2(a)(ii)b)4 marksCalculate α³ + β³.
  9. 2(a)(iii)4 marksFind a quadratic equation whose roots are α³ and β³.
  10. 2(b)(i)5 marksSolve for x the equation x^(1/3) - 4x^(-1/3) = 3.
  11. 2(b)(ii)5 marksFind x such that log₂(x + 3) + log₂(x - 1) = 1.
  12. 2(b)(iii)3 marksWithout the use of calculators or tables, evaluate log₁₀(1/2) + log₁₀(2/3) + log₁₀(3/4) + ... + log₁₀(8/9) + log₁₀(9/10).
  13. 3(a)(i)2 marksState the values of tan α and tan β.
  14. 3(a)(ii)4 marksWithout using tables or calculators, find the tangent of the angle between the two lines.
  15. 3(b)(i)3 marksProve that sin 2θ - tan θ cos 2θ = tan θ.
  16. 3(b)(ii)2 marksExpress tan θ in terms of sin 2θ and cos 2θ.
  17. 3(b)(iii)4 marksHence show, without using tables or calculators, that tan 22.5° = √2 - 1.
  18. 3(c)(i)a)3 marksProve that sin((A+B)/2) = cos(C/2).
  19. 3(c)(i)b)2 marksProve that sin B + sin C = 2 cos(A/2) cos((B-C)/2).
  20. 3(c)(ii)5 marksHence, show that sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2).
  21. 4(a)(i)8 marksFind the equation of the line which passes through M and is perpendicular to PQ.
  22. 4(a)(ii)9 marksHence, or otherwise, find the coordinates of the centre of the circle through P, O and Q.
  23. 4(b)(i)6 marksProve that the line y = x + 1 is a tangent to the circle x² + y² + 10x – 12y + 11 = 0.
  24. 4(b)(ii)2 marksFind the coordinates of the point of contact of this tangent to the circle.
  25. 5(a)4 marksFind lim (as x→3) (x² - 27) / (x² + x - 12).
  26. 5(b)6 marksFind the values of u and v.
  27. 5(c)(i)3 marksFind the equation of C.
  28. 5(c)(ii)7 marksFind the coordinates of the stationary points of C and determine the nature of EACH point.
  29. 5(c)(iii)5 marksSketch the graph of C and label the x-intercepts.
  30. 6(a)(i)3 marksDifferentiate with respect to x: x√(2x - 1).
  31. 6(a)(ii)4 marksDifferentiate with respect to x: sin²(x³ + 4).
  32. 6(b)(i)3 marksGiven that ∫(from 1 to 6) f(x) dx = 7, evaluate ∫(from 1 to 6) [2 - f(x)] dx.
  33. 6(b)(ii)4 marksFind the value of the constant k.
  34. 6(c)(i)a)3 marksShow that h = 45/r² - 2r/3.
  35. 6(c)(i)b)3 marksShow that A = 5πr²/3 + 90π/r, where A units is the external surface area of the can.
  36. 6(c)(ii)5 marksHence, find the value of r for which A is a minimum and the corresponding minimum value of A.

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