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CAPE Pure Mathematics Unit 1 · May/June 2007 · 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)3 marksFind all the real factors of g(x).
  2. 1(a)(ii)1 markFind all the real roots of g(x) = 0.
  3. 1(b)(i)3 marksExpress u^2 in terms of x.
  4. 1(b)(ii)6 marksBy writing f(x) = x^2 [x^2 - 9x + 28 - 36/x + 16/x^2] and using the result from (b)(i) above, show that if f(x) = 0, then u^2 - 9u + 20 = 0.
  5. 1(b)(iii)7 marksHence, determine the values of x ∈ R for which f(x) = 0.
  6. 2(a)4 marksFind the value of n for which 3S_{2n} = 11 S_n. Note: Σr = n(n+1)/2.
  7. 2(b)(i)2 marksExpress p and q in terms of α and β.
  8. 2(b)(ii)4 marksFind the values of α and β.
  9. 2(b)(iii)2 marksHence, determine the values of p and q.
  10. 2(c)8 marksProve, by Mathematical Induction, that n^2 > 2n for all integers n ≥ 3.
  11. 3(a)(i)2 marksDetermine the length of the radius of the circle.
  12. 3(a)(ii)1 markDetermine the equation of the circle.
  13. 3(a)(iii)6 marksDetermine the coordinates of the points A and B, at which the circle cuts the x-axis.
  14. 3(a)(iv)4 marksDetermine the equation of the tangent at B.
  15. 3(a)(v)2 marksDetermine the coordinates of P.
  16. 3(b)5 marksShow by calculation that PD = PB.
  17. 4(a)(i)4 marksProve that cos 2θ = (1 - tan^2 θ) / (1 + tan^2 θ).
  18. 4(a)(ii)7 marksHence, show, without using calculators, that tan 67.5° = 1 + √2.
  19. 4(b)(i)1 markDetermine the exact value of cos q.
  20. 4(b)(ii)1 markDetermine the exact value of sin p.
  21. 4(b)(iii)3 marksDetermine the exact value of sin r.
  22. 4(b)(iv)4 marksDetermine the exact value of cos (p + t).
  23. 5(a)(i)4 marksObtain dy/dx.
  24. 5(a)(ii)2 marksShow that y dy/dx = 5x.
  25. 5(a)(iii)4 marksHence, or otherwise, show that y d^2y/dx^2 + (dy/dx)^2 = 5.
  26. 5(b)(i)2 marksDetermine the height of the tide when high tide occurs for the first time.
  27. 5(b)(ii)3 marksDetermine the length of time which elapses between the first high tide and the first low tide.
  28. 5(b)(iii)5 marksDetermine the rate, in metres per minute, at which the tide is falling 75 minutes after high tide.
  29. 6(a)(i)2 marksUse the result ∫_0^a f(x)dx = ∫_0^a f(a-x)dx, a > 0, to show that if I = ∫_0^(π/2) sin^2 x dx, then I = ∫_0^(π/2) cos^2 x dx.
  30. 6(a)(ii)6 marksHence, or otherwise, show that I = π/4.
  31. 6(b)(i)4 marksSketch the curve y = x^2 + 4.
  32. 6(b)(ii)8 marksCalculate the volume created by rotating the plane figure bounded by x = 0, y = 4, y = 5 and the curve y = x^2 + 4 through 360° about the y-axis.

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