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CAPE Pure Mathematics Unit 1 · May/June 2008 · 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.

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  1. 1(a)(i)8 marksDetermine the values of the real number h for which the roots of the quadratic equation 4x² - 2hx + (8 – h) = 0 are real.
  2. 1(a)(ii)7 marksThe roots of the cubic equation x³ - 15x² + px – 105 = 0 are 5 - k, 5 and 5 + k. Find the values of the constants p and k.
  3. 1(b)(i)4 marksCopy the table below and complete by inserting the values for the functions f(x) = |x + 2 | and g(x) = 2 |x − 1 |.
  4. 1(b)(ii)4 marksUsing a scale of 1 cm to 1 unit on both axes, draw on the same graph f(x) and g(x) for −3 ≤ x ≤ 5.
  5. 1(b)(iii)2 marksUsing the graphs, find the values of x for which f(x) = g(x).
  6. 2(a)8 marksWithout using calculators or tables, evaluate (27¹⁰ + 9¹⁰) / (27⁴ + 9¹¹).
  7. 2(b)(i)4 marksProve that logₙm = log₁₀m / log₁₀n, for m, n ∈ N.
  8. 2(b)(ii)6 marksHence, given that y = (log₂ 3) (log₃ 4) (log₄ 5) ... (log₃₁ 32), calculate the exact value of y.
  9. 2(c)7 marksProve, by the principle of mathematical induction, that f(n) = 7ⁿ - 1 is divisible by 6, for all n ∈ N.
  10. 3(a)(i)1 markLet p = i - j. If q = λi + 2j, find values of λ such that q is parallel to p.
  11. 3(a)(ii)2 marksq is perpendicular to p.
  12. 3(a)(iii)5 marksthe angle between p and q is π/3.
  13. 3(b)6 marksShow that (1 - cos 2A + sin 2A) / (1 + cos 2A + sin 2A) = tan A.
  14. 3(c)(i)2 marksUsing the formula for sin A + sin B, show that if t = 2 cos θ then sin (n + 1) θ = t sin nθ – sin (n − 1) θ.
  15. 3(c)(ii)2 marksHence, show that sin 3θ = (t² − 1) sin θ.
  16. 3(c)(iii)7 marksUsing (c) (ii) above, or otherwise, find ALL solutions of sin 3θ = sin θ, 0 ≤ θ ≤ π.
  17. 4(a)(i)6 marksThe line x - 2y + 4 = 0 cuts the circle, x² + y² - 2x - 20y + 51 = 0 with centre P, at the points A and B. Find the coordinates of P, A and B.
  18. 4(a)(ii)b)2 marksFind the equation of circle C.
  19. 4(a)(ii)c)3 marksFind the distance, | PQ |, between the centres.
  20. 4(a)(ii)d)4 marksFind the distance | PM | if PQ cuts AB at M.
  21. 4(b)(i)3 marksShow that the Cartesian equation of the curve is (x-2)²/9 + (y-3)²/16 = 1.
  22. 4(b)(ii)5 marksShow that every point on the curve lies within or on the circle (x – 2)² + (y - 3)² = 25.
  23. 5(a)3 marksUse L'Hopital's rule to obtain lim (x→0) (sin 4x / sin 5x).
  24. 5(b)(i)a)4 marksFind dy/dx.
  25. 5(b)(i)b)2 marksShow that x² dy/dx = y².
  26. 5(b)(ii)3 marksHence, or otherwise, show that x² d²y/dx² + 2(x - y) dy/dx = 0.
  27. 5(c)(i)4 marksShow that h = 20/x - 3x/5.
  28. 5(c)(ii)9 marksFind the height of the box for which its volume V cm³ is a maximum.
  29. 6(a)6 marksUse the substitution u = 3x² + 1 to find ∫ x dx / √(3x² + 1).
  30. 6(b)4 marksFind the equation of C.
  31. 6(c)(i)6 marksFind the coordinates of A, B and C.
  32. 6(c)(ii)9 marksHence find the exact value of the area of the shaded region.

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