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CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 2

37 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)5 marksWithout the use of tables or a calculator, simplify √28 + √343 in the form k√7, where k is an integer.
  2. 1(b)6 marksSimplify (x⁴ - y⁴) / (x - y).
  3. 1(b)(ii)4 marksHence, or otherwise, show that (y+1)⁴ - y⁴ = (y+1)³ + (y+1)²y + (y+1)y² + y³.
  4. 1(b)(iii)2 marksDeduce that (y+1)⁴ - y⁴ < 4(y + 1)³.
  5. 1(c)8 marksSolve the equation log₂ x = 1 + log₂ 2x, x > 0.
  6. 2(a)6 marksWithout solving the equation, find a quadratic equation with roots 2/α and 2/β.
  7. 2(b)4 marksExpress f as a set of ordered pairs.
  8. 2(b)(ii)a)2 marksState TWO reasons why f is NOT a function.
  9. 2(b)(ii)b)4 marksHence, with MINIMUM changes to f, construct a function g : A → B as a set of ordered pairs.
  10. 2(b)(ii)c)2 marksDetermine how many different functions are possible for g in (ii) b) above.
  11. 2(c)3 marksFind the value of f[f(20)].
  12. 2(c)(ii)2 marksFind the value of f[f(8)].
  13. 2(c)(iii)2 marksFind the value of f[f(3)].
  14. 3(a)(i)2 marksState the radius and the coordinates of the centre of C.
  15. 3(a)(ii)4 marksFind the equation of the tangent at the point (6, 8) on C.
  16. 3(a)(iii)7 marksCalculate the coordinates of the points of intersection of C with the straight line y = 2x + 3.
  17. 3(b)(i)a)5 marksCalculate, in degrees, the size of the acute angle θ between p and q.
  18. 3(b)(i)b)2 marksHence, calculate the area of triangle POQ.
  19. 3(b)(ii)a)2 marksFind, in terms of i and j, the position vector of M, where M is the midpoint of PQ.
  20. 3(b)(ii)b)3 marksFind, in terms of i and j, the position vector of R, where R is such that PQRO, labelled clockwise, forms a parallelogram.
  21. 4(a)(i)4 marksShow that x = 4 cos θ + 9 sin θ.
  22. 4(a)(ii)6 marksBy expressing x in the form r cos (θ – α), where r is positive and 0 < α < π/2, find the MAXIMUM possible value of x.
  23. 4(b)(i)3 marksFind, without using tables or calculators, the EXACT values of sin (A + B).
  24. 4(b)(ii)3 marksFind, without using tables or calculators, the EXACT values of cos (A - B).
  25. 4(b)(iii)2 marksFind, without using tables or calculators, the EXACT values of cos 2A.
  26. 4(c)7 marksProve that tan (x/2 + π/4) = sec x + tan x.
  27. 5(a)5 marksFind lim (x³ - 8) / (x² - 6x + 8) as x → 2.
  28. 5(b)(i)2 marksSketch the graph of f(x) for the domain -1 ≤ x ≤ 2.
  29. 5(b)(ii)a)2 marksFind lim f(x) as x → 1+.
  30. 5(b)(ii)b)2 marksFind lim f(x) as x → 1-.
  31. 5(b)(iii)3 marksDeduce that f(x) is continuous at x = 1.
  32. 5(c)6 marksDifferentiate from first principles, with respect to x, the function y = 1/x².
  33. 5(d)5 marksFind the function f(x).
  34. 6(a)6 marksShow that d²y/dx² + 4y = 0.
  35. 6(b)6 marksGiven that ∫₀ᵃ (x + 1) dx = 3 ∫₀ᵃ (x - 1) dx, a > 0, find the value of the constant a.
  36. 6(c)(i)5 marksShow that the volume, V cm³, of the tray is given by V = 4(x³ - 13x² + 40x).
  37. 6(c)(ii)8 marksHence, find a possible value of x such that V is a maximum.

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