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CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 2

43 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)3 marksFind the exact value of (√75 + √12)² - (√75 - √12)²
  2. 1(a)(ii)3 marksFind the exact value of 27^(3/4) x 9^(3/8) x 81^(1/8)
  3. 1(b)(i)2 marksFind the value of p
  4. 1(b)(ii)4 marksFind the values of m and n
  5. 1(b)(iii)2 marksFind the x-coordinate of the point Q.
  6. 1(c)(i)6 marksBy substituting y = log₂x, or otherwise, solve, for x, the equation √log₂x = log₂(√x).
  7. 1(c)(ii)5 marksSolve, for real values of x, the inequality x² - |x| - 12 < 0.
  8. 2(a)(i)a)1 markExpress α + β in terms of p
  9. 2(a)(i)b)4 marksExpress α² + β² in terms of p
  10. 2(a)(ii)3 marksGiven that α² + β² = 33, find the possible values of p.
  11. 2(b)(i)4 marksFind the value of f(3)
  12. 2(b)(ii)2 marksFind the value of f(9)
  13. 2(b)(iii)3 marksFind the value of f(-3).
  14. 2(c)2 marksProve that the product of any two consecutive integers k and k + 1 is an even integer.
  15. 2(d)6 marksProve, by mathematical induction, that n(n² + 5) is divisible by 6 for all positive integers n.
  16. 3(a)(i)5 marksFind the value of (a+b)⋅(a-b).
  17. 3(a)(ii)5 marksIf 2b - a = 11i, determine the possible values of a and b.
  18. 3(b)(i)2 marksShow that L passes through the centre of C.
  19. 3(b)(ii)3 marksIf L intersects C at P and Q, determine the coordinates of P and Q.
  20. 3(b)(iii)3 marksFind the constants a, b and c such that x = b + a cos θ and y = c + a sin θ are parametric equations (in parameter θ) of C.
  21. 3(b)(iv)7 marksAnother circle C₂, with the same radius as C, touches L at the centre of C. Find the possible equations of C₂.
  22. 4(a)6 marksBy using x = cos²θ, or otherwise, find all values of the angle θ such that 8 cos³θ - 10 cos²θ + 3 = 0, for 0 ≤ θ ≤ π.
  23. 4(b)(i)2 marksFind, in terms of θ, the length of the side BC.
  24. 4(b)(ii)5 marksFind the value of θ if |BC| = 7 cm.
  25. 4(b)(iii)2 marksIs 15 a possible value for |BC|? Give a reason for your answer.
  26. 4(c)(i)3 marksShow that (1 - cos 2θ) / sin 2θ = tan θ.
  27. 4(c)(ii)a)3 marksShow that (1 - cos 4θ) / sin 4θ = tan 2θ.
  28. 4(c)(ii)b)2 marksShow that (1 - cos 6θ) / sin 6θ = tan 3θ.
  29. 4(c)(iii)2 marksEvaluate Σ (from r=1 to n) (tan rθ sin 2rθ + cos 2rθ) where n is a positive integer.
  30. 5(a)4 marksFind lim (as x→2) (x² + 5x + 6) / (x² - x - 6).
  31. 5(b)(i)2 marksDetermine f(2)
  32. 5(b)(ii)2 marksDetermine lim (as x→2⁺) f(x)
  33. 5(b)(iii)2 marksDetermine lim (as x→2⁻) f(x) in terms of the constant b
  34. 5(b)(iv)4 marksDetermine the value of b such that f is continuous at x = 2.
  35. 5(c)(i)6 marksFind the values of the constants p and q
  36. 5(c)(ii)3 marksFind the equation of the normal to the curve at T
  37. 5(c)(iii)2 marksFind the length of MN.
  38. 6(a)(i)8 marksFind the coordinates of each of the stationary points A and B
  39. 6(a)(ii)2 marksFind the equation of the normal to the curve f(x) = x(x² - 12) at the origin, O
  40. 6(a)(iii)6 marksFind the area between the curve and the positive x-axis.
  41. 6(b)(i)2 marksUse the result ∫(from 0 to a) f(x) dx = ∫(from 0 to a) f(a-x) dx, where a > 0, to show that ∫(from 0 to π) x sin x dx = ∫(from 0 to π) (π - x) sin x dx.
  42. 6(b)(ii)a)2 marksShow that ∫(from 0 to π) x sin x dx = ∫(from 0 to π) sin x dx - ∫(from 0 to π) x sin x dx
  43. 6(b)(ii)b)5 marksShow that ∫(from 0 to π) x sin x dx = π.

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