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CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 2

42 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)1 markFind
  2. 1(a)(i)7 marksthe values of the constants p and q
  3. 1(a)(ii)3 marksthe factors of f(x).
  4. 1(b)8 marksFind positive integers x and y such that (√x + √y)² = 16 + √240.
  5. 1(c)(i)5 marksSolve, for real values of x, the inequality |3x-7|≤5.
  6. 1(c)(ii)2 marksShow that no real solution, x, exists for the inequality |3x-7|+5≤0.
  7. 2(a)1 markFind
  8. 2(a)(i)3 marksin terms of x, f(f(x)).
  9. 2(a)(ii)6 marksDetermine the values of x for which f(f(x)) = f(x + 3).
  10. 2(b)(i)2 markswrite down the values of α + β and αβ
  11. 2(b)(ii)2 marksfind the value of α² + β²
  12. 2(b)(iii)5 marksobtain a quadratic equation whose roots are 2/α² and 2/β².
  13. 2(c)1 markWithout the use of calculators or tables, evaluate
  14. 2(c)(i)3 markslog₁₀(1/3) + log₁₀(3/5) + log₁₀(5/7) + log₁₀(7/9) + log₁₀(9/10)
  15. 2(c)(ii)4 marks∑_{r=1}^{99} log₁₀(r/(r+1)).
  16. 3(a)(i)7 marksprove that cos 3θ = 2 cos θ [cos² θ – sin² θ – 1/2].
  17. 3(a)(ii)5 marksUsing the appropriate formula, show that 1/2 [sin 6θ – sin 2θ] = (2 cos² 2θ – 1) sin 2θ.
  18. 3(a)(iii)5 marksHence, or otherwise, solve sin 6θ − sin 2θ = 0 for 0 ≤ θ ≤ π/2.
  19. 3(b)8 marksFind ALL possible values of cos θ such that 2 cot² θ + cos θ = 0.
  20. 4(a)(i)5 marksDetermine the Cartesian equation of the curve, C, defined by the parametric equations y = 3 sec θ and x = 3 tan θ.
  21. 4(a)(ii)9 marksFind the points of intersection of the curve y = √10x with C.
  22. 4(b)(i)2 marksExpress p and q in the form xi + yj.
  23. 4(b)(ii)2 marksObtain the vector p - q.
  24. 4(b)(iii)2 marksCalculate p.q.
  25. 4(b)(iv)5 marksLet the angle between p and q be θ. Use the result of (iii) above to calculate θ in degrees.
  26. 5(a)(i)2 marksFind the values of x for which (x³+8)/(x²-4) is discontinuous.
  27. 5(a)(ii)3 marksHence, or otherwise, find lim_{x→-2} (x³+8)/(x²-4).
  28. 5(a)(iii)5 marksBy using the fact that lim_{x→0} (sin x)/x = 1, or otherwise, find, lim_{x→0} (2x²+4x)/(sin 2x).
  29. 5(b)1 markFind
  30. 5(b)(i)a)2 markslim_{x→1+} f(x)
  31. 5(b)(i)b)4 marksthe value of the constant p such that lim_{x→1} f(x) exists.
  32. 5(b)(ii)1 markHence, determine the value of f(1) for f to be continuous at the point x = 1.
  33. 5(c)8 marksfind the values of u and v.
  34. 6(a)(i)3 marksGiven that y = √4x² – 7, show that y dy/dx = 4x.
  35. 6(a)(ii)3 marksHence, or otherwise, show that y d²y/dx² + (dy/dx)² = 4.
  36. 6(b)(i)4 marksFind the equation of C.
  37. 6(b)(ii)3 marksFind the coordinates of the stationary points of C.
  38. 6(b)(iii)3 marksDetermine the nature of EACH stationary point.
  39. 6(b)(iv)5 marksFind the coordinates of the points P and Q at which the curve C meets the x-axis.
  40. 6(b)(v)1 markHence, sketch the curve C, showing
  41. 6(b)(v)a)1 markthe stationary points
  42. 6(b)(v)b)4 marksthe points P and Q.

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