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CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 2

34 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 marksFind the values of the constant p such that x - p is a factor of f(x) = 4x³ - (3p + 2) x² - (p² – 1) x + 3.
  2. 1(b)8 marksSolve, for x and y, the simultaneous equations log (x-1) + 2 log y = 2 log 3 and log x + log y = log 6.
  3. 1(c)5 marksSolve, for x ∈ R, the inequality 2x-3 / x+1 > 5.
  4. 1(d)7 marksBy using y = 2ˣ, or otherwise, solve 4ˣ - 3 (2ˣ⁺¹) + 8 = 0.
  5. 2(a)(i)2 marksUse the fact that Sₙ = ∑r = 1 to n of r = (1/2) n (n + 1) to express S₂ₙ = ∑r = 1 to 2n of r in terms of n.
  6. 2(a)(ii)5 marksFind constants p and q such that S₂ₙ - Sₙ = pn² + qn.
  7. 2(a)(iii)5 marksHence, or otherwise, find n such that S₂ₙ - Sₙ = 260.
  8. 2(b)(i)4 marksWrite down the solution set of the inequality x² (3 - x) ≤ 0.
  9. 2(b)(ii)3 marksGiven that the equation x² (3 - x) = k has three real solutions for x, write down the set of possible values for k.
  10. 2(b)(iii)a)6 marksBy using (b) (ii) above, or otherwise, show that f has an inverse.
  11. 2(b)(iii)b)1 markBy using (b) (ii) above, or otherwise, show that g does NOT have an inverse.
  12. 3(a)(i)5 marksCalculate, in degrees, the angle between p and q.
  13. 3(a)(ii)a)5 marksFind a non-zero vector v such that p.v = 0.
  14. 3(a)(ii)b)1 markState the relationship between p and v.
  15. 3(b)(i)6 marksShow that the equation of C₁ is x² + y² + 2x – 6y + 5 = 0.
  16. 3(b)(ii)9 marksCalculate the coordinates of the points of intersection of C₁ and C₂.
  17. 4(a)(i)4 marksSolve the equation cos 3A = 0.5 for 0 ≤ A ≤ π.
  18. 4(a)(ii)6 marksShow that cos 3A = 4 cos³ A - 3 cos A.
  19. 4(a)(iii)4 marksThe THREE roots of the equation 4p³ – 3p – 0.5 = 0 all lie between -1 and 1. Use the results in (a) (i) and (ii) to find these roots.
  20. 4(b)(i)6 marksShow that tan (α – β) = hx / (x² + d (d+h)).
  21. 4(b)(ii)5 marksThe viewing angle of the painting, (α – β), is at a maximum when x = √h (d+h). Calculate the maximum viewing angle, in radians, when d = 3h.
  22. 5(a)(i)4 marksFind lim x→3 of (x²-9) / (x³-27).
  23. 5(a)(ii)5 marksFind lim x→0 of (tan x - 5x) / (sin 2x - 4x).
  24. 5(b)(i)a)2 marksFind lim x→4⁺ of f(x).
  25. 5(b)(i)b)2 marksFind lim x→4⁻ of f(x).
  26. 5(b)(ii)2 marksDeduce that f(x) is discontinuous at x = 4.
  27. 5(c)(i)6 marksEvaluate ∫ from -1 to 1 of (x - 1/x)² dx.
  28. 5(c)(ii)4 marksUsing the substitution u = x² + 4, or otherwise, find ∫ of x / √(x² + 4) dx.
  29. 6(a)(i)3 marksDifferentiate with respect to x: y = sin (3x + 2) + tan 5x.
  30. 6(a)(ii)4 marksDifferentiate with respect to x: y = (x² + 1) / (x³ - 1).
  31. 6(b)(i)4 marksFind ∫ from 1 to 4 of [3 f(x) + 4] dx.
  32. 6(b)(ii)4 marksUsing the substitution u = x + 1, evaluate ∫ from 0 to 3 of 2f(x + 1) dx.
  33. 6(c)(i)5 marksFind the coordinates of the points P and Q.
  34. 6(c)(ii)5 marksCalculate the area of the shaded portion of the diagram bounded by the curve and the straight line.

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