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CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 2

33 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)2 marksComplete the table below for the function |f(x)|, where f(x) = x(2-x).
  2. 1(a)(ii)4 marksSketch the graph of |f(x)| for -2 ≤ x ≤ 4.
  3. 1(b)6 marksFind the value(s) of the real number, k, for which the equation k(x² + 5) = 6 + 12x - x² has equal roots.
  4. 1(c)(i)4 marksIf 2^(x²) = 16^(x-1), find x.
  5. 1(c)(ii)4 marksWithout using calculators or tables, evaluate (√2 + 1)³ - (√2 - 1)³.
  6. 2(a)9 marksProve, by Mathematical Induction, that 10ⁿ - 1 is divisible by 9 for all positive integers n.
  7. 2(b)(i)3 marksFind the value of p for which the system has an infinite number of solutions.
  8. 2(b)(ii)3 marksFind the solutions for this value of p.
  9. 2(c)5 marksFind the set of real values of x for which (x+4)/(x-2) > 5.
  10. 3(a)5 marksExpress the equation of Q in the form (x - a)² + (y - b)² = c.
  11. 3(b)(i)2 marksHence, or otherwise, state the coordinates of the centre of Q.
  12. 3(b)(ii)1 markHence, or otherwise, state the radius of Q.
  13. 3(c)3 marksShow that the point A(4, 3) lies on Q.
  14. 3(d)5 marksFind the equation of the tangent to Q at the point A.
  15. 3(e)4 marksThe centre of Q is the midpoint of its diameter AB. Find the coordinates of B.
  16. 4(a)(i)1 markExpress the arc length ABC in terms of π.
  17. 4(a)(ii)a)3 marksHence, show that r = 7/6.
  18. 4(a)(ii)b)2 marksIf h cm is the height of the cone, then the exact value of h is (7√35)/6.
  19. 4(b)(i)5 marksShow that cos 3θ = 4cos³θ - 3cosθ.
  20. 4(b)(ii)5 marksBy using the identity in (b)(i) above, find the value of θ, 0 ≤ θ ≤ π/2, such that a and b are perpendicular.
  21. 4(c)4 marksFind the modulus of the complex number z = (25(2+3i))/(4+3i).
  22. 5(a)(i)1 markState the value of lim(u→0) (sin u)/u.
  23. 5(a)(ii)4 marksBy means of the substitution u = 3x, show that lim(x→0) (sin 3x)/x = 3.
  24. 5(a)(iii)4 marksHence, evaluate lim(x→0) (sin 3x)/(sin 5x).
  25. 5(b)4 marksIf y = A/x + Bx, where A and B are constants, show that x(d²y/dx²) + x(dy/dx) = y.
  26. 5(c)(i)3 marksFind the coordinates of P.
  27. 5(c)(ii)4 marksFind the volume of the solid generated by rotating the shaded area through 2π radians about the x-axis.
  28. 6(a)6 marksDifferentiate, with respect to x, (x² + 7)³ + sin 3x.
  29. 6(b)(i)3 marksDetermine the values of x for which the function has stationary points.
  30. 6(b)(ii)2 marksDetermine the values of x for which the function is increasing.
  31. 6(b)(iii)2 marksDetermine the values of x for which the function is decreasing.
  32. 6(c)(i)4 marksUse the substitution t = a - x to show that ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx.
  33. 6(c)(ii)3 marksIf ∫₀⁴ f(x) dx = 12, use the substitution t = x - 1 to evaluate ∫₁⁵ 3f(x-1) dx.

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