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CAPE Pure Mathematics Unit 1 · May/June 2000 · Paper 2

36 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)9 marksFind the coordinates of A and B.
  2. 1(b)(i)3 marksWrite down, in terms of θ and r, an expression for the area of S₁.
  3. 1(b)(ii)4 marksWrite down, in terms of θ and r, an expression for the area of S₂.
  4. 1(b)(iii)4 marksGiven that the area of S₂ is three times the area of S₁, show that 4θ = π + 2 sin θ.
  5. 1(b)(iv)5 marksShow that when r = 6, θ = π/3, the area of triangle PQR = 18√3.
  6. 2(a)(i)3 marksState the range of f.
  7. 2(a)(ii)3 marksSketch the graph of f.
  8. 2(a)(iii)3 marksExplain why the inverse function f⁻¹ of f exists.
  9. 2(a)(iv)2 marksState the domain of f⁻¹.
  10. 2(a)(v)4 marksSketch f⁻¹ on the same diagram as f.
  11. 2(b)4 marksState clearly the transformation which maps g onto f.
  12. 2(c)(i)3 marksFind an expression for f(h(x)).
  13. 2(c)(ii)3 marksWrite down the range for the function f(h(x)).
  14. 3(a)(i)4 marksExpress the complex number z = (11 - 2i) / (3 + 4i) in the form a + ib, where a and b are real numbers.
  15. 3(a)(ii)3 marksHence, express z² and iz in a similar form.
  16. 3(a)(iii)3 marksFind the modulus and principal value of the argument of z, where -π < arg z ≤ π.
  17. 3(a)(iv)3 marksFind the exact distance between the points on the Argand diagram represented by z² and iz.
  18. 3(b)(i)4 marksFind the cartesian equation of the curve, C.
  19. 3(b)(ii)3 marksDescribe the curve, C, in detail.
  20. 3(b)(iii)5 marksFind the equations of the tangent and normal to the curve, C, at the point given by θ = 0.
  21. 4(a)(i)2 marksExpress cos 4θ in terms of cos 2θ.
  22. 4(a)(ii)10 marksHence, solve the equation cos 4θ + 3 cos 2θ - 1 = 0, for 0 < θ < π.
  23. 4(b)(i)3 marksDetermine a unit vector in the direction of OC.
  24. 4(b)(ii)5 marksFind, by calculation, the position vector of D.
  25. 4(b)(iii)5 marksShow that OE and OC are perpendicular.
  26. 5(a)6 marksWith the aid of a diagram, show that for n = 4, ∫₀ᵇ f(x)dx ≈ (d/2)[y₀ + 2y₁ + 2y₂ + 2y₃ + y₄], where y₀ = f(a), y₁ = f(a + d), ..., y₄ = f(a + 4d).
  27. 5(b)(i)8 marksWrite down an appropriate differential equation connecting the length of the rod and the rate of decrease at time, t. Hence, derive an expression for L.
  28. 5(b)(ii)7 marksEvaluate the length, L, of the rod using the trapezium rule for five ordinates and strips of length 1 unit.
  29. 5(b)(iii)2 marksFind the time at which the rate of decrease vanishes.
  30. 5(b)(iv)2 marksShow that the rate of decrease is never negative.
  31. 6(a)5 marksFind the values of x for which h(x) = 0.
  32. 6(b)6 marksFind the stationary points of h.
  33. 6(c)(i)2 marksDetermine the value(s) of x where h has a local maximum.
  34. 6(c)(ii)2 marksDetermine the value(s) of x where h has a local minimum.
  35. 6(d)5 marksUsing the above, and any other information, sketch the graph of h.
  36. 6(e)5 marksFind the total area enclosed between the curve, the x-axis and the values x = -2 and x = 2.

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