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CAPE Pure Mathematics Unit 1 · May/June 1999 · Paper 2

35 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)8 marksSolve the simultaneous equations: (x - 2)² + (y+2)² = 4, y+x-2=0.
  2. 1(b)(i)1 markWrite down the size of angle PQR.
  3. 1(b)(ii)2 marksCalculate, in terms of r, the area of triangle PQR.
  4. 1(b)(iii)1 markWrite down the size of angle PSQ.
  5. 1(b)(iv)2 marksBy considering triangle PSR, or otherwise, show that r/(r+α) = √3/2.
  6. 1(b)(v)6 marksCalculate, in terms of r, the area of the shaded region.
  7. 2(a)(i)2 marksEvaluate f(-2).
  8. 2(a)(ii)4 marksCalculate the exact values of x which map onto themselves under the function f.
  9. 2(a)(iii)3 marksExpress f(x) in the form u(x - v)² + w, where u, v, w ∈ R.
  10. 2(a)(iv)3 marksBy sketching the graph of y = f(x), or otherwise, state the turning point P of the graph and indicate whether P is a maximum or a minimum.
  11. 2(b)(i)1 markDetermine the range of f.
  12. 2(b)(ii)1 markExplain why the function f has an inverse.
  13. 2(b)(iii)4 marksFind an expression for f⁻¹(x).
  14. 2(b)(iv)2 marksDescribe the geometrical relationship between the graphs y = f(x) and y = f⁻¹(x).
  15. 3(a)(i)3 marksFind the equation of the line which passes through the point (4, -1) and is perpendicular to the straight line y = -2x + 2.
  16. 3(a)(ii)3 marksCalculate the coordinates of the point of intersection of these two straight lines.
  17. 3(b)(i)5 marksCalculate the range of values of k for which the equation f(x) = 0 has no real roots.
  18. 3(b)(ii)3 marksSolve the equation f(x) = 0 for k = 1, giving your answer in the form a ± bi, where a, b ∈ R.
  19. 3(b)(iii)6 marksLet α and β be roots of f(x) = 0 when k = 8. Without first solving f(x) = 0, determine the equation whose roots are respectively 1/α and 1/β.
  20. 4(a)2 marksGiven that θ is an obtuse angle such that sin θ = 2/3, find the value of cos 2θ.
  21. 4(b)(i)3 marksObtain an expression for AD in terms of θ.
  22. 4(b)(ii)4 marksExpress AD in the form Rcos(θ + α), where R is positive and α is an acute angle.
  23. 4(c)(i)5 marksExpress cos 3θ in terms of cos θ.
  24. 4(c)(ii)6 marksHence, solve, for 0 < θ < 2π, the equation cos 3θ + 2 cos 2θ + 4 cos θ + 2 = 0.
  25. 5(a)2 marksSketch the curve traced out by the cutter.
  26. 5(b)6 marksUse the Trapezium rule to find the approximate area of a flat side of each disc by using eight (8) subintervals.
  27. 5(c)4 marksCompare this approximate area with the exact area of a side of each disc.
  28. 5(d)(i)2 marksSketch the solid that is generated by the rotation.
  29. 5(d)(ii)6 marksFind the volume of the solid that is generated.
  30. 6(a)3 marksFind g(0), g(1) and g(-1).
  31. 6(b)2 marksObtain an expression for the derivative g'(x) of g at x ∈ R.
  32. 6(c)4 marksFind the stationary points of g.
  33. 6(d)(i)2 marksDetermine the value(s) of x where g has a local maximum.
  34. 6(d)(ii)2 marksDetermine the value(s) of x where g has a local minimum.
  35. 6(e)7 marksUsing the above, and any other information, sketch the graph of g.

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