Quelpr

CAPE Pure Mathematics Unit 1 · May/June 2004 · Paper 2

32 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)10 marksFind the constants m and n, and the third factor of f(x).
  2. 1(b)8 marksFind the constants p, q and r such that 2y²-9y + 14 = p(y-1)(y-2) + q(y - 1) + r.
  3. 1(c)(i)5 marksFind the range of values of x ∈ R for which |2x-3|≤5.
  4. 1(c)(ii)1 markHence, determine the LEAST possible value of x + 1.
  5. 1(c)(iii)1 markHence, determine the GREATEST possible value of x + 1.
  6. 2(a)(i)7 marksExpress f(x) = 12x-2x² in the form A + B(x+p)² where A, B and p are real numbers, and find the maximum value of 12x - 2x².
  7. 2(a)(ii)5 marksHence, sketch the graph of f(x) = 12x - 2x², showing clearly its main features.
  8. 2(a)(iii)2 marksShow that f(x) = 12x - 2x² is NOT one-to-one.
  9. 2(b)(i)a)4 marksCopy and complete the following table for the function f(x) = sin x, 0≤x≤ 2π.
  10. 2(b)(i)b)4 marksSketch the graph of f.
  11. 2(b)(ii)4 marksOn a separate diagram, sketch the graph of f(x) = | sin x |, 0 ≤ x ≤ 2π.
  12. 2(b)(iii)3 marksBy comparing the diagrams in (b)(i) and (ii) above, determine the solution set of the equation sin x = | sin x |, 0≤ x ≤ 2π.
  13. 3(a)(i)4 marksFind the equation of the line AB.
  14. 3(a)(ii)4 marksFind the equation of the line PQ.
  15. 3(a)(iii)4 marksFind the coordinates of the point Q.
  16. 3(b)7 marksSolve, for 0° ≤ θ ≤ 180°, the equation 6 cos² θ + sin θ = 4.
  17. 3(c)6 marksSolve, for 0 ≤ x ≤ π, the equation sin x + sin 3x = 0.
  18. 4(a)8 marksExpress the complex number, w = (z-1)/(z+2), in a similar form.
  19. 4(b)(i)4 marksFind the equation connecting x and y in the form ax² + by² + cx + dy + f = 0 where a, b, c, d, f are integers.
  20. 4(b)(ii)3 marksShow that the equation in (i) represents a circle, C.
  21. 4(b)(iii)4 marksDetermine the centre and radius of the circle C.
  22. 4(c)6 marksFind the position vector of P.
  23. 5(a)4 marksEvaluate lim (x²-2x-3)/(x²-4x+3) as x approaches 3.
  24. 5(b)3 marksDetermine the values of x ∈ R for which the function (x+2)/(x(x+1)) is NOT continuous.
  25. 5(c)(i)5 marksFind dy/dx in terms of x.
  26. 5(c)(ii)5 marksShow that x(x² + 1)dy/dx - 4y = -4/(x² + 1).
  27. 5(d)8 marksBy investigating the sign of f'(x), determine the range of real values of x for which x³ - 5x + 3 is decreasing.
  28. 6(a)6 marksFind the stationary point(s) of the curve, f(x) = x³ - 3x + 2.
  29. 6(b)3 marksDetermine the nature of the stationary point(s).
  30. 6(c)4 marksShow that the curve f(x) touches the x-axis at x = 1.
  31. 6(d)6 marksSketch the curve, f(x) = x³ - 3x + 2, for -2 ≤ x ≤ 2.
  32. 6(e)6 marksFind the area bounded by this curve and the x-axis for -2 ≤ x ≤ 1.

More CAPE Pure Mathematics Unit 1 papers