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CAPE Pure Mathematics Unit 1 · May/June 2001 · Paper 2

40 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)4 marksShow that the inequality |x-3| ≥ |x-3| holds for any x ∈ R.
  2. 1(b)4 marksFactorise completely the polynomial, x³ + 2x² - x - 2.
  3. 1(c)4 marksFind the value of r such that x⁴ - 7x + 4r has a remainder -2 when it is divided by x + 3.
  4. 1(d)(i)3 marksExplain clearly why f is not 1 – 1.
  5. 1(d)(ii)5 marksCalculate and simplify g f(x).
  6. 1(e)(i)4 marksUsing a scale of 2 cm to represent 10 minutes on the x-axis and 2 cm to represent 20 bacteria on the y-axis, draw the growth curve for the treatment stage.
  7. 1(e)(ii)1 markHence, estimate the number of bacteria present after the first 35 minutes of treatment.
  8. 2(a)(i)3 marksExpress f(x) in the form p(x-q)² + r, where p, q, r ∈ R.
  9. 2(a)(ii)2 marksSketch the graph of f(x).
  10. 2(a)(iii)3 marksHence, or otherwise, find the turning point of the graph and determine whether it is a maximum or a minimum.
  11. 2(b)4 marksGiven that δ is an acute angle such that cos δ = (1/4)x, find the expression for sin 2δ in terms of x.
  12. 2(c)(i)3 marksExpress cot θ and cosec θ in terms of x and y.
  13. 2(c)(ii)2 marksFind an equation connecting x and y.
  14. 2(d)(i)6 marksSketch, in separate diagrams, the graphs of y = sin x and y = cos x for -2π ≤ x ≤ 2π.
  15. 2(d)(ii)2 marksState clearly the transformation which maps y = cos x onto y = sin x.
  16. 3(a)(i)3 marksFind the value(s) of p for which the system of equations above has a unique solution.
  17. 3(a)(ii)3 marksFind the value(s) of p for which the system of equations above has an infinite number of solutions.
  18. 3(a)(iii)3 marksFind the value(s) of p for which the system of equations above has no solution.
  19. 3(b)(i)5 marksSolve the following equations for 0 ≤ x ≤ π/2: 6 sin²x - cos x - 4 = 0.
  20. 3(b)(ii)5 marksSolve the following equations for 0 ≤ x ≤ π/2: √3 sin x + cos x = 2.
  21. 3(c)6 marksShow that, for A ≠ π/2, sec A – tan A = tan (π/4 - A/2).
  22. 4(a)7 marksExpress the complex number, (z-1)/(z+1), in a similar form.
  23. 4(a)(i)3 marksFind the equation connecting x and y.
  24. 4(a)(ii)3 marksShow that the equation represents a circle, C.
  25. 4(a)(iii)2 marksDetermine the centre and radius of C.
  26. 4(b)(i)5 marksFind the values of t such that the vector t r₁ + r₂ is perpendicular to the vector r₂ + r₃.
  27. 4(b)(ii)3 marksFind the position vector of C relative to O.
  28. 4(b)(iii)2 marksDetermine cos AÔC.
  29. 5(a)5 marksUsing differentiation, determine the range of real values of x for which the function f: x → 12 + 6x² - x³ is decreasing.
  30. 5(b)6 marksDifferentiate x³ with respect to x from first principles.
  31. 5(c)(i)1 markSimplifying your answers where possible, find expressions in terms of x for the length of TS.
  32. 5(c)(ii)4 marksSimplifying your answers where possible, find expressions in terms of x for the length of RS.
  33. 5(c)(iii)2 marksSimplifying your answers where possible, find expressions in terms of x for the area, A, of PRST.
  34. 5(c)(iv)5 marksSimplifying your answers where possible, find expressions in terms of x for the stationary value of x and show that it is a maximum.
  35. 5(c)(v)2 marksSimplifying your answers where possible, find expressions in terms of x for the maximum area of the sheep enclosure.
  36. 6(a)6 marksCalculate the volume generated when the finite region in the first quadrant bounded by the curve, y = 2x², the y-axis and the line y = 2 is rotated completely about the y-axis.
  37. 6(b)(i)5 marksCalculate the area of R.
  38. 6(b)(ii)5 marksThe area of R is estimated using the trapezium rule with 2 intervals of equal width. Show that this trapezium rule estimate differs by 1/3 from the exact value for the area of R found in (b)(i).
  39. 6(b)(iii)4 marksOn a carefully labelled sketch of y = x² + 1, shade in the 2 trapezia which are used to estimate the area of R.
  40. 6(b)(iv)5 marksAnother approximation for the area of R is obtained using 2 trapezia of unequal width. The first trapezium has width, h, and the second trapezium width, (2 - h) with the three ordinates occurring where x = 0, x = h and…

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