Quelpr

CAPE Pure Mathematics Unit 1 · May/June 2002 · Paper 2

28 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)3 marksFind p, q ∈ R such that f(x) = p + q / (x - 2).
  2. 1(b)6 marksShow that f is one-to-one.
  3. 1(c)7 marksDetermine whether there is an x ∈ A such that f(x) = 1.
  4. 1(d)(i)5 marksUse Part (c) above to determine the range of f.
  5. 1(d)(ii)4 marksUse Part (c) above to determine whether or not f is onto.
  6. 2(a)13 marksIf t = tan (θ/2), express cos θ and sin θ in terms of t. Hence, find tan (θ/2) when cos θ + 2 sin θ = 11/5.
  7. 2(b)(i)3 marksFind cos θ.
  8. 2(b)(ii)3 marksFind sin θ.
  9. 2(b)(iii)3 marksFind the length of BC.
  10. 2(b)(iv)3 marksFind the length of AC.
  11. 3(a)(i)4 marksFind dy/dx in terms of t.
  12. 3(a)(ii)5 marksShow that the Cartesian equation of the tangent to the curve at the point P with parameter T is Ty = x + 4T².
  13. 3(a)(iii)6 marksThe tangent in (ii) above meets the y-axis at the point Q and the x-axis at the point R. If O is the origin, show that the area of triangle OQR is 8T³ square units.
  14. 3(b)10 marksSolve for x ∈ R the inequality (2x + 3) / (3x + 4) < 1.
  15. 4(a)8 marksFind the two square roots of the complex number 5 - 12i in the form x + yi, where x, y ∈ R.
  16. 4(b)(i)5 marksIf z = x + yi, where x, y ∈ R, y ≠ 0, find the real and imaginary parts of z + 1/z.
  17. 4(b)(ii)5 marksFind and identify the locus of the points for which the imaginary part of z + 1/z is zero.
  18. 4(c)(i)4 marksFind |AB|.
  19. 4(c)(ii)3 marksFind the position vector of the mid-point of AB.
  20. 5(a)5 marksBy expressing x - 4 as (√x + 2)(√x - 2), find lim (x→4) (√x - 2) / (x - 4). Hence, find lim (x→4) (√x - 2) / (x² - 5x + 4).
  21. 5(b)(i)3 marksObtain an expression for f'(x).
  22. 5(b)(ii)2 marksFind the stationary point(s) of f.
  23. 5(b)(iii)5 marksDetermine the nature of the stationary point(s) of f.
  24. 5(b)(iv)5 marksSketch the curve.
  25. 5(b)(v)5 marksFind the area bounded by the curve and the interval of the x-axis, -2 < x < 0.
  26. 6(a)6 marksUsing the substitution u = x + 3 or otherwise, evaluate ∫ x√(x+3) dx.
  27. 6(b)(i)8 marksFind the volume generated by direct integration.
  28. 6(b)(ii)11 marksFind the volume generated by the trapezium rule, using five coordinates.

More CAPE Pure Mathematics Unit 1 papers