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CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1

39 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)(i)2 marksShow that (x - 1) is a factor of f(x) for all values of p.
  2. 1(a)(ii)2 marksIf (x - 2) is a factor of f(x), find the value of p.
  3. 1(b)4 marksGiven that sum_(r=1)^n r = n/2(n + 1), show that sum_(r=1)^n (3r + 1) = 1/2 n(3n + 5).
  4. 2(a)6 marksLet A = {x : 2 <= x <= 7} and B = {x : |x - 4| <= h}, h in R. Find the LARGEST value of h for which B subset of A.
  5. 2(b)3 marksLet x, y, k in R such that (x + 1/2 y)^2 + ky^2 = x^2 + xy + y^2. Find the value of k.
  6. 3(a)(i)2 marksFind a, b in R such that (3x)/(x + 1) - 2 = (ax + b)/(x + 1), where x != -1.
  7. 3(a)(ii)4 marksHence, find the range of values of x in R for which (3x)/(x + 1) > 2.
  8. 3(b)4 marksWithout the use of calculators or tables, show that 4^2 / (sqrt(2) * 8^(-1/3)) = 2^4 (sqrt(2)).
  9. 4(a)(i)2 marksFind the value of p and of q.
  10. 4(a)(ii)1 markFind the range of the function f(x) for the given domain.
  11. 4(b)(i)1 markDetermine whether f(x) is surjective (onto).
  12. 4(b)(ii)1 markDetermine whether f(x) is injective (one-to-one).
  13. 4(b)(iii)1 markDetermine whether f(x) has an inverse.
  14. 5(a)3 marksFind the values of m, n in R for which the system of equations possesses a unique solution.
  15. 5(b)2 marksFind the values of m, n in R for which the system of equations is inconsistent.
  16. 5(c)2 marksFind the values of m, n in R for which the system of equations possesses infinitely many solutions.
  17. 6(a)2 marksFind the equation of the line through P and Q.
  18. 6(b)3 marksFind the coordinates of the point Q.
  19. 6(c)2 marksFind the EXACT length of the line segment PQ.
  20. 7(a)5 marksFind the EXACT length of AC.
  21. 7(b)3 marksFind the EXACT length of AB.
  22. 8(a)5 marksSolve the equation 4 cos^2 theta - 4 sin theta - 1 = 0 for 0 <= theta <= pi.
  23. 8(b)3 marksShow that (1 - cos 2x)/(1 + cos 2x) = tan^2 x.
  24. 9(a)2 marksThe roots of the quadratic equation x^2 + 6x + k = 0 are -3 + 2i and -3 - 2i. Find the value of the constant k.
  25. 9(b)6 marksFind the real numbers u and v such that (u + 2i)/(3 - 4i) = 1 + vi.
  26. 10(a)7 marksFind x, y in R such that xp + yq = -3i - 11j.
  27. 10(b)2 marksShow that p and q are perpendicular.
  28. 11(a)3 marksFind lim_(x -> 1) (x^2 + x - 2)/(x^2 - 3x + 2).
  29. 11(b)4 marksFind the values of x in R such that the function f(x) = (9 - x^2)/((x^2 - 3)(|x| - 3)) is discontinuous.
  30. 12(a)6 marksDetermine the nature of the critical value(s) of f(x).
  31. 12(b)3 marksDifferentiate, with respect to x, f(x) = sin^2(x^2).
  32. 13(a)5 marksFind the coordinates of each of the stationary points, A and B.
  33. 13(b)4 marksFind the equation of the normal to the curve f(x) = x(x^2 - 12) at the origin.
  34. 14(a)1 markExpress the shaded area, A, as the difference of two definite integrals.
  35. 14(b)2 marksHence, show that A = 16 int_2^3 x^(-2) dx - 1/2 int_2^3 x dx + int_2^3 dx.
  36. 14(c)3 marksFind the value of A.
  37. 15(a)2 marksShow that int_0^pi x sin x dx = int_0^pi (pi - x) sin x dx.
  38. 15(b)(i)2 marksHence, show that int_0^pi x sin x dx = pi int_0^pi sin x dx - int_0^pi x sin x dx.
  39. 15(b)(ii)5 marksHence, show that int_0^pi x sin x dx = pi.

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