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CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1

34 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)2 marksFrom the graph, state the value of EACH of f(0) and f(2).
  2. 1(b)3 marksHence, or otherwise, find the value of EACH of the constants h and k.
  3. 1(c)4 marksFactorise f(x) completely.
  4. 2(a)4 marksFind the range of values of the real number x < 0 such that x^2 - 2|x| - 3 < 0.
  5. 2(b)4 marksShow that if x and y are real numbers such that x < y, then for any real number k < 0, kx > ky.
  6. 3(a)2 marksWithout using calculators or tables, show that (sqrt(11) + sqrt(7)) = 4 / (sqrt(11) - sqrt(7)).
  7. 3(b)(i)2 marksGiven that x + 1/x = 1, by considering (x + 1/x)^2, show that x^2 + 1/x^2 = -1.
  8. 3(b)(ii)5 marksHence, or otherwise, find the value of x^3 + 1/x^3.
  9. 47 marksSolve the following pair of equations simultaneously: x - 2y = -3 x^2 + 3y = 7
  10. 5(a)2 marksShow that f is one-to-one (injective).
  11. 5(b)5 marksFind the value(s) of x in R such that f(f(x)) = f(x) + 6.
  12. 6(a)(i)2 marksFind the coordinates of M.
  13. 6(a)(ii)2 marksFind the gradient of the line through A and B.
  14. 6(a)(iii)2 marksFind the equation of the line through M and N.
  15. 6(b)2 marksThe point P on AB divides AB internally such that the ratio AP : PB is 3 : 1. Find the coordinates of P.
  16. 7(a)5 marksExpress f(theta) = sqrt(2) cos theta - sin theta in the form R cos(theta + alpha).
  17. 7(b)1 markHence, find the minimum value of f(theta), where 0 <= theta <= 2pi.
  18. 7(c)2 marksDetermine the value of theta, 0 <= theta <= 2pi, at which the minimum value of f(theta) occurs.
  19. 8(a)4 marksFind the range of values of k for which the quadratic equation x^2 + 2kx + 9 = 0 has complex roots.
  20. 8(b)4 marksExpress the complex number (2 + 3i) / (3 + 4i) in the form x + yi, where x and y are real numbers.
  21. 9(a)3 marksExpress the position vector of EACH of A, B and C in terms of i and j.
  22. 9(b)6 marksIf vec(AB) = vec(CD), find the position vector of D in terms of i and j.
  23. 107 marksFind the values of theta, 0 <= theta <= 2pi, for which the vectors cos(theta)i + sqrt(3)j and (1/4)i + sin(theta)j are parallel.
  24. 11(a)5 marksUse the result that (sqrt(x + h) + sqrt(x))(sqrt(x + h) - sqrt(x)) = h to show that lim_{h -> 0} (sqrt(x + h) - sqrt(x)) / h = 1 / (2 sqrt(x)).
  25. 11(b)1 markDeduce, from first principles, the derivative with respect to x of y = sqrt(x).
  26. 12(a)3 marksFind the real values of x for which the function f(x) = x / (x^2 - 2x - 8) is discontinuous.
  27. 12(b)5 marksShow that the equation x^3 = 8 + 4x has a root in the closed interval [2, 3].
  28. 13(a)3 marksFind the value of the constant k.
  29. 13(b)2 marksFind the value of d^2y/dx^2 at P.
  30. 13(c)4 marksFind the equation of the normal to the curve at P.
  31. 14(a)6 marksFind the coordinates of the stationary points of the function f.
  32. 14(b)3 marksDetermine the nature of the stationary points of f.
  33. 15(a)4 marksFind the coordinates of EACH of the points P, Q and R.
  34. 15(b)4 marksFind the TOTAL area bounded by the curve shown above, the x-axis and the lines x = -1 and x = 2.

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