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CAPE Pure Mathematics Unit 1 · May/June 2003 · Paper 2

30 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)7 marksFind p and q. Hence, find the remainder when x² + px + q is divided by (x + 1).
  2. 1(b)(i)4 marksCalculate to 3 significant figures the length of BC.
  3. 1(b)(ii)4 marksCalculate to 3 significant figures the value of sin C.
  4. 1(c)(i)6 marksShow that the area of the shaded region is 2(π - 2√2).
  5. 1(c)(ii)4 marksUsing the cosine rule, show that the length of the chord AB is 4√(2 - √2).
  6. 2(a)(i)3 marksUse this diagram to assist you in sketching the function x = f(x - 1).
  7. 2(a)(ii)3 marksUse this diagram to assist you in sketching the function y = f(x) + 3.
  8. 2(a)(iii)3 marksUse this diagram to assist you in sketching the function y = |f(x)|.
  9. 2(b)(i)3 marksSketch the graph of f: A → B.
  10. 2(b)(ii)3 marksFind a set C such that C ⊂ A and f: C → B, is one-to-one.
  11. 2(b)(iii)4 marksBy considering the solutions of the equation f(x) = 8, show that f is NOT onto.
  12. 2(b)(iv)4 marksBy solving the equation f(x) = 0, show that f: A → B is NOT one-to-one.
  13. 2(b)(v)2 marksFind the range of values of y for which the equation f(x) = y possesses a solution.
  14. 3(a)7 marksFind the coordinates of A, B and C.
  15. 3(b)7 marksFind the equations of the lines CD and AD.
  16. 3(c)5 marksFind the coordinates of the point D.
  17. 3(d)6 marksCalculate the area of triangle ACD.
  18. 4(a)6 marksSolve cos 2θ - 3 cos θ = 1 for 0 ≤ θ < 2π.
  19. 4(b)6 marksIf cos A = 3/5, find tan (A/2).
  20. 4(c)5 marksProve that cos⁴A - sin⁴A + 1 = 2 cos²A.
  21. 4(d)8 marksGiven that sin A = 12/13 and sin B = 4/5, where A and B are acute angles, find cos (A - B) and sin (A + B).
  22. 5(a)7 marksShow that f(x) = 0 possesses a root in the interval [1/2, 1]. By considering suitable values of x greater than 1, show that there is another root of f(x) = 0 greater than 1.
  23. 5(b)(i)6 marksFind the coordinates of the stationary points of f(x).
  24. 5(b)(ii)5 marksFind the second derivative of f(x), and hence, determine which stationary point is a local maximum and which is a local minimum.
  25. 5(c)7 marksIf y = 1/(x² + 2), show that d²y/dx² = 2(3x² - 2)y³.
  26. 6(a)7 marksShow that the sum, S, of the areas of the rectangular strips is 1/(n+1) + 1/(n+2) + ... + 1/(2n).
  27. 6(b)(i)4 marksShow that for f(x) = x/(x² + 4), f'(x) = (4 - x²)/(x² + 4)².
  28. 6(b)(ii)4 marksHence, evaluate ∫₀² (12 - 3x²)/(x² + 4)² dx.
  29. 6(c)(i)3 marksSketch the curve y = x² + 1.
  30. 6(c)(ii)7 marksFind the volume obtained by rotating the portion of the curve between x = 0 and x = 1 through 2π radians about the y axis.

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