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CAPE Pure Mathematics Unit 1 · May/June 2018 · Paper 2

33 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)4 marksLet p and q be any two propositions. Complete the truth table below.
  2. 1(a)(ii)1 markHence, state whether the statements ~(p ∨ q) and ~p ∧ ~q are logically equivalent. Justify your response.
  3. 1(b)(i)1 markCalculate 5 ⊗ 2.
  4. 1(b)(ii)3 marksProve that ⊗ is closed in R.
  5. 1(b)(iii)3 marksDetermine whether ⊗ is commutative.
  6. 1(c)7 marksCalculate the values of a, b and c.
  7. 1(d)6 marksSolve the logarithmic equation log₄(2x + 2) - log₄(x + 1) = 1.
  8. 2(a)(i)2 marksOn the diagram above, sketch the inverse of f(x).
  9. 2(a)(ii)3 marksUse a graphical method to show that f is bijective.
  10. 2(b)(i)4 marksProve that |x - y| ≤ |x - z| + |z - y| for all x, y, z ∈ R.
  11. 2(b)(ii)8 marksSolve the inequality |6x - 2| + x² ≤ 5.
  12. 2(c)8 marksDetermine the equation whose roots are 1/(αβ), 1/(αγ) and 1/(βγ).
  13. 3(a)(i)8 marksShow that (sin 2θ - cos 2θ + 1) / (cos 2θ + sin 2θ - 1) = sec 2θ + tan 2θ.
  14. 3(a)(ii)5 marksHence, or otherwise, determine the general solution of (sin θ - cos θ + 1) / (cos θ + sin θ - 1) = 0.
  15. 3(b)(i)3 marksCalculate sin 2A.
  16. 3(b)(ii)3 marksCalculate cos(A + B).
  17. 3(c)6 marksSolve the equation sin θ - √3 cos θ = 1, for -π ≤ θ ≤ π.
  18. 4(a)(i)3 marksDetermine the centre and radius of the circle, C.
  19. 4(a)(ii)6 marksShow that (1, -2) is one of the points of intersection of the circle, C, and the straight line, L.
  20. 4(a)(iii)4 marksDetermine the equation of the tangent to the circle, C, at the point (1, -2).
  21. 4(b)9 marksShow that u and v are NOT parallel.
  22. 4(c)3 marksDetermine the vector equation of a plane which passes through the point (1, 3, 0) and which is perpendicular to the vector 2i + 4j + 5k.
  23. 5(a)6 marksUse the substitution u = x² + 2 to determine ∫(x² + 2)³ (4x³) dx.
  24. 5(b)6 marksCalculate the area of the region lying between the parabolas y = x² and x = (1/8)y².
  25. 5(c)(i)1 markDetermine f'.
  26. 5(c)(ii)1 markDetermine f''.
  27. 5(c)(iii)11 marksCalculate the x coordinates of the stationary points of f(x) = 3x⁴ - 2x³ - 6x² + 6x and determine the nature of these stationary points.
  28. 6(a)(i)4 marksDetermine whether or not the limit of f at x = 1 exists.
  29. 6(a)(ii)2 marksDetermine whether f is continuous at x = 1.
  30. 6(b)(i)3 marksDetermine (d/dθ)g(x), in terms of θ.
  31. 6(b)(ii)8 marksDetermine the equation of the normal to the curve at the point (√3, 5/2).
  32. 6(c)(i)5 marksFormulate an appropriate differential equation and find the equation of the curve family.
  33. 6(c)(ii)13 marksHence, or otherwise, determine the equation of the curve given that it passes through the point (1, 3).

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