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CAPE Pure Mathematics Unit 1 · May/June 2014 · 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)5 marksConstruct a truth table for the statement (p → q) ∧ (r → q).
  2. 1(b)(i)3 marksState, giving a reason for your answer, if ⊕ is commutative in R.
  3. 1(b)(ii)a)4 marksfind the value of a
  4. 1(b)(ii)b)3 marksfactorize f(x) completely.
  5. 1(c)10 marksUse mathematical induction to prove that 1² + 3² + 5² + ..... + (2n - 1)² = n/3 (4n² - 1) for n ∈ N.
  6. 2(a)(i)a)3 marksDetermine, in terms of x, f²(x)
  7. 2(a)(i)b)3 marksDetermine, in terms of x, f[g(x)].
  8. 2(a)(ii)1 markHence, or otherwise, state the relationship between f and g.
  9. 2(b)5 marksGiven that a² + b³ + 3a²b = 5ab², show that 3 log((a+b)/2) = log a + 2 log b.
  10. 2(c)(i)4 marksSolve EACH of the following equations: eˣ + 1/eˣ - 2 = 0
  11. 2(c)(ii)4 marksSolve EACH of the following equations: log₂(x + 1) – log₂(3x + 1) = 2
  12. 2(d)5 marksWithout the use of a calculator, show that (√3-1)/(√3+1) + (√3+1)/(√3-1) + (√2-1)/(√2+1) + (√2+1)/(√2-1) = 10.
  13. 3(a)(i)4 marksProve that (cot y - cot x)/(cot x + cot y) = sin(x - y)/sin(x + y).
  14. 3(a)(ii)8 marksHence, or otherwise, find the possible values for y in the trigonometric equation (cot y - cot x)/(cot x + cot y) = 1, 0 ≤ y ≤ 2π, when sin x = 1/2, 0 ≤ x ≤ π/2.
  15. 3(b)(i)4 marksExpress f(θ) = 3 sin 2θ + 4 cos 2θ in the form r sin (2θ + α) where r > 0 and 0 < α < π/2.
  16. 3(b)(ii)a)4 marksHence, or otherwise, determine the value of θ, between 0 and 2π radians, at which f(θ) is a minimum
  17. 3(b)(ii)b)5 marksthe minimum and maximum values of 1/(7-f(θ))
  18. 4(a)(i)3 marksShow that the coordinates of the centre of the circle, C, where L₁ and L₂ intersect are (2, 3).
  19. 4(a)(ii)3 marksdetermine the coordinates of B.
  20. 4(a)(iii)3 marksA point, p, moves in the x - y plane such that its distance from C (2, 3) is always √2 units. Determine the locus of p.
  21. 4(b)6 marksDetermine the Cartesian equation of the curve, S.
  22. 4(c)(i)4 marksExpress EACH of the vectors PQ, QR and RP in the form xi + yj + zk.
  23. 4(c)(ii)6 marksHence, find the value of λ, given that PQR is right-angled with the side PQ as hypotenuse.
  24. 5(a)(i)4 marksFind the value of a if f(x) is continuous at x = 3.
  25. 5(a)(ii)5 marksfind the value of b.
  26. 5(b)(i)8 marksLet y = 1/√x. Using first principles, find dy/dx.
  27. 5(b)(ii)4 marksIf y = x/√(1+x), determine an expression for dy/dx. Simplify the answer FULLY.
  28. 5(c)4 marksFind dy/dx in terms of θ. Simplify the answer as far as possible.
  29. 6(a)(i)a)4 marksFind the equation of the curve
  30. 6(a)(i)b)8 marksFind the coordinates of the stationary points and determine their nature.
  31. 6(a)(ii)4 marksSketch the curve in (a) (i) a) above, clearly marking ALL stationary points and intercepts.
  32. 6(b)(i)5 marksEvaluate ∫₀³ f(x) dx.
  33. 6(b)(ii)4 marksFind the volume generated by rotating the area bounded by the curve in (b) (i) above, the x-axis, and the lines x = 0 and x = 2 about the x-axis.

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