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CAPE Pure Mathematics Unit 1 · May/June 2016 · 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)(i)7 marksShow that p = -25 and q = -12.
  2. 1(a)(ii)6 marksHence, solve the equation f(x) = 0.
  3. 1(b)6 marksUse mathematical induction to prove that 6ⁿ - 1 is divisible by 5 for all natural numbers n.
  4. 1(c)(i)4 marksComplete the truth table below:
  5. 1(c)(ii)2 marksState, giving a reason for your response, whether the following statements are logically equivalent: p → q and ¬(p ∨ q) → (p ∧ q).
  6. 2(a)6 marksSolve the following equation for x: log₂(10 - x) + log₂x = 4.
  7. 2(b)8 marksDetermine whether f is bijective, that is, both one-to-one and onto.
  8. 2(c)(i)3 marksState the values of α + β + γ, αβ + αγ + βγ and αβγ.
  9. 2(c)(ii)8 marksHence, or otherwise, determine an equation with integer coefficients which has roots 1/α², 1/β² and 1/γ². Note: (αβ)² + (αγ)² + (βγ)² = (αβ + αγ + βγ)² – 2αβγ (α + β + γ) and α² + β² + γ² = (α + β + γ)² – 2 (αβ + αγ +…
  10. 3(a)(i)4 marksShow that sec²θ = cosecθ / (cosecθ - sinθ).
  11. 3(a)(ii)5 marksHence, or otherwise, solve the equation cosecθ / (cosecθ - sinθ) = 4/3 for 0 ≤ θ ≤ 2π.
  12. 3(b)(i)5 marksExpress the function f(θ) = sinθ + cosθ in the form r sin(θ + α), where r > 0 and 0 ≤ α ≤ π/2.
  13. 3(b)(ii)5 marksHence, find the maximum value of f and the smallest non-negative value of θ at which it occurs.
  14. 3(c)6 marksProve that tan(A + B + C) = (tanA + tanB + tanC - tanA tanB tanC) / (1 - tanA tanB - tanA tanC - tanB tanC).
  15. 4(a)(i)3 marksGiven that sinθ = x, show that tanθ = x / √(1 - x²) where 0 < θ < π/2.
  16. 4(a)(ii)5 marksHence, or otherwise, determine the Cartesian equation of the curve defined parametrically by y = tan(2t) and x = sin(t) for 0 < t < π/2.
  17. 4(b)(i)3 marksCalculate the lengths of u and v respectively.
  18. 4(b)(ii)4 marksFind cosθ where θ is the angle between u and v in ℝ³.
  19. 4(c)5 marksDetermine the Cartesian equation of the locus of P.
  20. 4(d)5 marksDetermine the points of intersection of the circle C and the line L.
  21. 5(a)4 marksUse an appropriate substitution to find ∫x(x + 1)³ dx.
  22. 5(b)5 marksCalculate the volume of the solid that results from rotating R about the y-axis.
  23. 5(c)6 marksShow that ∫₀¹ (eˣ / (eˣ + e¹⁻ˣ)) dx = 1/2.
  24. 5(d)(i)7 marksSolve an appropriate differential equation to show that the number of bacteria present at any time can be modelled by the equation y = 10,000 e⁰.⁰²ᵗ.
  25. 5(d)(ii)3 marksDetermine the time required for the bacteria population to double in size.
  26. 6(a)4 marksFind the equation of the tangent to the curve f(x) = 2x³ + 5x² – x + 12 at the point where x = 3.
  27. 6(b)(i)4 marksCalculate the lim (x→0⁻) f(x) and lim (x→0⁺) f(x).
  28. 6(b)(ii)5 marksHence, determine the values of a and b such that f(x) is continuous at x = 0.
  29. 6(b)(iii)6 marksIf the value of b = 3, determine a such that f'(0) = lim (t→0) (f(0 + t) - f(0)) / t.
  30. 6(c)6 marksUse first principles to differentiate f(x) = √x with respect to x.

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