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CAPE Pure Mathematics Unit 1 · May/June 2017 · Paper 2

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)(i)1 markWrite EACH of the statements below in terms of p and q. It is not raining or John is sick.
  2. 1(a)(ii)1 markWrite EACH of the statements below in terms of p and q. If it is raining then John is not sick.
  3. 1(b)(i)1 markProve that * is commutative.
  4. 1(b)(ii)2 marksShow that the identity element of * is 3.
  5. 1(c)(i)4 marksShow that a = 2 and b = -30.
  6. 1(c)(ii)9 marksHence, solve ax³ + 9x² – 11x + b = 0.
  7. 1(d)7 marksUse mathematical induction to prove that 8+16+24+ 32 + ... + 8n = 4n(n + 1) for all n ∈ N.
  8. 2(a)(i)5 marksGiven that a² + b² = 14ab, prove that ln((a+b)/4) = (1/2)(ln a + ln b).
  9. 2(a)(ii)6 marksSolve the equation 2ˣ + 3(2⁻ˣ) = 4. [Your response may be expressed in terms of logarithms.]
  10. 2(b)(i)2 marksOn the diagram, insert the asymptotes for the function f.
  11. 2(b)(ii)4 markssketch the graph of f⁻¹, the inverse of f showing the asymptotes for f⁻¹.
  12. 2(c)8 marksGiven that α, β and γ are the roots of the equation x³ + 3x + 2 = 0, form an equation whose roots are βγ, αγ and αβ.
  13. 3(a)(i)4 marksProve the identity tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)).
  14. 3(a)(ii)6 marksGiven that sin A = 3/5 and cos B = -1/2, where angle A is acute and angle B is obtuse, express tan (A + B) in the form a + b√3, where a and b are real numbers.
  15. 3(b)6 marksSolve the equation sin²θ - 2cos²θ + 3cosθ + 5 = 0 for 0 ≤ θ ≤ 4π.
  16. 3(c)(i)3 marksExpress f(θ) = 6cosθ + 8sinθ in the form r sin(θ + α) where 0 ≤ α ≤ 90°.
  17. 3(c)(ii)6 marksHence, or otherwise, find the general solution of f(θ) = 2.
  18. 4(a)(i)3 marksExpress the equation of C₂ in the form (x – h)² + (y - k)² = k.
  19. 4(a)(ii)7 marksThe equation of the line L₁ is x + 3y = 3. Determine whether L₁ is a tangent to the circle, C₁, in a (i) on page 16.
  20. 4(b)(i)2 marksExpress the vector PQ in the form xi + yj + zk.
  21. 4(b)(ii)6 marksDetermine the Cartesian equation of the plane which passes through the point Q and is perpendicular to PQ.
  22. 4(c)(i)5 marksShow that L₁ and L₂ intersect.
  23. 4(c)(ii)2 marksHence, determine the coordinates of the point of intersection of the two lines.
  24. 5(a)4 marksDetermine the value of k for which f(x) = (x²-1)/(x-1) for x ≠ 1 and f(x) = k for x = 1 is continuous for all values of x.
  25. 5(b)(i)3 marksFind dy/dx in terms of t.
  26. 5(b)(ii)6 marksHence, determine all points of C such that dy/dx = 0.
  27. 5(c)(i)(a)9 marksShow that y(dy/dx) - 2x = 0.
  28. 5(c)(i)(b)9 marksShow that d²y/dx² - 4/y³ = 0.
  29. 5(c)(ii)3 marksHence, find the value of d²y/dx² when x = 0.
  30. 6(a)(i)1 markOn the axes below, sketch triangle PQR.
  31. 6(a)(ii)7 marksDetermine the equations of EACH of the following: PQ, QR, PR.
  32. 6(a)(iii)7 marksHence, use integration to determine the area of triangle PQR.
  33. 6(b)5 marksThe voltage in a circuit, V, satisfies the equation dV/dt + V/2.5 = 0. Given that V = 25 volts when t = 0 seconds, write an expression for V in terms of t.
  34. 6(c)5 marksGiven that ∫[-1,3] (3f(x) + g(x)) dx = 5 and ∫[-1,3] (5f(x) - 2g(x)) dx = 1, determine ∫[-1,3] f(x) dx and ∫[-1,3] g(x) dx.

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