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Complex Numbers · CAPE Pure Mathematics Unit 2

96 past-paper questions on Complex Numbers, part of Complex Numbers and Calculus II, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Complex Numbers
  1. 1(b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Given that log_a(bc) = x, log_b(ca) = y, log_c(ab) = z and a != b != c, show that a^x b^y c^z = (abc)^2.
  2. 5(c)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Calculate the value of p.
  3. 5(c)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Determine the probability that there are more than 3 accidents in a week.
  4. 6(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Express the complex number \frac{2 - 3i}{5 - i} in the form \lambda(1 - i).
  5. 6(a)(ii)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2State the value of \lambda.
  6. 6(a)(iii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Verify that \left(\frac{2 - 3i}{5 - i}\right)^4 is a real number and state its value.
  7. 6(b)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Show that z + \bar{z} = 6 z\bar{z}.
  8. 6(b)(ii)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Show that as t varies, T lies on a circle, and state the coordinates of the centre of this circle.
  9. 5(c)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Find the values of h, k \in \mathbb{R} such that 3 + 4\mathrm{i} is a root of the quadratic equation z^2 + hz + k = 0.
  10. 5(c)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Use De Moivre's theorem for (\cos \theta + \mathrm{i} \sin \theta)^3 to show that \cos 3\theta = 4\cos^3 \theta - 3\cos \theta.
  11. 5(c)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find complex numbers u = x + \mathrm{i}y such that x and y are real numbers and u^2 = -15 + 8\mathrm{i}.
  12. 5(c)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Hence, or otherwise, solve for z the equation z^2 - (3 + 2\mathrm{i})z + (5 + \mathrm{i}) = 0.
  13. 5(c)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Express the complex number (2 + 3i) + \frac{i - 1}{i + 1} in the form a + ib, where a and b are both real numbers.
  14. 5(c)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Given that 1 - i is the root of the equation z^3 + z^2 - 4z + 6 = 0, find the remaining roots.
  15. 5(c)(i) a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that (1 - i) is one of the square roots of -2i.
  16. 5(c)(i) b)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the other square root of -2i.
  17. 5(c)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Hence, find the roots of the quadratic equation z^2 - (3 + 5i)z + (8i - 4) = 0.
  18. 6(a)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Draw the points A and B on an Argand diagram.
  19. 6(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, or otherwise, show that the argument of \frac{(1 + \sqrt{2} + i)}{1 - i} is EXACTLY \frac{3\pi}{8}.
  20. 6(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find ALL complex numbers, z, such that z^2 = i.
  21. 6(b)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, find ALL complex roots of the equation z^2 - (3 + 5i)z - (4 - 7i) = 0.
  22. 6(c)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Use de Moivre's theorem to show that \cos 6\theta = \cos^6 \theta - 15\cos^4 \theta \sin^2 \theta + 15\cos^2 \theta \sin^4 \theta - \sin^6 \theta.
  23. 1(c)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Use de Moivre's theorem to prove that \cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.
  24. 1(d)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Write the complex number z = (-1 + i)^7 in the form r e^{i\theta}, where r = |z| and \theta = \arg z.
  25. 1(d)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Hence, prove that (-1 + i)^7 = -8(1 + i).
  26. 1(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Determine the nature of the roots of the equation.
  27. 1(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Express \alpha and \beta in the form r e^{i\theta}, where r is the modulus and \theta is the argument, where -\pi < \theta \le \pi.
  28. 1(b)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Using de Moivre's theorem, or otherwise, compute \alpha^3 + \beta^3.
  29. 1(b)(iv)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Hence, or otherwise, obtain the quadratic equation whose roots are \alpha^3 and \beta^3.
  30. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The conjugate of the complex number 7 + \frac{1}{2}i is
  31. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The complex number z = \sqrt{3} + i can be expressed as
  32. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1One square root of 3 - 4i is
  33. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The complex number z = \frac{1}{1-i} can be represented on an Argand diagram as
  34. Q151 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The argument of the complex number z = -\frac{1}{2} + i\frac{\sqrt{3}}{2} is
  35. 1(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Express the quotient \frac{z_3}{z_2} in the form x + iy where x, y \in \mathbb{R}.
  36. 1(a)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Given that \arg w = \arg z_3 - [\arg z_1 + \arg z_2], |z_1| = 1 and \arg z_1 = \frac{\pi}{12}, rewrite w = \frac{z_3}{z_1 z_2} in the form r e^{i\theta} where r = |w| and \theta = \arg w.
  37. 1(b)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2A complex number v = x + iy is such that v^2 = 2 + i. Show that x^2 = \frac{2 + \sqrt{5}}{2}.
  38. 6(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Construct a tree diagram to show the probabilities that Alicia arrives at school.
  39. 6(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2What is the probability that Alicia is at school on any given school day?
  40. 6(a)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Given that Alicia is at school today, determine the probability that it is a rainy day.
  41. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The complex number z = \frac{1}{1 - i} can be represented on an Argand diagram as
  42. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1Which of the following is a sketch of the locus of the point represented by the complex number z, given that |z + 5i| = 3?
  43. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The expression i [(1 + i)^2 - (1 - i)^2] is equal to
  44. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The value of \left(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2}\right)^2 is
  45. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1One square root of 3 - 4i is
  46. 1(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Calculate (alpha + beta) and (alpha * beta).
  47. 1(a)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Hence, show that an equation with roots 1/(alpha - 2) and 1/(beta - 2) is given by 10x^2 + 2x + 1 = 0.
  48. 1(b)(i)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Complete the Argand diagram to illustrate u.
  49. 1(b)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2On the same Argand plane, sketch the circle with equation |z - u| = 3.
  50. 1(b)(iii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Calculate the modulus and principal argument of z = (u / v)^5.
  51. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Expressed in the form a + bi, where a, b, \in \mathbb{R}, \frac{-2 + 2i}{1 + i} =
  52. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The complex number z = \sqrt{3} + i can be expressed as
  53. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Which of the following is a sketch of the locus of the point represented by the complex number z, given that |z + 5i| = 3?
  54. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1One square root of 3 - 4i is
  55. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1\bar{z} is the conjugate of z. Which of the following are always true? I. |\bar{z}| = |z| II. \arg z = \arg \bar{z} III. z\bar{z} is real
  56. Q131 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The argument of the complex number z = -\frac{1}{2} + i\frac{\sqrt{3}}{2} is
  57. 1(c)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Find the complex numbers u = x + \mathrm{i}y such that x and y are real and u^2 = -15 + 8\mathrm{i}.
  58. 1(c)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, or otherwise, solve the equation z^2 - (3 + 2\mathrm{i})z + (5 + \mathrm{i}) = 0, for z.
  59. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Which of the following is a sketch of the locus of the point represented by the complex number z, given that |z - 5i| = 3?
  60. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The roots of the equation x^2 + 1 = 0 are
  61. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Given that z = -1 + \sqrt{3}i, then the exponential form of the complex number z is
  62. Q61 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The locus of the points described by a complex number z is given by |z - 1 - 2i| = 3. The locus describes a circle with
  63. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The complex number z = \frac{1}{1 - i} can be represented on an Argand diagram as
  64. Q91 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The expression i[(1 + i)^2 - (1 - i)^2] is equal to
  65. Q121 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1One square root of 3 - 4i is
  66. Q141 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The principal value of the argument of the complex number -2 + 2i is
  67. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Which of the following statements is true?
  68. 1(c)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Use De Moivre's theorem to show that sin(5θ) = sin^5(θ) - 10 sin^3(θ) cos^2(θ) + 5 cos^4(θ) sin(θ).
  69. 1(d)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Write the complex number z = (1 - i) in the form r e^(iθ), where r = |z| and θ = arg(z).
  70. 1(d)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Hence, show that (1 - i)^9 = 16(1 - i).
  71. Q11 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Expressed in the form a + bi, where a, b, c \in \mathbb{R}, \frac{-2 + 2i}{1 + i} =
  72. Q21 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The quadratic equation with roots 2 \pm i\sqrt{3} is
  73. Q31 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The modulus of the complex number \frac{1}{2} - \frac{1}{2}i is
  74. Q41 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The value of \left[\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right]^4 is
  75. Q51 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1If |z - 5 + 2i| = 3, the locus of the point (x, y) is
  76. Q71 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The complex number z = \sqrt{3} + i can be expressed as
  77. Q81 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that z = -1 + \sqrt{3}\,i, then the exponential form of the complex number z is
  78. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Which of the following statements is true?
  79. 1(a)4 marks· Pure Mathematics · Unit 2 Q1 1(a)Express the complex number 5-3i / 4+2i in the form x + yi where x and y are real numbers.
  80. 1(a)7 marks· Pure Mathematics · Unit 2 Q1 1(a)Using DeMoivre's theorem, prove that sin 5θ / sin θ = 5 - 20 sin² θ + 16 sin⁴ θ.
  81. 1(a)(i)3 marks· Pure Mathematics · Unit 2 Q1 1(a)(i)Sketch the locus of z on the Argand diagram below.
  82. 1(a)(i)6 marks· Pure Mathematics · Unit 2 Q1 1(a)(i)Show that the complex number z₁/z₂ = (√5)/6 e^(i7π/12).
  83. 1(a)(ii)4 marks· Pure Mathematics · Unit 2 Q1 1(a)(ii)Determine the Cartesian equation of the locus of z.
  84. 1(a)(ii)3 marks· Pure Mathematics · Unit 2 Q1 1(a)(ii)Hence, without using a calculator, determine the value of (z₁/z₂)².
  85. 1(b)7 marks· Pure Mathematics · Unit 2 Q1 1(b)Given that 3 + 5i is a root of the quadratic equation z² + pz + q = 0, determine the values of p, q ∈ R.
  86. 1(b)4 marks· Pure Mathematics · Unit 2 Q1 1(b)One root of a quadratic equation is given as 4 – 7i. Determine the quadratic equation with real coefficients which has the root 4 – 7i.
  87. 1(b)6 marks· Pure Mathematics · Unit 2 Q1 1(b)Use de Moivre's theorem to prove that sin 5x = 16 sin⁵ x - 20 sin³ x + 5 sin x.
  88. 1(b)(i)a)2 marks· Pure Mathematics · Unit 2 Q1 1(b)(i)a)Calculate the EXACT value of |z|.
  89. 1(b)(i)b)2 marks· Pure Mathematics · Unit 2 Q1 1(b)(i)b)Calculate the EXACT value of arg z.
  90. 1(b)(ii)4 marks· Pure Mathematics · Unit 2 Q1 1(b)(ii)Hence, use de Moivre's theorem to determine the value of z^12.
  91. 1(c)5 marks· Pure Mathematics · Unit 2 Q1 1(c)A complex number, z, is such that arg(z-2) = π/2 and arg(z) = π/3. Determine the complex number z.
  92. 1(c)7 marks· Pure Mathematics · Unit 2 Q1 1(c)Given that 3z² - pz + 2q = 0 has root 3 - 5i, determine the values of p and q where p, q ∈ R.
  93. 1(c)(i)3 marks· Pure Mathematics · Unit 2 Q1 1(c)(i)Write the complex number z = (-1 + √3 i)⁷ in the form re^(iθ), where r = |z| and θ = arg z.
  94. 1(c)(ii)6 marks· Pure Mathematics · Unit 2 Q1 1(c)(ii)Hence, prove that (-1 + √3 i)⁷ = 64 (-1 + √3 i).
  95. 1(d)6 marks· Pure Mathematics · Unit 2 Q1 1(d)Using DeMoivre's theorem, determine the square root of 1 - i√3, expressing your answer in radians.
  96. 2(b)7 marks· Pure Mathematics · Unit 2 Q2 2(b)Use DeMoivre's theorem to show that (1 + 3i)⁴ = 28 - 96i.

Related topics in Complex Numbers and Calculus II