Browse Complex numbers

79 questions at your level, grouped by concept

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Complex numbers in Cartesian form

17 questions

LessonNot started
  1. 1
    Expand and simplify (3+2i)(4−5i)(3 + 2i)(4 - 5i), giving your answer in the form a+bia + bi. (2 marks)
    Level 2 · 2 marksTry it
  2. 2
    The complex number z=5−2iz = 5 - 2i, and z∗z^{*} denotes its conjugate.
    (a) Find zz∗zz^{*}. (2 marks)
    (b) Write down z+z∗z + z^{*} and z−z∗z - z^{*}. (2 marks)
    Level 2 · 4 marksTry it

The Argand diagram, modulus and argument

15 questions

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  1. 1
    Find the modulus and argument of z=1+3 iz = 1 + \sqrt{3}\,i, giving the argument as an exact multiple of π\pi. (3 marks)
    Level 2 · 3 marksTry it
  2. 2
    The complex number z=2+5iz = 2 + 5i is shown on an Argand diagram.
    (a) Describe the geometric relationship between the points representing zz and z∗z^{*}. (1 mark)
    (b) Describe the geometric relationship between the points representing zz and −z-z. (1 mark)
    (c) Find ∣z∣|z|, giving your answer as an exact surd. (2 marks)
    Level 2 · 4 marksTry it

Complex roots of polynomial equations

15 questions

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  1. 1
    Solve the equation z2+2z+5=0z^2 + 2z + 5 = 0, giving your answers in the form a+bia + bi. (3 marks)
    Level 2 · 3 marksTry it
  2. 2
    Solve the equation z2−6z+25=0z^{2} - 6z + 25 = 0, giving your answers in the form a+bia + bi. (3 marks)
    Level 2 · 3 marksTry it

Loci in the Argand diagram

16 questions

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  1. 1
    Find the Cartesian equation of the locus ∣z−3i∣=2|z - 3i| = 2, and describe it geometrically. (3 marks)
    Level 2 · 3 marksTry it
  2. 2
    The complex number zz satisfies ∣z−4∣=3|z - 4| = 3.
    (a) Describe the locus of zz fully. (2 marks)
    (b) Find the least and greatest values of ∣z∣|z|. (2 marks)
    Level 2 · 4 marksTry it

Modulus–argument (polar) form

16 questions

LessonNot started
  1. 1
    Express each of the following in the form r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta), where r>0r > 0 and −π<θ≤π-\pi < \theta \le \pi.
    (a) 3+i\sqrt{3} + i (2 marks)
    (b) −1+i-1 + i (2 marks)
    Level 2 · 4 marksTry it
  2. 2
    The complex number z=4(cos⁡2π3+isin⁡2π3)z = 4\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right).

    Express zz in the form a+bia + bi, giving aa and bb exactly. (2 marks)
    Level 2 · 2 marksTry it
  3. 3
    Express each of the following in the form r(cos⁡θ+isin⁡θ)r(\cos\theta + i\sin\theta) with r>0r > 0 and −π<θ⩽π-\pi < \theta \leqslant \pi:
    (a) z=−5iz = -5i (2 marks)
    (b) z=−4z = -4 (2 marks)
    Level 2 · 4 marksTry it