Browse Complex numbers

79 questions at your level, grouped by concept

DifficultyAllLevel 1Level 2Level 3Level 4Level 5

Complex numbers in Cartesian form

17 questions

LessonNot started
  1. 1
    (a) The complex number zz is given by z=2+4i1−iz = \dfrac{2+4\mathrm{i}}{1-\mathrm{i}}. Express zz in the form x+iyx + \mathrm{i}y, and find ∣z∣|z| and arg⁡z\arg z, giving the argument in radians correct to 33 decimal places. (5 marks)
    (b) The complex number ww satisfies ∣w−3+2i∣=2|w - 3 + 2\mathrm{i}| = 2. Describe the locus of ww in an Argand diagram, and find the greatest and least possible values of ∣w∣|w|, giving your answers in an exact form. (4 marks)
    Level 5 · 9 marksTry it
  2. 2
    The complex number z=x+yiz = x + yi satisfies

    ∣z∣=5andRe((1+i)z)=1|z| = 5 \qquad \text{and} \qquad \mathrm{Re}\big((1 + i)z\big) = 1

    Find the two possible values of zz. (6 marks)
    Level 5 · 6 marksTry it

Complex roots of polynomial equations

15 questions

LessonNot started
  1. 1
    It is given that 2−i2 - i is a root of the equation z4−6z3+10z2+2z−15=0z^4 - 6z^3 + 10z^2 + 2z - 15 = 0, whose coefficients are all real. Find the other three roots of the equation. (6 marks)
    Level 5 · 6 marksTry it
  2. 2
    The equation z3+az+b=0z^{3} + az + b = 0, where aa and bb are real, has a root 2+i2 + i.
    (a) Explain why the third root is −4-4. (3 marks)
    (b) Find the values of aa and bb. (3 marks)
    Level 5 · 6 marksTry it
  3. 3
    It is given that 3i3i is a root of the equation

    z4−4z3+14z2−36z+45=0z^{4} - 4z^{3} + 14z^{2} - 36z + 45 = 0
    (a) Write down another root. (1 mark)
    (b) Find a real quadratic factor of the quartic, and hence find the other quadratic factor. (3 marks)
    (c) Solve the equation completely. (2 marks)
    Level 5 · 6 marksTry it

Loci in the Argand diagram

16 questions

LessonNot started
  1. 1
    The complex number zz satisfies ∣z−4−3i∣=2|z - 4 - 3i| = 2.

    Find the least and greatest values of arg⁡z\arg z, giving your answers in radians to 33 decimal places. (5 marks)
    Level 5 · 5 marksTry it
  2. 2
    The region RR in the Argand diagram consists of the points satisfying both

    ∣z∣⩽4and∣z−4∣⩽∣z∣|z| \leqslant 4 \qquad \text{and} \qquad |z - 4| \leqslant |z|
    (a) Describe the two conditions geometrically. (2 marks)
    (b) Find the exact area of RR. (4 marks)
    Level 5 · 6 marksTry it

Modulus–argument (polar) form

16 questions

LessonNot started
  1. 1
    The complex numbers z1=1+3 iz_1 = 1 + \sqrt{3}\,i and z2=−1+iz_2 = -1 + i.
    (a) Express z1z_1 and z2z_2 in modulus–argument form, with arguments in (−π,π](-\pi, \pi]. (3 marks)
    (b) Hence find the modulus and argument of z1z2z_1 z_2, giving the argument in (−π,π](-\pi, \pi]. (2 marks)
    (c) Hence find the modulus and argument of z1z2\dfrac{z_1}{z_2}. (2 marks)
    Level 5 · 7 marksTry it
  2. 2
    Let z1=1+iz_{1} = 1 + i and z2=3+iz_{2} = \sqrt{3} + i.
    (a) Express z1z_{1} and z2z_{2} in polar form. (4 marks)
    (b) Write down the modulus and argument of z1z2z_{1}z_{2}. (2 marks)
    (c) By also expanding z1z2z_{1}z_{2} in Cartesian form, show that

    cos⁡5π12=3−122\cos\frac{5\pi}{12} = \frac{\sqrt{3} - 1}{2\sqrt{2}} (2 marks)
    Level 5 · 8 marksTry it
  3. 3
    The complex number z=−1+3 iz = -1 + \sqrt{3}\,i.
    (a) Express zz in polar form. (3 marks)
    (b) Find z2z^{2} in the form a+bia + bi, giving aa and bb exactly. (3 marks)
    (c) Show that z3z^{3} is real, and state its value. (2 marks)
    Level 5 · 8 marksTry it