A-level · Complex numbers

Complex numbers in Cartesian formExam-style questions

The imaginary unit i with i² = −1, real and imaginary parts, equality of complex numbers, and the four operations in the form a + bi — addition, subtraction, multiplication, and division by realising the denominator with the conjugate.

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17 questions · A Level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 1Level 2Level 3Level 4Level 5
  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