A-level · Complex numbers

The Argand diagram, modulus and argumentExam-style questions

Picturing a complex number z = a + bi as a point (or vector) on the Argand diagram, its modulus |z| = √(a²+b²) as the distance from the origin, and its argument arg z as the angle from the positive real axis — taken as the principal value in (−π, π], which means getting the quadrant right, not just arctan(b/a). Geometric effects: the conjugate reflects in the real axis, addition is vector addition, and |z₁ − z₂| is the distance between the two points.

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

DifficultyAllLevel 1Level 2Level 3Level 4
  1. 1
    A complex number zz has modulus 2\sqrt2 and argument −π4-\dfrac{\pi}{4}. Express zz in the form a+bia + bi. (3 marks)
    Level 4 · 3 marksTry it
  2. 2
    On an Argand diagram, the points OO, AA and BB represent 00, z1=4+2iz_{1} = 4 + 2i and z2=−1+3iz_{2} = -1 + 3i respectively.

    Find the area of triangle OABOAB. (4 marks)
    Level 4 · 4 marksTry it
  3. 3
    Given z1=3+4iz_{1} = 3 + 4i and z2=5−12iz_{2} = 5 - 12i:
    (a) Find ∣z1+z2∣|z_{1} + z_{2}|, giving your answer as a simplified surd. (3 marks)
    (b) Verify that ∣z1+z2∣<∣z1∣+∣z2∣|z_{1} + z_{2}| < |z_{1}| + |z_{2}|, and state the geometric fact this illustrates. (2 marks)
    Level 4 · 5 marksTry it
  4. 4
    z=5+10i1−2iz = \frac{5 + 10i}{1 - 2i}
    (a) Express zz in the form a+bia + bi. (3 marks)
    (b) Find ∣z∣|z| and arg⁡z\arg z, giving the argument in radians to 22 decimal places. (3 marks)
    Level 4 · 6 marksTry it