CAIE A-Level Mathematics 3.9.1 Complex-Number Terminology and Forms

CAIE A-Level Mathematics 3.9.1 Complex-Number Terminology and Forms
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise using real and imaginary parts, modulus, argument and conjugates and converting complex numbers between Cartesian, modulus–argument and exponential forms.

How this is tested

  • equate real and imaginary parts separately after writing each number as x + iy
  • calculate |z| and arg z with the correct quadrant and specified argument interval
  • use conjugates and argument laws to express quotients or products in exact exponential form

Question 4(a)

[Maximum number: 2]

The complex number u is given by u=−1−i3u=-1-\mathrm{i} \sqrt{3}.

Express u in the form r(cos⁡θ+isin⁡θ)r(\cos \theta+\mathrm{i} \sin \theta), where r>0 and −π<θ⩽π-\pi<\theta \leqslant \pi. Give the exact values of r and θ\theta.
The complex number v is given by v=5(cos⁡16π+isin⁡16π)v=5\left(\cos \frac{1}{6} \pi+\mathrm{i} \sin \frac{1}{6} \pi\right).