Complex numbers questions in Paper 3 mix algebra with geometry: arithmetic, roots of equations, polar form, and loci on an Argand diagram. The reports show a split between students who are very comfortable with the topic and students who leave questions blank. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Using the calculator. The reports are clear that calculators should not be used in complex number questions, because working has to be shown to justify the result.
- Argand diagrams without scales. The most common error in drawing diagrams is omitting a scale or showing it on only one axis. The scales on both axes must be the same, otherwise a circle becomes an ellipse and the sketch means nothing. One report says the sketch does not need to be large but should clearly show the features that matter.
- Loci: circles. The centre was in the wrong place, the radius was wrong, the circle was drawn in the wrong quadrant, or a circle was drawn when the scales required an ellipse. Reports also note that it was often unclear that the radius was 1 because the points on it could not be identified.
- Loci: lines. The line y = 3 drawn instead of x = 3, a full line when a half-line was needed, half-lines starting at the origin instead of at the given point (such as 1 + 2i), an argument inequality measured from the origin rather than from the point, a horizontal bisector where a vertical one was needed (or the reverse), and a half-line that did not bisect the angle at the origin.
- Loci: regions. The perpendicular bisector was drawn for the wrong pair of points, the wrong side was shaded, or an extra region was shaded.
- The argument. An argument given in the wrong quadrant, because the calculator's angle was used without checking where the number lies. Reports describe students who found r and θ first, then doubled an incorrect θ to get a value that looked right but earned no marks. Students also have difficulty with an exact method for the argument of a square root, and lose marks when combining angles.
- Right-angled triangle confusion. Students confused where the right angle was in a triangle involving a tangent to a circle, and used tan⁻¹ where sin⁻¹ was needed, or the reverse.
- Algebra. Errors when multiplying by the conjugate, such as 1 + (2i)² instead of 1 − (2i)², arithmetic errors when collecting real and imaginary parts, and claiming that (1 − √3 i)³ equals 1 − (√3 i)³.
- Roots of polynomials. Students were unsure how to interpret the roots of a quartic, gave only one root, made sign errors in b and c, or gave c = 1 instead of i.
- Reflections. Students knew that u* is a reflection of u, but often omitted the line of reflection or gave the wrong line.
- Blank answers. In at least one recent session many students left a complex-number question blank.
What examiners advise
- Do not use the calculator in these questions.
- Draw a quick Argand diagram when you work with arguments, so you can see which of the two possible trigonometric answers is correct.
- Put the same scale on both axes and mark the key points on the diagram.
- Equate real and imaginary parts when you have an equation between two complex numbers.
- State the line of reflection, the half-line's starting point, and the region you are shading.
Habits that help
- Write z = x + iy and state what the locus is in Cartesian form first, then draw it.
- Check each argument against the quadrant of the point.
- Check a locus by testing a point from your shaded region.
Questions students ask
Can I use my calculator for complex numbers in Paper 3?
The reports are clear that calculators should not be used in complex number questions, because working has to be shown to justify the result.
Why must both axes of an Argand diagram have the same scale?
Otherwise a circle becomes an ellipse and the sketch means nothing. Omitting the scale, or showing it on only one axis, is the most common error in drawing diagrams.
How do I find the argument correctly?
Draw a quick Argand diagram. A calculator's angle can be in the wrong quadrant, and the sketch shows which of the two trigonometric answers is correct.
What mistakes happen with loci?
Circles with the wrong centre, radius or quadrant; the line y = 3 drawn instead of x = 3; a full line when a half-line was needed; half-lines starting at the origin instead of the given point; the perpendicular bisector for the wrong pair of points; the wrong side shaded.
How do I solve an equation between two complex numbers?
Equate real and imaginary parts. Also be careful with the conjugate: 1 + (2i)² is not what you get, and (1 − √3 i)³ does not equal 1 − (√3 i)³.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.