CAIE A-Level Mathematics 3.9.4 Argand Diagrams

Practise plotting complex numbers on Argand diagrams and using coordinates, arguments, gradients and distances to establish angles, axes conditions and geometric relationships.

Exam points

plot x + iy at coordinates (x, y) using a consistent real-axis and imaginary-axis scale
use argument differences or line gradients to calculate an angle at the origin
translate equal lengths, axes positions or equilateral geometry into algebraic conditions

[Maximum number: 2]

The complex numbers u and v are defined by u=-4+2 i and v=3+i.

Prove that angle $A O B=\frac{3}{4} \pi$ .

Use angle $A O B=\arg u-\arg v=\arg \frac{u}{v}$
M1
Obtain the given answer
A1
Alternative method for question 8(d)
Obtain $\tan A O B$ from gradients of O A and O B and the $\tan (A \pm B)$ formula
M1
Obtain the given answer
A1
Alternative method for question 8(d)
Obtain $\cos A O B$ by using the cosine rule or a scalar product
M1
Obtain the given answer
A1

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3.9.4—Complex numbers question 1