Edexcel A-Level Mathematics AS Fp1 1 1 Definition of Complex Numbers Questions

Practise recognising complex numbers in a + ib and polar-style forms, including modulus, argument, conjugates and real or imaginary parts.

Exam points

find modulus and argument from a + ib, choosing the correct quadrant or interval
use the conjugate z* to simplify an expression and compare it with kz
express a complex number in r(cosθ + i sinθ) form with exact r where possible

[Maximum number: 2]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

\mathrm{f}(z)=4 z^{3}+p z^{2}-24 z+108

where p is a constant.
Given that -3 is a root of the equation f(z)=0

determine the modulus of the complex roots of f(z)=0

Product of complex roots is $\frac{36}{4}=9$ , so modulus is $\sqrt{9}$ .
or
$\text{Modulus}=\sqrt{\left(\frac{5}{2}\right)^2+\left(\frac{\sqrt{11}}{2}\right)^2}$
Hence modulus is 3

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Edexcel A-Level Mathematics AS Fp1 1 1 Definition of Complex Numbers Questions question 1