Complex numbers 2 — Question 19

QCAA 2024, Paper 1 · 6 marks

Q19 · 2024 · Technology-freeComplex unfamiliar6 marks

QUESTION 19 (6 marks)

Consider complex numbers of the form

w=x+i

, where

x

is a positive real number. If

\operatorname{Re}(w^7)=0

, determine all possible values of

x

WORKED SOLUTION

6 marks

ANSWER

x\in\left\{\cot\frac{\pi}{14},\cot\frac{3\pi}{14},\cot\frac{5\pi}{14}\right\}

Worked solution

Let

w=r\operatorname{cis}(\theta)

. By De Moivre’s theorem,

w^7=r^7\operatorname{cis}(7\theta)

. Since

w=x+i

\theta=\tan^{-1}(1/x).

The condition

\operatorname{Re}(w^7)=0

requires

7\theta=(2n+1)\frac{\pi}{2},

x=\cot\left(\frac{(2n+1)\pi}{14}\right).

Because

x>0

, the admissible first-quadrant angles yield

x=\cot\frac{\pi}{14},\quad\cot\frac{3\pi}{14},\quad\cot\frac{5\pi}{14}.

correctly uses De Moivre’s theorem

[1 mark]

correctly determines an expression representing

\arg(w)

in terms of

x

[1 mark]

determines a relationship involving

\arg(w^7)

using the condition

\operatorname{Re}(w^7)=0

[1 mark]

determines a general expression representing possible values of

x

[1 mark]

determines one value of

x

[1 mark]

evaluates the reasonableness of the solution by determining the remaining two values of

x

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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