2021HSCShort answer

Q11 (d) (i)

2 marks

Question

Find the two square roots of $-i$ , giving the answers in the form $x+iy$ , where $x$ and $y$ are real numbers.

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Solution

Criteria	Marks
Provides correct solution	2
Attempts to use the square of $(x+iy)$ , or equivalent merit OR attempts to use the modulus/argument form of $-i$ OR plots $-i$ on an Argand diagram and attempts to locate the square roots	1

Sample answer:

Let $(x+iy)^2=-i$ where $x,y\in\mathbb R$ .

$x^2-y^2=0\quad\text{and}\quad2xy=-1$
$y=-\frac1{2x}$
$x^2-\left(-\frac1{2x}\right)^2=0$
$4x^4-1=0$
$x^4=\frac14$
$x^2=\frac12\quad(x\in\mathbb R)$
$x=\pm\frac1{\sqrt2}$

$\therefore\text{ square roots are }\frac1{\sqrt2}-\frac{i}{\sqrt2}\text{ and }-\frac1{\sqrt2}+\frac{i}{\sqrt2}$

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Syllabus alignment

Complex Numbers

Arithmetic of complex numbersRepresent and calculate with complex numbers in Cartesian form, including conjugates, division, reciprocals and square roots.
Geometric representation of a complex numberRepresent complex numbers in the complex plane and use modulus, argument and polar form with their identities.

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