Consider the solutions of the equation $z^4 = -9$ - HSC - SSCE Mathematics Extension 2 - Question 9 - 2024 - Paper 1
Question 9
Consider the solutions of the equation $z^4 = -9$.
What is the product of all the solutions that have a positive principal argument?
A. 3
B. -3
C. 3i
D. -3i
Worked Solution & Example Answer:Consider the solutions of the equation $z^4 = -9$ - HSC - SSCE Mathematics Extension 2 - Question 9 - 2024 - Paper 1
Step 1
Find all solutions of the equation $z^4 = -9$
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Answer
To find the solutions for the equation z4=−9, we can start by expressing −9 in polar form. We know that −9 can be written as:
9extcis(180°)
Using De Moivre's theorem, the general solutions for z are given by:
oot{4}{9} ext{cis} \left( \frac{180° + 360°k}{4} \right), k = 0, 1, 2, 3$$
Calculating the magnitude, we find:
$$\root{4}{9} = 9^{1/4} = 3^{1/2} = \sqrt{3}$$
The angles for the four solutions can be calculated as follows:
1. For $k = 0$: $$\frac{180° + 360° \cdot 0}{4} = 45°$$
2. For $k = 1$: $$\frac{180° + 360° \cdot 1}{4} = 135°$$
3. For $k = 2$: $$\frac{180° + 360° \cdot 2}{4} = 225°$$
4. For $k = 3$: $$\frac{180° + 360° \cdot 3}{4} = 315°$$
Thus, the four solutions are:
1. $z_1 = \sqrt{3} \text{cis}(45°)$
2. $z_2 = \sqrt{3} \text{cis}(135°)$
3. $z_3 = \sqrt{3} \text{cis}(225°)$
4. $z_4 = \sqrt{3} \text{cis}(315°)$
Step 2
Determine which solutions have a positive principal argument
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Answer
The solutions with a positive argument are:
z1=3cis(45°)
z4=3cis(315°) (not positive)
z2=3cis(135°) (positive)
The only solutions that have a positive principal argument are:
3cis(45°)
3cis(135°)
Step 3
Calculate the product of the solutions with positive principal argument
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Answer
To find the product of the solutions with positive arguments:
z1⋅z2=3cis(45°)⋅3cis(135°)
Using the properties of multiplication in polar form we have:
