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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 the solutions to the equation $z^4 = -9$
Answer
To solve the equation , we can rewrite it as z^4 = 9e^{i( rac{3 heta}{2})}, where represents the complex number in polar form. Here, the magnitude is 3 (since ) and the argument is heta = rac{3 heta}{2} for the angle corresponding to .
Next, we find the fourth roots by using: z_k = r^{1/n} e^{i( heta + 2k rac{ heta}{n})} where and .
Step 2
Calculate the roots
Answer
We have:
- Magnitude: r = |z| = oot{4}{9} = rac{3}{2}.
- Argument for :
- z_0 = rac{3}{2} e^{i( rac{3 heta}{8})}
- z_1 = rac{3}{2} e^{i( rac{3 heta}{8} + rac{ heta}{2})}
- z_2 = rac{3}{2} e^{i( rac{3 heta}{8} + heta)}
- z_3 = rac{3}{2} e^{i( rac{3 heta}{8} + rac{3 heta}{2})}
Determining the specific angles will give us the positive principal argument solutions.
Step 3
Step 4
Calculate the product of positive argument solutions
Answer
After identifying the roots with positive arguments, we can compute their product.
For the roots found, if we denote the roots with positive arguments as and , the product can be computed as:
As per the calculations, this product simplifies to .