5 Questions
- If $x = 2 + 3i$ and $y = 2 - 3i$, what is the value of $x^3 + y^3$?
Answer: $x^3 + y^3 = (2 + 3i)^3 + (2 - 3i)^3$
- Express $5 + i$ in the form $2 + 3i$ ($\overline{z}^3$).
Answer: $5 + i = (2 + 3i)^3$
- Express $1 + i\sqrt{3}$ in polar form.
Answer: $1 + i\sqrt{3} = r(\cos \theta + i\sin \theta)$
- Find the modulus and argument of $1 + \cos \theta + i\sin \theta$.
Answer: Modulus: $|1 + \cos \theta + i\sin \theta|$, Argument: $\arg(1 + \cos \theta + i\sin \theta)$
- Check the existence of $\lim_{z \to 0} (x^3 + y + i(x^2 - y^2))$.
Answer: Check the limit of $(x^3 + y + i(x^2 - y^2))$ as $z$ approaches $0$.
This quiz on Complex Variables covers topics such as finding the sum of cubes, expressing complex numbers in different forms, representing complex numbers in polar form, calculating modulus and argument, and analyzing the existence of limits. Test your understanding of complex variables with these challenging questions!
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