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# Challenging Quiz on Complex Variables

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!

Created by
@NiceOnyx

### 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 $1 + \cos \theta + i\sin \theta$.

<p>Answer: Modulus: $|1 + \cos \theta + i\sin \theta|$, Argument: $\arg(1 + \cos \theta + i\sin \theta)$</p> Signup and view all the answers

### Check the existence of $\lim_{z \to 0} (x^3 + y + i(x^2 - y^2))$.

<p>Answer: Check the limit of $(x^3 + y + i(x^2 - y^2))$ as $z$ approaches $0$.</p> Signup and view all the answers

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