Complex Numbers
JUPEB Mathematics

Simplify $i^7$, where $i = \sqrt{-1}$.

Simplify $(3 + 2i) + (1 - 5i)$.

Simplify $(2 + 3i)(1 - i)$.

Find the modulus of the complex number $5 - 12i$.

Express $\dfrac{1 + i}{1 - i}$ in the form $a + bi$.

If $z = 3 - 4i$ and $\bar{z}$ is its conjugate, find $z\bar{z}$.

Find the argument of the complex number $-1 + i$.

Solve the equation $z^2 + 4z + 13 = 0$.

Express $1 + \sqrt{3}\,i$ in polar form.

If $(x + yi)(2 - i) = 5$, where $x$ and $y$ are real, find $x$ and $y$.

Use De Moivre's theorem to evaluate $(\cos 30^\circ + i\sin 30^\circ)^6$.

Find the square roots of the complex number $3 + 4i$.