Question 1:
If $4x + i(3x - y) = 3 + i(-6)$, where $x$ and $y$ are real numbers, then find the values of $x$ and $y$.
Question 2:
Express the following in the form of $a + bi$: (i) $(-5i)(\frac{1}{8}i)$ (ii) $(-i)(2i)(-\frac{1}{8}i)^3$
Question 3:
Express $(5 - 3i)^3$ in the form $a + ib$.
Question 4:
Express $(-\sqrt{3} + \sqrt{-2})(2\sqrt{3} - i)$ in the form of $a + ib$.
Question 5:
Find the multiplicative inverse of $2 - 3i$.
Question 6:
Express the following in the form $a + ib$: (i) $\frac{5 + \sqrt{2}i}{1 -
\sqrt{2}i}$ (ii) $i^{-35}$
Question 7:
Find the conjugate of $\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}$.
Question 8:
If $x + iy = \frac{a + ib}{a - ib}$, prove that $x^2 + y^2 = 1$.
Question 9:
Express the complex number $(5i)(-\frac{3}{5}i)$ in the form $a + ib$.
Question 10:
Express the complex number $i^9 + i^{19}$ in the form $a + ib$.
Question 11:
Express the complex number $i^{-39}$ in the form $a + ib$.
Question 12:
Express the complex number $3(7 + i7) + i(7 + i7)$ in the form $a + ib$.
Question 13:
Express the complex number $(1 - i) - (-1 + i6)$ in the form $a + ib$.
Question 14:
Express the complex number $(\frac{1}{5} + i\frac{2}{5}) - (4 + i\frac{5}{2})$ in the form $a + ib$.
Question 15:
Express the complex number $[(\frac{1}{3} + i\frac{7}{3}) + (4 + i\frac{1}{3})] - (-\frac{4}{3} + i)$ in the form $a + ib$.
Question 16:
Express the complex number $(1 - i)^4$ in the form $a + ib$.
Question 17:
Express the complex number $(\frac{1}{3} + 3i)^3$ in the form $a + ib$.
Question 18:
Express the complex number $(-2 - \frac{1}{3}i)^3$ in the form $a + ib$.
Question 19:
Find the multiplicative inverse of the complex number $4 - 3i$.
Question 20:
Find the multiplicative inverse of the complex number $\sqrt{5} + 3i$.
Question 21:
Find the multiplicative inverse of the complex number $-i$.
Question 22:
Express the following expression in the form of $a + ib$: $\frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})}$
Question 23:
Evaluate: $[i^{18} + (\frac{1}{i})^{25}]^3$.
Question 24:
For any two complex numbers $z_1$ and $z_2$, prove that Re$(z_1 z_2) = \text{Re } z_1 \text{Re } z_2 - \text{Im } z_1 \text{Im } z_2$.
Question 25:
Reduce $(\frac{1}{1 - 4i} - \frac{2}{1 + i})(\frac{3 - 4i}{5 + i})$ to the standard form.
Question 26:
If $x - iy = \sqrt{\frac{a - ib}{c - id}}$ prove that $(x^2 + y^2)^2 = \frac{a^2 + b^2}{c^2 + d^2}$.
Question 27:
If $z_1 = 2 - i$, $z_2 = 1 + i$, find $|\frac{z_1 + z_2 + 1}{z_1 - z_2 + 1}|$.
Question 28:
If $a + ib = \frac{(x + i)^2}{2x^2 + 1}$, prove that $a^2 + b^2 = \frac{(x^2 + 1)^2}{(2x^2 + 1)^2}$.
Question 29:
Let $z_1 = 2 - i$, $z_2 = -2 + i$. Find (i) Re$(\frac{z_1 z_2}{\overline{z}_1})$ (ii) Im$(\frac{1}{z_1 \overline{z}_1})$.
Question 30:
Find the real numbers $x$ and $y$ if $(x - iy)(3 + 5i)$ is the conjugate of $-6 - 24i$.
Question 31:
Find the modulus of $\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}$.
Question 32:
If $(x + iy)^3 = u + iv$, then show that $\frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2)$.
Question 33:
If $\alpha$ and $\beta$ are different complex numbers with $|\beta| = 1$, then find $|\frac{\beta - \alpha}{1 - \overline{\alpha}\beta}|$.
Question 34:
Find the number of non-zero integral solutions of the equation $|1 - i|^x = 2^x$.
Question 35:
If $(a + ib)(c + id)(e + if)(g + ih) = A + iB$, then show that $(a^2 + b^2)(c^2 + d^2)(e^2 + f^2)(g^2 + h^2) = A^2 + B^2$.
Question 36:
If $(\frac{1 + i}{1 - i})^m = 1$, then find the least positive integral value of $m$.