Algebra of Complex Numbers

Let $A=\left\{\frac{1967+1686i \sin \theta }{7-3i \cos \theta }: \theta \in R\right\}$. If $A$ contains exactly one positive integer $n$, then the value of $n$ is

Express $\frac{\left(2+i\right)}{\left(3-i\right)\left(1+2i\right)}$ in the form of $a+ib, a,b\in R, i=\sqrt{-1}$. State the values of $a$ and $b$.

Let $a\ne b$ be two non-zero real numbers. Then the number of elements in the set $X=\left\{z\in C:\mathrm{Re}\left(a{z}^2+bz\right)=a \mathrm{and} \mathrm{Re}\left(b{z}^2+az\right)=b\right\}$ is equal to

Express $\frac{\left(2+i\right)}{\left(3-i\right)\left(1+2i\right)}$ in the form of $a+ib, a,b\in R, i=\sqrt{-1}$. State the values of $a$ and $b$.

Find the inverse complex number of $\frac{2+3i}{4-3i}$

Multiplicative inverse of $\left(3+i\right)$

Express $\left(-i\right)\left(2i\right){\left(-\frac{1}{8}i\right)}^3$ in the form of $a+bi$.

Multiplicative inverse of the complex number $\left(\sin \theta ,\cos \theta \right)$ is

The value of ${\left[\frac{5i}{\left(1-i\right)\left(2-i\right)\left(3-i\right)}\right]}^{50}$ is equal to

Convert $\frac{1+2i}{1-3i}$ into $a+ib$ form.

Prove that if $Z_1, Z_2\in C, \text{then}$

$Re\left(Z_1{Z}_2\right)=Re\left(Z_1\right)Re\left(Z_2\right)-Im\left(Z_1\right)Im\left(Z_2\right)$

$Im\left(Z_1{Z}_2\right)=Re\left(Z_1\right)Im\left(Z_2\right)+Im\left(Z_1\right)Re\left(Z_2\right)$

If $x=3+4i \text{and} y=3-4i$ then find the value of $x^3+y^3$

Find real $\theta$ such that $\frac{3+2i \sin \theta }{1-2i \sin \theta }$ is purely real.

Express $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}$ in the form of $a+bi$.$$

Find the multiplicative inverse of $2-3i$.

Express $\left(-\sqrt{3}+\sqrt{-2}\right)\left(2\sqrt{3}-i\right)$ in the form of $a+ib$

Express ${\left(5-3i\right)}^3$ in the form $a+ib$.

Express $\left(-5i\right)\left(\frac{1}{8}i\right)$ in the form of $a+bi$.

The conjugate of $\frac{{\left(2+i\right)}^2}{3+i}$, in the form of $a+ib$, is

Multiplicative inverse of $-1+i$ is