Algebra of Complex Numbers

For positive integers  $n_1, n_2$  the value of expression ${\left(1+i\right)}^{n_1}+{\left(1+i^3\right)}^{n_1}+$${\left(1+i^5\right)}^{n_2}+{\left(1+i^7\right)}^{n_2}$ where  $i=\sqrt{-1}$ , is a real number if and only if

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

If $2-i\sqrt{3}$ where $i=\sqrt{-1}$ is a root of the equation $x^2+ax+b=0$, then what is the value of $\left(a+b\right)$?

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

Find the value of $\frac{i^{9096}+i^{4397}+i^{2941}+i^{7986}}{i^{7976}+i^{4387}+i^{2931}+i^{5086}}$

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 $\frac{2+3i}{4-3i}$

Find the value of $\frac{1+i}{1-i}$

If $\mathrm{\alpha }=\cos \left(\frac{8\mathrm{\pi }}{11}\right)+\mathrm{i} \sin \left(\frac{8\mathrm{\pi }}{11}\right)$ then find the $\mathrm{Re}(\mathrm{\alpha }+{\mathrm{\alpha }}^2+{\mathrm{\alpha }}^3+{\mathrm{\alpha }}^4+{\mathrm{\alpha }}^5)$.

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

Express $i^{-35}$ in the form of $a+bi$.

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

Let z be a complex number satisfying $z^2+z+1=0$. If n is not a multiple of 3, then the value of $z^n+z^{2n}$ is equal to

If $3i\left(2\times \frac{1}{11}-y\right)=i$ then find the value of $y$.

The value of $\frac{2^n}{{\left(1+i\right)}^{2n}}+\frac{{\displaystyle {\left(1+i\right)}^{2n}}}{2^n}, n\in I$, is equal to 

If $g\left(x\right)=x^4-x^3+x^2+3x-5$, then find $g\left(2+3i\right)$

If $x=1+2i$, then the value of $x^3+2x^2-3x+5$ is

If ${\left(1-i\right)}^n=2^n$, then $n=$

Express $i^3$ in the form $a+ib$.