Modulus and Argument of Complex Numbers

_{If area of the region on complex plane by complex number $z$, such that $\frac{\pi }{4}\le \arg \left(\frac{z-2}{z+2}\right)\le \frac{3\pi }{4}$ is $a\pi +b$ then $\left(b-a\right)$ equals}

Let $\omega =e^{i\left(\frac{\mathrm{\pi }}{3}\right)}$ and $a,b,c,x,y,z$ be non-zero complex number such that

$a+b+c=x$

$a+b\omega +c{\omega }^2=y$

$a+b{\omega }^2+c\omega =z$

Then the value of $\frac{{\left|x\right|}^2+{\left|y\right|}^2+{\left|z\right|}^2}{{\left|a\right|}^2+{\left|b\right|}^2+{\left|c\right|}^2}$ is

If both the roots of the quadratic equation $a{x}^2+bx+c=0; a, b, c\in R-\left\{0\right\}$ are nonreal and if ' $\theta$ ' denotes the principle argument of the complex number $Z=a+i c,$ then which of the following options is necessarily true?

The argument of complex number$\text{'}z\text{'}$ satisfying the equation $z+\overline{z}=6$, can not be

If $\alpha , \beta , \gamma$ are three distinct non-zero complex numbers such that $\left|\beta \right|=\frac{1}{\left|\gamma \right|},$ then the value of $\left|\frac{\beta \gamma -\alpha }{1-\overline{\alpha }\beta \gamma }\right|$ will be

Consider three complex numbers $z_1, z_2, z_3$ such that $\left|z_1\right|=1,\left|z_2\right|=2,\left|z_3\right|=3.$ The value of $\left|\frac{36z_1+9z_2+4z_3}{\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}}\right|$

Find the principal argument of $2\sqrt{3}-2i.$

If $z_1$ and $z_2$ are complex numbers such that ${\left|5z_1-3z_2\right|}^2+{\left|3z_1+5z_2\right|}^2=\lambda \left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)$ then $\lambda$ is equal to

The argument of $-3i·|5-13i|$ is

The argument of $-2+2\sqrt{3}i$ is

If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, then $\arg \left(\frac{z_1}{z_3}\right)+\arg \left(\frac{z_2}{z_4}\right)$ is

The sum of principal arguments of complex numbers $1+i,-1+i\sqrt{3},-\sqrt{3}-i,\sqrt{3}-i,i,-3i,2,-1$ is

If $z=4+3\sqrt{20}i, i=\sqrt{-1},$ then the positive value of $(\sqrt{z}+\sqrt{\overline{z}}{)}^2$ is equal to

If $z=\frac{-4+2\sqrt{3}i}{5+\sqrt{3}i}$, then the value of $\arg \left(z\right)$ is

The argument of the complex number $(1+i{)}^4$ is

If $\left|z_1\right|=2,\left|z_2\right|=3,\left|z_3\right|=4$ and $\left|2z_1+3z_2+4z_3\right|=4$, then the expression $\left|8z_2{z}_3+27z_3{z}_1+64z_1{z}_2\right|$ equals

The value of $\left|\frac{1}{2+i}-\frac{1}{2-i}\right|$ is

The modulus of $i^{25}+(i+2{)}^3$ is

The modulus of $\overline{6+i^3}+\overline{9-i}+\overline{9+i^2}$ is

If $z=4-9 i$, then $z\overline{z}$ is