If $z$ is a complete number, then $z^2+{\overline{z}}^2=2$ represents

Important Questions on Complex Numbers and Quadratic Equations

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let $A=\left\{\theta \in \left(-\frac{\pi }{2},\pi \right):\frac{3+2i\hspace{0.17em}\mathrm{s}\mathrm{i}\mathrm{n}\theta }{1-2i \mathrm{s}\mathrm{i}\mathrm{n}\theta }\mathrm{ }\text{is purely imaginary} \right\}.$ Then the sum of the elements in $A$ is:

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Amplitude of $\frac{1+\sqrt{3}i}{\sqrt{3}+1}$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The value of sum $\sum _{n=1}^{13}\left(i^n+i^{n+1}\right), \text{where} i=\sqrt{-1},$equals

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $z$ and $\omega$ are two complex numbers such that $\left|z\omega \right|=1$ and $arg\left(z\right)-arg\left(\omega \right)=\frac{\mathrm{\pi }}{2}$, then:

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

For any real number $r,$ let $A_r=\left\{e^{i\pi rn}:n\right.$ is a natural number $\}$ be a set of complex numbers. Then

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $a>0$ and $z=\frac{{(1+i)}^2}{a-i}$ , has magnitude $\sqrt{\frac{2}{5}}$ , then $\stackrel{-}{z}$ is equal to:

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $a$ and $b$ are real numbers and ${\left(a+ib\right)}^{11}=1+3i$$,$ then ${\left(b+ia\right)}^{11}$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $2+i$ is a root of $x^2-4x+c=0,$ where $c$ is a real number, then the value of $c$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $z_1=2-i$ and $z_2=1+i$$,$ then $\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The value of ${\left(i^{18}+{\left(\frac{1}{i}\right)}^{25}\right)}^3$ is equal to

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If the equation $x^2+bx+45=0, b\in R$ has conjugate complex roots and they satisfy $\left|z+1\right|=2\sqrt{10},$ then

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $\frac{z - \alpha }{z + \alpha }\left(\alpha \in R\right)$ is a purely imaginary number and $\left|z\right|=2$, then a value of $\alpha$ is :

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $z$ is a complex number satisfying $\left|Re\left(z\right)\right|+\left|Im\left(z\right)\right|=4,$ then $\left|z\right|$ cannot be

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The set of all $\alpha \in R$, for which $w=\frac{1+\left(1-8\alpha \right)z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $\left|z\right|=1$ and $\mathrm{R}\mathrm{e}\left(\mathrm{z}\right)\ne 1$, is :

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let $z$ be a complex number such that $\left|\frac{z-i}{z+2i}\right|=1$ and $\left|z\right|=\frac{5}{2}$ . Then, the value of $\left|z+3i\right|$ is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

If $z=x-iy$ and $z^{\frac{1}{3}}=p+iq$, then $\frac{1}{p^2+q^2}\left(\frac{x}{p}+\frac{y}{q}\right)$ is equal to

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The principal argument of the complex number $z=\frac{1+\sin \pi -i\cos \pi }{1+\sin \pi +i\cos \pi }$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

The modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let $z={\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^5+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^5.$ If $R\left(z\right)$ and $I\left(z\right)$ respectively denote the real and imaginary parts of $z,$ then

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>The Modulus and the Conjugate of a Complex Number

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3\left|z_1\right|=4\left|z_2\right|.$ If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ then maximum value of $\left|z\right|$ is