If ${\left(\frac{1+i}{1-i}\right)}^3-{\left(\frac{1-i}{1+i}\right)}^3=x+iy,$ then $(x,y)$ is

Important Questions on Complex Numbers and Quadratic Equations

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $z\in C$ with $\mathrm{I}\mathrm{m}\left(z\right)=10$ and it satisfies $\frac{2z-n}{2z+n}=2i-1$ for some natural number $n.$ Then

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $z_1$ and $z_2$ be the roots of the equation $z^2+pz+q=0$ where $p, q$ are real. The points represented by $z_1, z_2$ and the origin form an equilateral triangle, if

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

A value of $\theta$ for which $\frac{2+3i\sin \theta }{1-2i\sin \theta }$ is purely imaginary, is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

The imaginary part of ${\left(3+2\sqrt{-54}\right)}^{\frac{1}{2}}-{\left(3-2\sqrt{-54}\right)}^{\frac{1}{2}}$, can be

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

The imaginary part of conjugate of ${\left(\frac{1+i}{1-i}\right)}^5$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Simplified form of $\frac{10i}{1+2i}$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $z=1+ai$, be a complex number, $a>0$, such that $z^3$ is a real number. Then, the sum $1+z+z^2+\dots .+z^{11}$ is equal to :

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $z\in C$ and $i=\sqrt{-1}$, if $a, b, c \in \left(0,1\right)$ be such that $a^2+b^2+c^2=1$ and $b+ic=\left(1+a\right)z$, then $\frac{1+iz}{1-iz}=$

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

The area of the triangle in the complex plane formed by $z, iz$ and $z+iz$ is

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $w\left(Im w\ne 0\right)$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w-\overline{w}z=k\left(1-z\right)$, for some real number $\mathit{k}$, is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If ${\left(\frac{1+i}{1-i}\right)}^{\frac{m}{2}}={\left(\frac{1+i}{i-1}\right)}^{\frac{n}{3}}=1, \left(m, n\in N\right)$ then the greatest common divisor of the least values of $\mathrm{m}$ and $\mathrm{n}$ is

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Find the real part of the complex number ${\left(1-2i\right)}^{-3}$

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

Let $z=x+iy$ be a non-zero complex number such that $z^2=i|z{|}^2$, where $i=\sqrt{-1}$, then $z$ lies on the

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If $\mathrm{R}\mathrm{e}\left(\frac{z-1}{2z+i}\right)=1,$ where $z=x+iy,$ then the point $\left(x,y\right)$ lies on a

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If $z=x+iy, z^{\frac{1}{3}}=a-ib$, then $\frac{x}{a}-\frac{y}{b}=k\left(a^2-b^2\right)$ where $k$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If ${\left(\frac{1-i}{1+i}\right)}^{96}=a+ib$, then $\left(a,b\right)$ is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If $A=\left\{z=x+iy/\mathrm{real} \mathrm{part} \mathrm{of} \frac{\overline{z}-1}{z-i}=2\right\}$ then the locus of the point $P\left(x,y\right)$ in the Cartesian plane is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If $\frac{3}{2+\cos \theta +i\sin \theta }=a+ib$, then $[{\left(a-2\right)}^2+b^2]$ is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

If $a$ and $b$ are unit vectors and $\left|a+b\right|=1$, then $\left|a-b\right|$ is equal to.

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Algebra of Complex Numbers

For all complex numbers $z$ of the form $1+i\alpha , \alpha \in R,$ if $z^2=x+iy,$ then