Complex Numbers

Assertion (A) If $a_1, a_2, \ldots, a_n$ are the $n$ distinct roots of the equation $x^n-2=0$, then $1+\left(1-a_1\right)\left(1-a_2\right) \ldots \left(1-a_{n-1}\right)\left(1-a_n\right)=0$
Reason (R) If $\alpha_1, \alpha_2, \ldots, \alpha...

Sum of the modulii of the complex roots of the equation $\left(x^2+\frac{1}{x^2}\right)-5\left(x+\frac{1}{x}\right)+6=0$ is

If $\omega$ is a non-real cube root of unity and $x=\omega^2-\omega-3$, then the value of $x^4+6 x^3+10 x^2-12 x-19$ is

The roots of the equation $(x-1)^5=32(x+1)^5$ are

$z_1, z_2$ are two fixed points on the Argand plane. If $z$ is a complex number such that $\left|z-z_1\right|+\left|z-z_2\right|=\lambda$, then the locus of $z$ is

For $n>1$ and $n \in \mathbf{N}$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n=z^n$, then $\sum_{i=1}^n \frac{\cot ^{-1}\left(2\left|\operatorname{Im} z_i\right|\right)-1}{2 \operatorname{Re} z_i}=$

If the amplitude of $z-2-3 i$ is $\pi / 4$, then the locus of $z=x+i y$ is

Let $z$ be a complex number such that $|z|-z=2+i$, where $i=\sqrt{-1}$. Then, $|z|=$

$A\left(z_1=2+2 i\right), B\left(z_2\right), C\left(z_3\right)$ are three points on the Argand plane satisfying $\left|z_k-2 i\right|=2,(k=1,2,3)$. If $\triangle A B C$ encloses the maximum area, then the sum of the imaginary parts of $z_2$...

Let $A_r=\left(x+\frac{1}{x}\right)^3 \cdot\left(x^2+\frac{1}{x^2}\right)^3 \cdot\left(x^3+\frac{1}{x^3}\right)^3 \cdots\left(x^r+\frac{1}{x^r}\right)^3$. If $x^2+x+1=0$, then $\frac{1}{A_3}+\frac{1}{A_6}+\frac{1}{A_9}+\frac{1}{A_{12}}+\ldo...

For $n \in \mathbf{N}$, If $A_n=\cos \left(\frac{\pi}{2^n}\right)+i \sin \left(\frac{\pi}{2^n}\right)$, then $\left(A_1 A_2 A_3 A_4\right)^4=$

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$, then $\frac{\alpha^3+\beta^3+\gamma^3+\delta^3}{\alpha^6+\beta^6+\gamma^6+\delta^6}=$

Let the roots of the equation $E_1 \equiv x^3+x^2+l x+n=0$ be $x_i,(i=1,2,3)$ and the roots of $E_2 \equiv x^3+a x^2+b x+c=0$ be $\frac{x_i-1}{2}$. If the equation $E_2=0$ is a equation of class one, then the roots of these two equations ex...

If $z$ is a complex number such that $z^2+z+1=0$, then $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots . .+\left(z^{2020}+\frac{1}{z^{2020}}\right)^3=$

If $(\sqrt{3}+i)^{10}=a+b i, a, b \in \mathbf{R}$, then the values of $a$ and $b$ are respectively

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4 i}\right)=0$ and $\lim \left(\frac{z-2}{z-4 i}\right)=1$ simultaneously is

If $z, \bar{z},-z,-\bar{z}$ forms a rectangle of area $2 \sqrt{3}$ square units, then one such $z$ is

\(\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^8+\left(\frac{1+\cos \theta-i \sin \theta}{1+\cos \theta+i \sin \theta}\right)^{16}=\)

The solutions of the equation $z^2\left(1-z^2\right)=16, z \in \mathbf{C}$, lie on the curve

Let $z=x+i y$ be a complex number, $A=\{z /|z| \leq 2\}$ and $B=\{z /(1-i) z+(1+i) \bar{z} \geq 4\}$ Then which one of the following options belongs to $A \cap B$ ?

If $z_1=x_1+i y_1, z_2=x_2+i y_2, z_3=x_1+\frac{i x_2}{2}, z_4=2 y_1+i y_2$ are complex numbers such that $\left|z_1\right|=1,\left|z_2\right|=2$ and $\operatorname{Re} \left(\begin{array}{ll}z_1 & z_2\end{array}\right)=0$, then

If $\omega$ is a complex cube root of unity, then $\sum_{x=1}^{10}\left((\omega x+2)\left(\omega^2 x+2\right)-3\right)$

\(\text { If }\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021}=x+i y,\)
then the value of $x+y$ at $\theta=\frac{\pi}{2}$ i...

Assertion (A) If $z$ is a complex number such that $|z| \geq 3$, then the least value of $\left|z+\frac{3}{z}\right|$ is 1 .
Reason (R) $\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|$, for any two complex numbers $z_1, z_2$
The...