Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 3: Level 3

If the least and the largest real values of $\alpha ,$ for which the equation $z+\alpha \left|z-1\right|+2i=0$ $\left(z\in C \text{and} i=\sqrt{-1}\right)$ has a solution, are $p$ and $q$ respectively; then $\mathrm{4}\left(p^2+q^2\right)$ is equal to_______.

If $\left|z_1\right|=1, \left|z_2\right|=2$ and $\left|z_3\right|=3$, then the largest real value $\mu$ such that $\left|9z_1{z}_2+z_2{z}_3+4z_3{z}_1\right|\ge \mu \left|z_1+z_2+z_3\right|$ is

If $\cos \theta +i\sin \theta$ is a root of the equation $x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots +a_{n-1}x+a_n=0$, then the value of $\sum _{r=1}^n{a}_r\cos r\theta$ is

Let $i=\sqrt{-1}.$ If $\frac{{\left(-1+i\sqrt{3}\right)}^{21}}{{\left(1-i\right)}^{24}}+\frac{{\left(1+i\sqrt{3}\right)}^{21}}{{\left(1+i\right)}^{24}}=k,$ and $n=\left[\left|k\right|\right]$ be the greatest integral part of $\left|k\right|.$

Then $\sum _{j=0}^{n+5}{\left(j+5\right)}^2-\sum _{j=0}^{n+5}\left(j+5\right)$ is equal to ________.

If $1, z_1, z_2, z_3, \dots , z_{n-1}$ is the $n^{\mathrm{th}}$ root of unity and $w$ is a non-real complex cube root of unity, then the product of $\prod _{r=1}^{n-1}\left(\omega -z_r\right)$ cannot be equal to

Number of ordered pairs $\left(z,\omega \right)$ of the complex numbers $z$ and $\omega$, satisfying the system of equations, $z^3+(\overline{\omega }{)}^7=0$ and $z^5·{\omega }^{11}=1$ is

If a variable circle $S$ touches $S_1:\left|z-z_1\right|=r_1$ internally and $S_2:\left|z-z_2\right|=r_2$ externally while the curves $S_1\&$ $S_2$ touch internally to each other. Then the eccentricity of the locus of the centre of the curve $S$ is equal to

The eccentricity of conic represented by $\left|z-2\right|+\left|z-2+17 i\right|=19$ is $e$, then ${\left(19e\right)}^2$ is equal to