Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 3: EXERCISE-3

Let $A=\left\{a\in R\mid \right.$ the equation $(1+2i)x^3-2(3+i)x^2+(5-4i)x+2a^2=0$ has at least one real root$\}$. Find the value of $\frac{1}{5}\sum _{a\in A}{a}^2$.

Let $\omega$ and $z$ be complex numbers such that $\left|\omega \right|=1$ and $\left|z\right|=10$. Let $\theta =\arg \left(\frac{\omega -z}{z}\right)$. The maximum possible value of ${\tan }^2\theta$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $\frac{1}{8}(p+q)$.

If the biquadratic $x^4+a{x}^3+b{x}^2+cx+d=0 (a,b,c,d\in R)$ has $4$ non real roots, two with sum $3+4 i$ and the other two with product $13+i$. Find the value of ' $b$ '.

Number of ordered pair $(z,\omega )$ satisfying $z+\frac{1}{\overline{z}}=\omega$ and $\omega +\frac{1}{\overline{\omega }}=z$ is/ are equal to

Complex number $a,b$ and $c$ zeros of the polynomial $P\left(z\right)=z^3+qz+r($ where $q,r\in R)$ and $|a{|}^2+|b{|}^2+|c{|}^2=250$ If the points corresponding to $a, b$ and $c$ in the complex plane are the vertices of a right angled triangle with hypotenuse $h$, then $\left(\frac{h^2}{125}\right)$ is

If $\frac{z-(1+i)}{2+i}$ is purely real, then value of $2lm\left(z\right)-Re\left(z\right)$ is equal to

If $z_1$ and $z_2$ be two distinct complex numbers satisfying $\left|z_1^2-z_2^2\right|=\left|{\overline{z}}_1^2+{\overline{z}}_2^2-2{\overline{z}}_1{\overline{z}}_2\right|$ and if $\left(\arg z_1-\arg z_2\right)=\frac{a\pi }{b}$, then find the least possible value of $\frac{|a+b|}{2}$($a,b\in$ Integer)

The least value of $|z-3-4i{|}^2+|z+2-7i{|}^2+|z-5+2i{|}^2$occurs when $z=x+iy$ then the real part of $z$ is