Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 2: Exercise

Dividing $f\left(z\right)$ by $z-i,$ we obtain the remainder $i$ and dividing it by $z+i,$ we get the remainder $1+i,$ then remainder upon the division of $f\left(z\right)$ by $z^2+1$ is

$z_1$ and $z_2$ are two distinct points in an argand plane. If $a\left|z_1\right|=b\left|z_2\right|$ (where $a, b\in R$), then the point $\left(\frac{a{z}_1}{b{z}_2}\right)$ $+\left(\frac{b{z}_2}{a{z}_1}\right)$ is a point on the

If $|2z-1|=|z-2|$ and $z_1, z_2$ are complex numbers such that $\left|z_1-\alpha \right|<\alpha , \left|z_2-\beta \right|<\beta ,$ then $\left|\frac{z_1+z_2}{\alpha +\beta }\right|$

The complex number associated with the vertices $A, B, C$ of $\Delta ABC$ are $e^{i\theta }, \omega , \overline{\omega }$, respectively [where $\omega , \overline{\omega }$ are the complex cube roots of unity and $\cos \theta >Re\left(\omega \right)],$ then the complex number representing the point where angle bisector of $A$ meets the circumcircle of the triangle is

If $\left|z_2+i{z}_1\right|=\left|z_1\right|+\left|z_2\right|$ and $\left|z_1\right|=3, \left|z_2\right|=4,$ then area (in sq. units) of $\Delta ABC,$ if affixes of $A, B$ and $C$ are $z_1, z_2$ and $\left[z_2-i{z}_1\right]/\left[\right(1-i\left)\right]$, respectively, is

If $\omega$ is an imaginary cube root of unity, then expression $(2-\omega )\left(2-{\omega }^2\right)+2(3-\omega )\left(3-{\omega }^2\right)+\dots ..+(n-1)(n-\omega )\left(n-{\omega }^2\right)$ is equal to

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

If $w\ne 1$ is $n^{\mathrm{th}}$ root of unity, then value of $\sum _{k=0}^{n-1}{\left|z_1+w^k{z}_2\right|}^2$ is