Subject Experts Solutions for Chapter: Complex Numbers, Exercise 1: PRACTICE EXERCISE

What is the real part of $(\sin x+i\cos x{)}^3$, where $i=\sqrt{-1}$?

$\left(x^3-1\right)$ can be factorised as, where, $\omega$ is one of the cube roots of unity.

If $z_1$ and $z_2$ are complex numbers with $\left|z_1\right|=\left|z_2\right|$, then which of the following is/are correct?

I. $z_1=z_2$

II. Real part of $z_1=$ Real part of $z_2$

III. Imaginary part of $z_1=$ Imaginary part of $z_2$

Select the correct answer using the code given below

If the point $z_1=1+i$, where $i=\sqrt{-1}$, is the reflection of a point $z_2=x+iy$ in the line $i\overline{z}-iz=5$, then the point $z_2$ is

$z\overline{z}+(3-i)z+(3+i)\overline{z}+1=0$ represents a circle with

Suppose, $\omega$ is a cube root of unity with $\omega \ne 1$. Suppose, $P$ and $Q$ are the points on the complex plane defined by $\omega$ and ${\omega }^2$. If $O$ is the origin, then what is the angle between $OP$ and $OQ$?

Suppose, ${\omega }_1$ and ${\omega }_2$ are two distinct cube roots of unity different from $1$. Then, what is ${\left({\omega }_1-{\omega }_2\right)}^2$ equal to?

If $z=x+iy={\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)}^{-25}$, where $i=\sqrt{-1}$, then what is the fundamental amplitude of $\frac{z-\sqrt{2}}{z-i\sqrt{2}}$?