For a positive integer $n$, find the value of $(1-i)^n\left(1-\frac{1}{i}\right)^n$.
Evaluate $\sum_\limits{n=1}^{13}\left(i^n+i^{n+1}\right)$, where $n \in N$
If $\left(\frac{1+i}{1-i}\right)^3-\left(\frac{1-i}{1+i}\right)^3=x+i y$, then find $(x, y)$
If $\frac{(1+i)^2}{2-i}=x+i y$, then find the value of $x+y$
If $\left(\frac{1-i}{1+i}\right)^{100}=a+i b$, then find $(a, b)$
If $a=\cos \theta+i \sin \theta$, then find the value of $\frac{1+a}{1-a}$
If $(1+i) z=(1-i) \bar{z}$, then show that $z=-i \bar{z}$.
If $z=x+i y$, then show that $z \bar{z}+2(z+\bar{z})+b=0$, where $b \in R$, represents a circle.
If the real part of $\frac{\bar{z}+2}{\bar{z}-1}$ is 4 , then show that the locus of the point representing $z$ in the complex plane is a circle.
Show that the complex number $z$, satisfying the condition arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ lies on a circle.
Solve the equation $|z|=z+1+2 i$
12 If $|z+1|=z+2(1+i)$, then find the value of $z$.
If $\arg (z-1)=\arg (z+3 i)$, then find $x-1: y$, where $z=x+i y$.
Show that $\left|\frac{z-2}{z-3}\right|=2$ represents a circle. Find its centre and radius.
If $\frac{z-1}{z+1}$ is a purely imaginary number $(z \neq-1)$, then find the value of $|z|$.
$z_1$ and $z_2$ are two complex numbers such that $\left|z_1\right|=\left|z_2\right|$ and $\arg \left(z_1\right)+\arg \left(z_2\right)=\pi$, then show that $z_1=-\bar{z}_2$.
If $\left|z_1\right|=1\left(z_1 \neq-1\right)$ and $z_2=\frac{z_1-1}{z_1+1}$, then show that the real part of $z_2$ is zero.
If $z_1, z_2$ and $z_3, z_4$ are two pairs of conjugate complex numbers, then find arg $\left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$
If $\left|z_1\right|=\left|z_2\right|=\cdots=\left|z_n\right|=1$, then show that $$ \left|z_1+z_2+z_3+\cdots+z_n\right|=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}+\cdots+\frac{1}{z_n}\right| \text {. }$$
If the complex numbers $z_1$ and $z_2, \arg \left(z_1\right)-\arg \left(z_2\right)=0$, then show that $\left|z_1-z_2\right|=\left|z_1\right|-\left|z_2\right|$.
Solve the system of equations $\operatorname{Re}\left(z^2\right)=0,|z|=2$.
Find the complex number satisfying the equation $z+\sqrt{2}|(z+1)|+i=0$.
Write the complex number $z=\frac{1-i}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ in polar form.
If $z$ and $w$ are two complex numbers such that $|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$, then show that $\bar{z} w=-i$.
Fill in the blanks of the following.
(i) For any two complex numbers $z_1, z_2$ and any real numbers $a, b$, $\left|a z_1-b z_2\right|^2+\left|b z_1+a z_2\right|^2=\cdots$
(ii) The value of $\sqrt{-25} \times \sqrt{-9}$ is ...
(iii) The number $\frac{(1-i)^3}{1-i^3}$ is equal to ...
(iv) The sum of the series $i+i^2+i^3+\cdots$ upto 1000 terms is $\ldots$
(v) Multiplicative inverse of $1+i$ is $\ldots$
(vi) If $z_1$ and $z_2$ are complex numbers such that $z_1+z_2$ is a real number, then $z_1=\cdots$
(vii) $\arg (z)+\arg \bar{z}$ where, $(\bar{z} \neq 0)$ is $\ldots$
(viii) If $|z+4| \leq 3$, then the greatest and least values of $|z+1|$ are $\ldots$ and ...
(ix) If $\left|\frac{z-2}{z+2}\right|=\frac{\pi}{6}$, then the locus of $z$ is ...
(x) If $|z|=4$ and $\arg (z)=\frac{5 \pi}{6}$, then $z=\cdots$
Match the statements of Column A and Column B.
| Column A | Column B | ||
|---|---|---|---|
| (i) | The polar form of $i+\sqrt{3}$ is | (a) | Perpendicular bisector of segment joining $(-2,0)$ and $(2,0)$. |
| (ii) | The amplitude of $-1+\sqrt{-3}$ is | (b) | On or outside the circle having centre of $(0,-4)$ and radius 3. |
| (iii) | It $|z+2|=|z-2|$, then locus of z is | (c) | $\frac{2\pi}{3}$ |
| (iv) | It $|z+2i|=|z-2i|$, then locus of z is | (d) | Perpendicular bisector of segment joining $(0,-2)$ and $(0,2)$. |
| (v) | Region represented by | (e) | $2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ |
| (vi) | Region represented by $|z+4|\le3$ is | (f) | On or inside the circle having centre $(-4,0)$ and radius 3 units. |
| (vii) | Conjugate of $\frac{1+2i}{1-i}$ lies in | (g) | First quadrant |
| (viii) | Reciprocal of $1-i$ lies in | (h) | Third quadrant |
What is the conjugate of $\frac{2-i}{(1-2 i)^2}$ ?
If $\left|z_1\right|=\left|z_2\right|$, is it necessary that $z_1=z_2$.
If $\frac{\left(a^2+1\right)^2}{2 a-i}=x+i y$, then what is the value of $x^2+y^2$ ?
Find the value of $z$, if $|z|=4$ and $\arg (z)=\frac{5 \pi}{6}$.
Find the value of $\left|(1+i) \frac{(2+i)}{(3+i)}\right|$.
Find the principal argument of $(1+i\sqrt3)^2$.
Where does $z$ lie, if $\left|\frac{z-5 i}{z+5 i}\right|=1$ ?
State true or false for the following.
The order relation is defined on the set of complex numbers.
State true or false for the following.
Multiplication of a non-zero complex number by $-i$ rotates the point about origin through a right angle in the anti-clockwise direction.
State true or false for the following.
For any complex number $z$, the minimum value of $|z|+|z-1|$ is 1 .
State true or false for the following.
The locus represented by $|z-1|=|z-i|$ is a line perpendicular to the join of the points $(1,0)$ and $(0,1)$.
State true or false for the following.
If $z$ is a complex number such that $z \neq 0$ and $\operatorname{Re}(z)=0$, then, $\operatorname{Im}\left(z^2\right)=0$.
State true or false for the following.
The inequality $|z-4|<|z-2|$ represents the region given by $x>3$.
State true or false for the following.
Let $z_1$ and $z_2$ be two complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then $\arg \left(z_1-z_2\right)=0$.
State true or false for the following.
2 is not a complex number.
$\sin x+i \cos 2 x$ and $\cos x-i \sin 2 x$ are conjugate to each other for
The real value of $\alpha$ for which the expression $\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely real is
where, $n\in N$
If $z=x+i y$ lies in the third quadrant, then $\frac{\bar{z}}{z}$ also lies in the third quadrant, if
The value of $(z+3)(\bar{z}+3)$ is equivalent to
If $\left(\frac{1+i}{1-i}\right)^x=1$, then where, $n \in N$
A real value of $x$ satisfies the equation $\left(\frac{3-4 i x}{3+4 i x}\right)=\alpha-i \beta(\alpha, \beta \in R)$, if $\alpha^2+\beta^2$ is equal to
Which of the following is correct for any two complex numbers $z_1$ and $z_2$ ?
The point represented by the complex number $(2-i)$ is rotated about origin through an angle $\frac{\pi}{2}$ in the clockwise direction, the new position of point is
If $x, y \in R$, then $x+i y$ is a non-real complex number, if
If $a+i b=c+i d$, then
The complex number $z$ which satisfies the condition $\left|\frac{i+z}{i-z}\right|=1$ lies on
If $z$ is a complex number, then
$\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$ is possible, if
The real value of $\theta$ for which the expression $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is a real number is
The value of $\arg(x)$, when $x < 0$ is
If $f(z)=\frac{7-z}{1-z^2}$, where $z=1+2 i$, then $|f(z)|$ is equal to