Complex Numbers

If $w$ is a complex cube root of unity , then the value of $w^{10} - w^7 + w^5 - w^2 + 1$ is.

If $i = \sqrt{-1}$ then $\left[i^{18} + \left(\dfrac{1}{i}\right)^{25}\right]^3 =$

The smallest positive integer $n$ for which $\dfrac{(1 + i)^n}{(1 - i)^{n-2}}$ is a real number, is ...

Point A$(5, 12)$ rotated about the origin O in the XY-plane through an angle of $30^\circ$ in the anticlockwise direction to a new position B. The ordinate of point B is...

If $z = \sum_{n=0}^{2026} i^n$, where $i = \sqrt{-1}$, then one of the values of $\sqrt{z}$ is...

The value of $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^{18} + \left(\dfrac{-1-i\sqrt{3}}{2}\right)^{18}$ is

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $\alpha^{200} + \beta^{206} + 2$ is equal to

$\dfrac{(\cos 2\theta + i\sin 2\theta)^7}{(\cos 4\theta + i\sin 4\theta)^3} =$

If $(1 + i) \cdot (1 + 2i) \ldots\ldots\ldots (1 + ni) = x + iy$ (Where $i = \sqrt{-1}$ ), then the value of $(2) \cdot (5) \cdot (10)\ldots\ldots\ldots(1 + n^2)$

If $n$ is a positive integer, then $(1 + i\sqrt{3})^{2n} + (1 - i\sqrt{3})^{2n}$ is equal to .........

If $\omega$ is a complex cube root of unity, then the value of the expression $2\left(1+\dfrac{1}{\omega}\right)\left(1+\dfrac{1}{\omega^2}\right) + 3\left(2+\dfrac{1}{\omega}\right)\left(2+\dfrac{1}{\omega^2}\right) + \ldots + (n+1)\left(n...

If $\omega$ is a complex cube root of unity, then the value of $\sin\left[\pi(\omega^{10} + \omega^{23}) - \dfrac{\pi}{4}\right] =$

The polar form of the complex number $z = \dfrac{1}{1 + i}$, (where $i = \sqrt{-1}$) is

If $x = \sqrt{-1-\sqrt{-1-\sqrt{-1-\ldots\infty}}}$, where $\omega$ is a non-real complex cube root of unity, then the value of $x$ is...

If $-1 + \sqrt{-3} = r e^{i\theta}$, then the value of $\theta$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + x + 1 = 0$ then $\alpha^{2026} + \beta^{2026} = $

The area of the triangle whose vertices are $i, \omega$ and $\omega^2$ is (Where $\omega$ is a complex cube root of unity other than $1, i$ is an imaginary number)__________ sq.units

The modulus of the square root of the complex number $6+8 \mathrm{i}$ (where $\mathrm{i}=\sqrt{-1}$ ) is

Let z be the complex number such that $|z|+z=3+i$ where $i=\sqrt{-1}$, then $|z|=$

Let $z$ be the complex number with $\operatorname{Im}(z)=10$ and satisfying $\frac{2 \mathrm{z}-\mathrm{n}}{2 \mathrm{z}+\mathrm{n}}=2 \mathrm{i}-1$, where $\mathrm{i}=\sqrt{-1}$, for some natural number ' $n$ ' then

A particle P starts from the point $\mathrm{Z}_0=1+2 \mathrm{i}$ where $\mathrm{i}=\sqrt{-1}$. It moves first horizontally away from the origin by 5 units and then vertically upwards parallel to positive Y -axis by 3 units to reach a point ...

If $x=-2+\sqrt{-3}$, then the value of $2 x^4+5 x^3+7 x^2-x+38$ is equal to

Argument of the complex number $z=\frac{13-5 i}{4-9 i}, i=\sqrt{-1}$ is

If $\mathrm{z}=x+\mathrm{i} y$ is a complex number, then the equation $\left|\frac{z+i}{z-i}\right|=\sqrt{3}$ represents the

$\mathrm{z}=\frac{3+2 \mathrm{i} \sin \theta}{1-2 \mathrm{i} \sin \theta},(\mathrm{i}=\sqrt{-1})$ will be purely imaginary if $\theta=$

The complex numbers $\sin x+i \cos 2 x$ and $\cos x$ - $\mathrm{i} \sin 2 x,(\mathrm{i}=\sqrt{-1})$ are conjugate to each other for,

The locus of the points represented by $|z+3|-|z-3|=6$, where $z$ is a complex number, is ….

The equation $|z+1-i|=|z-1+i|$ represents a (where z is a complex number)

The value of $\frac{(\cos \theta+i \sin \theta)^4}{(\sin \theta+i \cos \theta)^5}=$ where $\mathrm{i}=\sqrt{-1}$

The modulus of the square root of the conjugate of $-7+24 \sqrt{-1}$ is __________

$$\begin{aligned} & \mathrm{f}(x)=(\cos x+\mathrm{i} \sin x) \cdot(\cos 3 x+\mathrm{i} \sin 3 x) \cdots {[\cos (2 \mathrm{n}-1) x+\mathrm{i} \sin (2 \mathrm{n}-1) x] \mathrm{n} \in \mathbb{N}} \end{aligned}$$
Then $\mathrm{f}^{\prime \prim...

If $\frac{z-1}{2 z+1}$ is an imaginary number and if it represents a circle then its radius is

If $Z=\frac{-2}{1+\sqrt{3} i}, i=\sqrt{-1}$, then the value of $\arg Z$ is

Let $\left(-2-\frac{1}{3} \mathrm{i}\right)^3=\frac{x+\mathrm{i} y}{27}, \mathrm{i}=\sqrt{-1}$, where $x$ and $y$ are real numbers, then $(y-x)$ has the value

If $\left|\frac{\mathrm{z}}{1+\mathrm{i}}\right|=2$, where $\mathrm{z}=x+\mathrm{i} y, \mathrm{i}=\sqrt{-1}$ represents a circle, then centre ' $C$ ' and radius ' $r$ ' of the circle are

If $|z|=1$ and $w=\frac{z-1}{z+1}$ (where $\left.z \neq-1\right)$, then $\operatorname{Re}(w)$ is

If $z^2+z+1=0$ then $\left(z^3+\frac{1}{z^3}\right)^2+\left(z^4+\frac{1}{z^4}\right)^2=$ where $z=w=$ complex cube root of unity

If $\mathrm{a}>0$ and $\mathrm{z}=\frac{(1+\mathrm{i})^2}{\mathrm{a}-\mathrm{i}}, \mathrm{i}=\sqrt{-1}$, has magnitude $\sqrt{\frac{2}{5}}$ then $\bar{z}$ is equal to

If $\mathrm{P}(x, y)$ denotes $\mathrm{z}=x+\mathrm{i} y x, y \in \mathbb{R}$ and $\mathrm{i}=\sqrt{-1}$ in Argand's plane and $\left|\frac{z-1}{z+2 i}\right|=1$, then the locus of P is

Let $\mathrm{z}=x+\mathrm{i} y$ be a complex number, where $x$ and $y$ are integers and $i=\sqrt{-1}$. Then the area of the rectangle whose vertices are the roots of the equation $\overline{z z}^3+\overline{\mathrm{zz}}^3=350$ is

If $z_1=5-2 i$ and $z_2=3+i$, where $i=\sqrt{-1}$, then $\arg \left(\frac{z_1+z_2}{z_1-z_2}\right)$ is

Let $z$ be a complex number such that $|z|+z=2+i$, where $i=\sqrt{-1}$, then $|z|$ is equal to

If the complex number $z=x+i y$, where $i=\sqrt{-1}$, satisfies the condition $|z+1|=1$, then $z$ lies on

Let $Z$ be a complex number such that $|Z|+Z=2+i$ (where $i=\sqrt{-1})$, then $|Z|$ is equal to

Let $\omega=-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}, \mathrm{i}=\sqrt{-1}$, then the value of $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1-\omega^2 & \omega^2 \\ 1 & \omega^2 & \omega^4\end{array}\right|$ is

If $\mathrm{w}=\frac{-1+i \sqrt{3}}{2}$, where $\mathrm{i}=\sqrt{-1}$, then the value of $\left(3+w+3 w^2\right)^4$ is

If \(Z_1=4 i^{40}-5 i^{35}+6 i^{17}+2, Z_2=-1+i\), where \(i=\sqrt{-1}\), then \(\left|Z_1+Z_2\right|=\)

If \(Z_1=2+i\) and \(Z_2=3-4 i\) and \(\frac{\overline{Z_1}}{\overline{Z_2}}=a+b i\), then the value of \(-7 a+b\) is (where \(i=\sqrt{-1}\) and \(a, b \in R)\)

If \(a>0\) and \(z=\frac{(1+i)^2}{a-i}, i=\sqrt{-1}\), has magnitude \(\frac{2}{\sqrt{5}}\), then \(\bar{z}\) is

Let \(z \in C\) with \(\operatorname{Im}(z)=10\) and it satisfies \(\frac{2 z-n}{2 z+n}=2 i-1, i=\sqrt{-1}\) for some natural number \(\mathrm{n}\), then