If $\left|Z_1-3-4 i\right|=5$ and $\left|Z_2\right|=15$, then the sum of the maximum and minimum values of $\left|Z_1-Z_2\right|$ is

If $Z=r(\cos \theta+i \sin \theta),\left(\theta \neq-\frac{\pi}{2}\right)$ is solution of $x^3=i$, then $r^9(\cos \theta+i \sin \theta)^9=x^{3-}=i$

If $\omega \neq 1$ is a cube root of unity, then one root among the 7th roots of $(1+\omega)$ is

If $1+2 i$ is a root of the equation $x^4-3 x^3+8 x^2-7 x+5=0$, then sum of the squares of the other roots is

$ \left(\frac{1+i}{1-i}\right)^{228}= $

Let $z=x+i y$ represent a point of $P(x, y)$ in the argand plane. If $z$ satisfies the condition that amplitude of $\frac{z-3}{z-2 i}=-\frac{\pi}{2}$ then the locus of $P$ is

$ (1-i \sqrt{3})^{2025}= $

One of the roots of the equation $(x+1)^4+81=0$ is

The amplitude of the complex number $\frac{(\sqrt{3}+i)(1-\sqrt{3} i)}{(-1+i)(-1-i)}$ is

If a complex number $z=x+i y$ represents a point $p(x, y)$ in the argand plane and $z$ satisfies the condition that the imaginary part of $\frac{z-3}{z+3 i}$ is zero, then the locus of the point $P$ is

$ (\sqrt{3}+i)^{10}+(\sqrt{3}-i)^{10}= $

Number of real values of $(-1-\sqrt{3 i})^{3 / 4}$ is

One of the values of $\sqrt{24-70 i}+\sqrt{-24+70 i}$ is

The set of all values of $\theta$ such that $\frac{1-i \cos \theta}{1+2 i \sin \theta}$ is purely imaginary is

If $\alpha$ is a root of the equation $x^2-x+1=0$, then

$\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3+\ldots$ to 12 terms $=$

$\omega$ is a complex cube root of unity and $Z$ is a complex number satisfying $|Z-1| \leq 2$. The possible values of $r$ such that $|Z-1| \leq 2$ and $\left|\omega Z-1-\omega^2\right|=r$ have no common solution are

If $|Z|=2, Z_1=\frac{Z}{2} e^{i \alpha}$ and $\theta$ is the $\operatorname{amp}(Z)$, then $\frac{Z_1^n-Z_1^{-n}}{Z_1^n+Z_1^{-n}}=$

If $n, K \in N$ such that $n \neq 3 K$, then $(\sqrt{3}+i)^{2 n}+(\sqrt{3}-i)^{2 n}=$

In argand plane, no value of $\sqrt[3]{1-i \sqrt{3}}$ lie in

If $\frac{2+3 i}{i-2}-\frac{4 i-3}{3+4 i}=x+i y$, then $3 x+y=$

Let $z=x+i y$ and $P(x, y)$ be a point on the argand plane. If $z$ satisfies the condition $\arg \left(\frac{z-3 i}{z+2 i}\right)=\frac{\pi}{4}$, then the locus of $P$ is

If $\omega$ is a complex cube root of unity and $x=\omega^2-\omega+2$, then

The product of all the values of $(\sqrt{3}-i)^{\frac{3}{7}}$ is

If $\omega$ is the complex cube root of unity and

$\left(\frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}\right)^{k}+\left(\frac{a+b \omega+c \omega^{2}}{b+a \omega^{2}+c \omega}\right)^{l}=2$, then $2 k+l$ is always

If $\sqrt{5}-i \sqrt{15} \doteqdot r(\cos \theta+i \sin \theta),-\pi<\theta<\pi$, then $r^2\left(\sec \theta+3 \operatorname{cosec}^2 \theta\right)=$

The point $P$ denotes the complex number $z=x+i y$ in the argand plane. If $\frac{2 z-i}{z-2}$ is a purely real number, then the equation of the locus of $P$ is

$x$ and $y$ are two complex numbers such that $|x|=|y|=1$.

If $\arg (x)=2 \alpha, \arg (y)=3 \beta$ and $\alpha+\beta=\frac{\pi}{36}$, then $x^6 y^4+\frac{1}{x^6 y^4}=$

If $x=a+b, y=a \alpha+b \beta, z=a \beta+b \alpha$ and $\alpha, \beta$ are the complex cube roots of unity, then $x^3+y^3+z^3=$

If $z=\frac{3+2 i \cos \theta}{1-2 i \sin \theta}$ is a purely imaginary number, then

$ \sin ^2 \theta+\cos ^2 3 \theta= $

If $z=x+i y$ is a complex number such that $z \bar{z}^3+\bar{z} z^3=350$ and $x, y$ are integers, then $|z|=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$, then $(\alpha+\beta)^2+\left(\alpha^2+\beta^2\right)^2+\left(\alpha^3+\beta^3\right)^2+\ldots+\left(\alpha^{12}+\beta^{12}\right)^2=$

The least positive integral value of $n$ such that $\left[\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right]^n=1$ is

If a polynomial $P(x)$ given by

$P(x)=2 x^4+a x^3+b x^2+c x+d$ is such that $P(1)=4$,

$P(2)=7, P(3)=12$ and $P(4)=19$, then $P(5)=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+1=0$, then match the items of List I with those of List II

Then, the correct match is

If $i=\sqrt{-1}$, then $\operatorname{Arg}\left[\frac{(1+i)^{2025}}{(1-i)^{2022}}\right]=$

The locus of $z$ such that $\left|\frac{z-i}{z+i}\right|=2$, where $z=x+i y$, is

If $x_n=\cos \frac{\pi}{2^n}+i \sin \frac{\pi}{2^n}$, then $\prod_{n=1}^{\infty} x_n=$

If the roots of the equation $z^2-i=0$ are $\alpha$ and $\beta$, then $|\arg \beta-\arg \alpha|=$

$\alpha, \beta, \gamma$ are the roots of the equation $x^3+2 x^2-x-2=0$, then $\alpha^6+\beta^6+\gamma^6=$

If $\frac{3 x+2}{(x+1)\left(2 x^2+3\right)}=\frac{A}{x+1}+\frac{B x+C}{2 x^2+3}$, then $A-B+C=$

If $x=\log \left(y+\sqrt{y^2+1}\right)$, then $y=$

If $i^2=-1$, then $(1+\sqrt{3} i)^{2022}-(\sqrt{3}-i)^{2022}=$

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^4+\left(\frac{\sqrt{3}-i}{\sqrt{3}+i}\right)^4=r$ cis $\theta$, then one of the values of $\sqrt{r \operatorname{cis} \theta}$ is

If $z=x+i y$ and the point $P$ in the argand plane represents $z$, then the locus of $z$ satisfying the equation $|z-2|+|z-2 i|=4$ is

One of the values of $(\sqrt{3}-i)^{2 / 5}$ is

If $\alpha, \beta, \gamma$ and $\delta$ are the roots of the equation $x^4+x^2+1=0$ such that $\alpha+\beta=-1, \gamma+\delta=1, \alpha^2=\beta$ and $\gamma^2=-\delta$, then $\alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022}=$

If $\alpha, \beta, \gamma$ are the roots of the equation $2 x^3+x^2-13 x+6=0$, then $\alpha^3+\beta^3+\gamma^3=$

If $\alpha, \beta, \gamma$ are the real roots of the equation $18 x^3-15 x^2-4 x+4=0$ such that $\alpha=\beta$ and $\alpha>\gamma$, then $\alpha+\beta^2+\gamma^3=$

If $\alpha$ is a multiple root of the equation $x^5-6 x^4+11 x^3-2 x^2-12 x+8=0$, then $3 \alpha^2-2 \alpha+1=$

When $3^{2023}$ is divided by 16 , the remainder obtained is

If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$, then $k=$

Let $z=x+i y$ be a point in the argand plane. If the amplitude of $\left(\frac{z-3}{z+2 i}\right)$ is $\frac{\pi}{2}$, then the locus of $z$ is

If a point $P$ denotes the complex number $z=x+i y$ in the argand plane and if $\frac{z-(2+i)}{z+(1-2 i)}$ is purely real, then the locus of $P$ is

If $i$ is the root of the equation $x^2+1=0$, then

$ (1+\sqrt{3} i)^{2023}+(1-\sqrt{3} i)^{2023}= $

One of the values of $(\sqrt{3}-i)^{\frac{1}{6}}$ is

If $a x^2-x y-3 y^2-5 x+20 y+c=0$ represents a pair of lines passing through the point $(2,3)$, then $a-c=$

$\operatorname{Arg}\left(\sin \frac{6 \pi}{5}+i\left(1+\cos \frac{6 \pi}{5}\right)\right)=$

$ \text { If } x+i y=\sqrt{\frac{3+i}{1+3 i}}, \text { then }\left(x^2+y^2\right)^2= $

If the imaginary part of $\frac{2 z+1}{i z+1}$ is -2, then the locus of the point representing $z$ in the Argand plane is

If $i=\sqrt{-1}$, then $(1+i)^{10}+(1-i)^{10}=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 x+2=0$, then $\alpha^{2020}+\beta^{2020}=$

If $z=\frac{-1-i \sqrt{3}}{2}$, then $\sum_{k=1}^{2022}\left(z^k+\frac{1}{z^k}\right)^2=$

$\{x \in[0,2 \pi] / \sin x+i \cos 2 x$ and $\cos x-i \sin 2 x$ are conjugate to each other} $=$

If $|x+i y|=\sqrt{x^2+y^2}$, then $\left|(1-\sqrt{3} i)^9+(\sqrt{3}+i)^9\right|=$

If $1, \omega, \omega^2$ are the cube roots of unity and $1, \alpha, \alpha^2, \alpha^3$ are the fourth roots of unity in usual notation, then $\alpha+\alpha \omega-\alpha^3 \omega^2=$

If $z=\alpha+i \beta$ satisfies the equation $|z|-z=1+2 i$ and $|z|=\sqrt{\alpha^2+\beta^2}$, then $z \bar{z}=$

If $-i$ and $\alpha$ are the roots of the equation $i z^2-2(i+1) z+(2-i)=0, \tan \theta=\frac{-1}{2}$ and $\theta \in 4$ th quadrant, then $5^3 \cos 6 \theta=$

If $1, \alpha_1, \alpha_2, \alpha_3, \ldots \alpha_{n-1}$ are $n$th roots of unity then $\sum\limits_{1 \le i < f \le n - 1}^{} {} {a_i}{a_j} = $

If $(2-i)$ is one of the roots of the equation $x^4-9 x^3+31 x^2-49 x+30=0$ and $\alpha, \beta(\alpha<\beta)$ are its real roots, then $2 \alpha-\beta=$

If $e^{i t}=\cos t+i \sin t$ and $e^{-i t}=\cos t-i \sin t$, then $\cosh (x+i y)-\cosh (x-i y)=$

If $(2 x-y+1)+i(x-2 y-1)=2-3 i$, then the multiplicative inverse of $(x-i y)$ is

If $\cos \alpha$ is the common value of $(-1)^{\frac{1}{4}}$ and $(-i)^{\frac{1}{2}}$ then $\tan \alpha=$

The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ is

If the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i =\frac{2 y+i(1-3 y)}{i-3}$, then $x+y=$

1. If $\cos \alpha+\cos \beta+\cos \gamma=0$ and $\sin \alpha+\sin \beta+\sin \gamma=0$ then $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=$

One of the values of $(-32 i)^{\frac{2}{5}}$ is

$ \sqrt{(-3+4 i)(8+6 i)}= $

If $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^m=1,2022 < m < 2029$, then $m=$

If $1, \omega, \omega^2$ are the cube roots of unity, $n \in N$ and $n>2$ then the least value of $n$ such that $1+\omega$ is a root of $x^n-x=0$ is

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4 i}\right)=0$ and $\lim \left(\frac{z-2}{z-4 i}\right)=1$ simultaneously is

If $(\sqrt{3}+i)^{10}=a+b i, a, b \in \mathbf{R}$, then the values of $a$ and $b$ are respectively

If $z$ is a complex number such that $z^2+z+1=0$, then $\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots . .+\left(z^{2020}+\frac{1}{z^{2020}}\right)^3=$

Let the roots of the equation $E_1 \equiv x^3+x^2+l x+n=0$ be $x_i,(i=1,2,3)$ and the roots of $E_2 \equiv x^3+a x^2+b x+c=0$ be $\frac{x_i-1}{2}$. If the equation $E_2=0$ is a equation of class one, then the roots of these two equations excluding the common roots are

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4+x^2+1=0$, then $\frac{\alpha^3+\beta^3+\gamma^3+\delta^3}{\alpha^6+\beta^6+\gamma^6+\delta^6}=$

Let $z$ be a complex number such that $|z|-z=2+i$, where $i=\sqrt{-1}$. Then, $|z|=$

If the amplitude of $z-2-3 i$ is $\pi / 4$, then the locus of $z=x+i y$ is

For $n>1$ and $n \in \mathbf{N}$, if $z_1, z_2, \ldots, z_n$ are the roots of the equation $(z+1)^n=z^n$, then $\sum_{i=1}^n \frac{\cot ^{-1}\left(2\left|\operatorname{Im} z_i\right|\right)-1}{2 \operatorname{Re} z_i}=$

If $z_1=x_1+i y_1, z_2=x_2+i y_2, z_3=x_1+\frac{i x_2}{2}, z_4=2 y_1+i y_2$ are complex numbers such that $\left|z_1\right|=1,\left|z_2\right|=2$ and $\operatorname{Re} \left(\begin{array}{ll}z_1 & z_2\end{array}\right)=0$, then

Assertion (A) If $z$ is a complex number such that $|z| \geq 3$, then the least value of $\left|z+\frac{3}{z}\right|$ is 1 .

Reason (R) $\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|$, for any two complex numbers $z_1, z_2$

The correct option among the following is

$ \text { If }\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021}=x+i y, $

then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

If $\omega$ is a complex cube root of unity, then $\sum_{x=1}^{10}\left((\omega x+2)\left(\omega^2 x+2\right)-3\right)$

Let $z=x+i y$ be a complex number, $A=\{z /|z| \leq 2\}$ and $B=\{z /(1-i) z+(1+i) \bar{z} \geq 4\}$ Then which one of the following options belongs to $A \cap B$ ?

The solutions of the equation $z^2\left(1-z^2\right)=16, z \in \mathbf{C}$, lie on the curve

If $z, \bar{z},-z,-\bar{z}$ forms a rectangle of area $2 \sqrt{3}$ square units, then one such $z$ is

$ \left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^8+\left(\frac{1+\cos \theta-i \sin \theta}{1+\cos \theta+i \sin \theta}\right)^{16}= $