Complex Numbers Questions (654)

If $z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2}$ ($i = \sqrt{-1}$), then $(1 + iz + z^5 + iz^8)^9$ is equal to

Number of integral values of \( k \) for which the expression \( \dfrac{k + 2i}{k - 1 + i} \) is purely real is:

The number of complex numbers satisfying \( z^2 = 2\bar{z} \) is:

If \(\alpha, \beta\) are the roots of \(z^2-z+1=0\), then \(\alpha^{101}+\beta^{107}\) is:

For all complex numbers \(z\) with \(|z-3-4i|=1\), the maximum value of \(|z|\) is:

The number of solutions of \(z^2 + 3\bar{z} = 0\) is:

The value of \(|1+\omega+\omega^2+\cdots+\omega^n|\) where \(\omega=e^{2\pi i/(n+1)}\) is:

If z and \bar{z} represent adjacent vertices of a regular polygon of n sides with centre at origin and if \frac{\text{Im}(z)}{\text{Re}(z)} = \sqrt{2} - 1, the value of n is equal to

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$, $z\in\mathbb{C}$. Then $4(\alpha^2+\beta^2)$ is equal to:

For the regular hexagon ABCDEF with incircle, the area of the region formed by points $P(z)$ lying inside the incircle and satisfying $\frac{\pi}{3} \leq \arg(z) \leq \frac{5\pi}{6}$ is $\frac{m\pi}{n}$, where $m$ and $n$ are relatively prime natural numbers. Find $m + n$.

Let $\alpha,\beta$ be the roots of the equation $x^2-\sqrt{6}x+3=0$ such that $\text{Im}(\alpha)>\text{Im}(\beta)$. Let $a,b$ be integers not divisible by 3 and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+ib)$, $i=\sqrt{-1}$. Then $n+a+b$ is equal to ______.

If \(\dfrac{3\pi}{2}

The real value of \(\alpha\) for which \(\dfrac{1-i\sin\alpha}{1+2i\sin\alpha}\) is purely real is (\(n\in\mathbb{Z}\)):

Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1| = 9$ and $|z_2 - 3 - 4i| = 4$. Then the minimum value of $|z_1 - z_2|$ is

Let $z \in \mathbb{C}$ be such that $|z| < 1$. If $\omega = \dfrac{5 + 3z}{5(1 - z)}$, then

The equation $|z - i| = |z - 1|$, $i = \sqrt{-1}$, represents

Let $z = \left(\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}\right)^5 + \left(\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}\right)^5$. If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z$, then

Let $z \in \mathbb{C}$ with $\text{Im}(z) = 10$ and it satisfies $\dfrac{2z - n}{2z + n} = 2i - 1$ for some natural number $n$, then

A value of $\theta$ for which $\dfrac{2 + 3i\sin\theta}{1 - 2i\sin\theta}$ is purely imaginary, is

927. Let \(A\) and \(B\) be two sets of complex numbers such that \(A = \{z \mid z^3 - 2|z|^2 + 3i|z| - 6i = 0\}\) and \(B = \{z \mid |z^2 - 4| + |z^2 + 4| \leq 4|z|\}\). Find the area of the figure enclosed by joining the points lying in \(A \cap B\).

Let $p,q\in\mathbb{R}$ and $(1-\sqrt{3}i)^{200}=2^{199}(p+iq)$, $i=\sqrt{-1}$. Then $p+q+q^2$ and $p-q+q^2$ are roots of the equation:

Let $z$ be a complex number such that $\left|\dfrac{z-2i}{z+i}\right|=2$, $z\neq-i$. Then $z$ lies on the circle of radius 2 and centre:

For two non-zero complex numbers $z_1$ and $z_2$, if $\text{Re}(z_1z_2)=0$ and $\text{Re}(z_1+z_2)=0$, then which of the following are possible? (A) $\text{Im}(z_1)>0$ and $\text{Im}(z_2)>0$; (B) $\text{Im}(z_1)<0$ and $\text{Im}(z_2)>0$; (C) $\text{Im}(z_1)>0$ and $\text{Im}(z_2)<0$; (D) $\text{Im}(z_1)<0$ and $\text{Im}(z_2)<0$.

Let $\alpha=8-14i$, $A=\left\{z\in\mathbb{C}:\dfrac{\alpha z-\overline{\alpha}\bar{z}}{z^2-(\bar{z})^2-112i}=1\right\}$ and $B=\{z\in\mathbb{C}:|z+3i|=4\}$. Then $\displaystyle\sum_{z\in A\cap B}(\text{Re}\,z-\text{Im}\,z)$ is equal to ___.

Let $z=1+i$ and $z_1=\dfrac{1+i\bar{z}}{\bar{z}(1-z)+\dfrac{1}{z}}$. Then $\dfrac{12}{\pi}\arg(z_1)$ is equal to ___.

For all $z\in C$ on the curve $C_1:|z|=4$, let the locus of the point $z+\dfrac{1}{z}$ be the curve $C_2$. Then:

If \(\dfrac{1+i}{1-i}=x+iy\), then \((x,y)\) is:

If $z = x + iy$ and $|z| + |w| = 4$, find the minimum and maximum values of $|z|$.

In $\triangle ABC$: $A=2025\omega+2024i$, $B=2024i\omega^2+2025$, $C=2024\omega^2-2025i$ where $\omega$ is non-real cube root of unity with positive imaginary part. The internal bisector of $\angle ACB$ meets $AB$ at $D$ and $\angle BDC=\frac{k\pi}{24}$. Then $k$ is

Let $S_1=\{z\in\mathbb{C}:|z-2|\leq|\text{Re}(z)+2|\}$, $S_2=\{z\in\mathbb{C}: z(1+i)+\bar{z}(1-i)-12\leq 0\}$, $S_3=\{z\in\mathbb{C}:\text{Re}(z)\geq 0,\text{Im}(z)\geq 0\}$ and $S=S_1\cap S_2\cap S_3$. The maximum value of $|z-2i|^2$; $z\in S$ is

If \(|z-2+3i|=9\), then the greatest value of \(|z|\) is:

The number of solution(s) of the equation $z^2 - z - |z|^2 - \frac{64}{|z|^3} = 0$ is ___________.

Let $w, \bar{w}$ is complex cube root of unity and $P(z)$ is point on a circle $|z| = 4$ such that $|z - 1|$ is maximum and centroid of triangle formed by $z_1 - w, w - \bar{w}$ is $\alpha$ then $-7 \text{Re}(\alpha)$ is___________.

Among the statements $(S_1)$: The set $\{z \in \mathbb{C} - \{-i\} : |z| = 1 \text{ and } \frac{z-i}{z+i} \text{ is purely real}\}$ contains exactly two elements, and $(S_2)$: The set $\{z \in \mathbb{C} - \{-1\} : |z| = 1 \text{ and } \frac{z-1}{z+1} \text{ is purely imaginary}\}$ contains infinitely many elements.

If $\omega \neq 1$ is a cube root of unity and $z$ is a complex number such that $|z| = 1$ then $\left|\frac{2 + 3\omega + 4z\omega^2}{4\omega + 3\omega^2 z + 2z}\right| = $ ____.

Let $z_i, i = 1, 2, \ldots, 6$ be the roots of $z^6 + z^4 = 2$ then $\sum_{i=1}^{6} |z_i|^4$ is equal to ____.

If $z\bar{z} = 1$, then the value of $\left|2 + \frac{1}{z}\right| + |2 - z|^2$ is ____.

Number of complex number $z$ satisfying $z^3 = \bar{z}$ is ____.

Let $\alpha$ and $\beta$ be two complex numbers satisfying $|\alpha + i| = 1$ and $|\beta - 2 - 3i| = 6$. Then the value of $6|\alpha|_{\max} - |\beta|_{\max}$ is ____.

The number of complex numbers which are conjugate of their own cube, is ____.

Let $z$ be a complex number with nonzero imaginary part and $a=z^2+z+1$ is real. Then '$a$' cannot take the value

In $\triangle ABC$: $A=2025\omega+2024i$, $B=2024i\omega^2+2025$, $C=2024\omega^2-2025i$ where $\omega$ is non-real cube root of unity with positive imaginary part. The internal bisector of $\angle ACB$ meets $AB$ at $D$ and $\angle BDC=\frac{k\pi}{24}$. Then $k$ is

If |z - 2| - |z - 3| = 2 represents a circle, then its radius is equal to

If \(x = \omega - \omega^2 - 2\), then the value of \(x^4 + 3x^3 + 2x^2 - 11x - 6\) is (where \(\omega\) is cube root of unity) ___.

Let $z_1=3+3i$, $z_2=9-6\sqrt2+(6\sqrt2-3)i$, $\triangle ABC$ right-angled at $C$. Min and max possible values of $\arg(z_3)$ are $a,b$ ($\arg z_3\in(-\pi,\pi]$). Then $b-a=\dfrac{\pi}{k}$. Find $k$.

Let z be a complex number such that |z| = 1. If$2+k$$z = kz$, k$\i_n R, then the maximum distance of$k + ik$from 2$¯$k+ z$the circle |$z - (1 + 2i)$| = 1 is:

If$\alpha is a root of the equation$x + x + 1 = 0$and$$\sum 2 n ($$\alpha$k + 1$)$2 = 20$, then n is equal to$k=1$k$$\alpha$

Let $z_1,z_2,z_3$ be three complex numbers such that $|z_1|=|z_2|=|z_3|=1$ and $\dfrac{z_1}{z_2z_3}+\dfrac{z_2}{z_3z_1}+\dfrac{z_3}{z_1z_2}=-1$. Number of possible integral values of $|z_1+z_2+z_3|$ is

Number of distinct quadratic equations with real roots such that the equation remains unchanged if their roots are cubed is equal to

If \( |z| = \left| z - \dfrac{4}{z} \right| + \dfrac{4}{|z|} \), then the maximum value of \( |z| \) is