Complex Numbers Questions - JEE

Let $ z_1, z_2, z_3 $ be distinct complex numbers with $ |z_1|=|z_2|=|z_3|=a $ and centroid at the origin. Then $ |z_1 z_2 + z_2 z_3 + z_3 z_1| $ equals:

Let $\omega$ be a complex cube root of unity. The value of $(1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^8)\cdots$ (up to $2n$ factors) relates to ___.

If $z_1$ lies on $|z-3| + |z+3| = 8$ such that $\arg z_1 = \pi/6$, then $37|z_1|^2 =$ ___.

Let $|z|=2$ and $\text{Re}(z^2)=0$. Then $z$ equals:

The complex number $ z $ satisfying $ |z - 1| = |z + 1| = |z - i| $ lies at:

Let $z_1=2+3i, z_2=3+4i$. If $z=1+yi$ and $z_1,z_2,z$ are in AP, then $\text{Im}(z)$ is:

The number of solutions of $z^4+4=0$ lying in the first quadrant is:

Let $\omega = e^{2\pi i/3}$. The number of distinct complex numbers $z$ satisfying $|z+1|=|z+\omega|=|z+\omega^2|$ is ___.

If $z$ satisfies $|z-3|=|z+3i|$, then the locus is:

$ z = \dfrac{1+2i}{1-(1-i)^2} $ equals:

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

The least value of $|z|+|z-1|$ is:

The complex number z satisfies $z + |z| = 2 + 8i$. The value of $|z|$ is ___.

The equation $ |z+1-i| = |z-1+i| $ represents:

Express the following complex numbers in $a + ib$ form: (b) $\dfrac{2 - \sqrt{-25}}{1 - \sqrt{-16}}$

If $\alpha$ and $\beta$ be the roots of the equation $x^2 - 2x + 2 = 0$, then the least value of $n$ for which $(\alpha/\beta)^n = 1$ is __________.

The number of points in the complex plane satisfying both $|z-2|=2$ and $z(1-i)+\bar{z}(1+i)=4$ (where $i=\sqrt{-1}$) is

If $\omega$ is the imaginary cube root of unity, then the number of pairs of integers $(a, b)$ such that $|a\omega + b| = 1$ is ___.

Let $1, \omega, \omega^2$ be the cube roots of unity. The least possible degree of a polynomial with real coefficients having roots $2\omega, (2+3\omega), (2+3\omega^2), (2-\omega-\omega^2)$ is ___.

Consider the following two statements: Statement I: For any two non-zero complex numbers $z_1,z_2$, $(|z_1|+|z_2|)\left|\dfrac{z_1}{|z_1|}+\dfrac{z_2}{|z_2|}\right|\leq2(|z_1|+|z_2|)$. Statement II: If $x,y,z$ are three distinct complex numbers and $a,b,c$ are three positive real numbers such that $\dfrac{a}{|y-z|}=\dfrac{b}{|z-x|}=\dfrac{c}{|x-y|}$, then $\dfrac{a^2}{y-z}+\dfrac{b^2}{z-x}+\dfrac{c^2}{x-y}=1$. Between the above two statements,

Given that the two curves $\arg(z) = \pi/6$ and $|z - 2\sqrt{3}\, i| = r$ intersect in two distinct points, then

There is only one real number $ x $ for which $ (x-1)^2 + x^2 + (x+1)^2 = 0 $ has a solution. Then $ M - N $ (where $M$ and $N$ are specific expressions from the problem) is:

If $ \alpha, \beta $ are the roots of $ x^2 - 2x + 2 = 0 $, and $ n $ is the least positive integer such that $ \left(\dfrac{\alpha}{\beta}\right)^n = 1 $, then $ n $ is:

970. A regular heptadecagon $P_1P_2P_3\ldots P_{17}$ is inscribed in a unit circle. Find $\displaystyle\prod_{n=2}^{17} P_1P_n$.

The sum of the square of the modulus of the elements in the set $\{z=a+ib: a,b\in\mathbb{Z},\,z\in\mathbb{C},\,|z-1|\leq1,\,|z-5|\leq|z-5i|\}$ is ________.

Let $z$ be a complex number such that the real part of $\dfrac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z-(6+8i)|$ is equal to:

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 $P(z)$, $R(z^4)$ and $Q(z^2)$ be three points in the Argand plane such that $PR + RQ = PQ$; ($z\neq 0,1$). Then $z$ lies on

If $ z = \dfrac{1-i}{\sqrt{2}} $, then $z^{2022}+\left(\dfrac{1}{z}\right)^{2022}$ equals:

The minimum value of $ |z| + |z-1| + |z-2| $ for $ z \in \mathbb{C} $ is:

Let the complex number $z=x+iy$ be such that $\dfrac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$ is equal to

Let $C$ be the circle in the complex plane with centre $z_0=\dfrac{1}{2}(1+3i)$ and radius $r=1$. Let $z_1=1+i$ and the complex number $z_2$ be outside circle $C$ such that $|z_1-z_0||z_2-z_0|=1$. If $z_0,z_1$ and $z_2$ are collinear, then the smaller value of $|z_2|^2$ is equal to

Let $\omega=e^{2\pi i/3}$. Find $N$ as described in the problem. ___

Let z$\i_n C be such that$z +3i = 2 + 3i$. Then the sum of all possible values of z is 2$z-2$+i$

Let$A = {z$$\i_n C$: |$z - 2 - i$| = 3},$B = {z$$\i_n C$: Re($z - iz) = 2}$and$S = A$$\cap B. Then$$\sum is equal to 2$|z| z$\inS ________. 2$

If the locus of $z \in \mathbb{C}$, such that $\text{Re}\left(\frac{z-1}{2z+i}\right) + \text{Re}\left(\frac{\bar{z}-1}{2\bar{z}-i}\right) = 2$ is a circle of radius $r$ and center $(a, b)$, then $\frac{15ab^2}{r}$ is equal to:

The sum of all possible values of $\theta\in[-\pi,2\pi]$, for which $\dfrac{1+i\cos\theta}{1-2i\cos\theta}$ is purely imaginary, is equal to:

If $\omega$ is a non-real cube root of unity, then the value of $\dfrac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} + \dfrac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2}$ is equal to:

If $z=x+iy$, $xy\ne 0$, satisfies the equation $z^2+i\bar{z}=0$, then $|z|^2$ is equal to:

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B\alpha+C\alpha^2$, $A,B,C\ge 0$, then $5(3A-2B-C)$ is equal to ______

Let $S = \{z \in \mathbb{C} : |z-1|=1 \text{ and } (\sqrt{2}-1)(z+\bar{z}) - i(z-\bar{z}) = 2\sqrt{2}\}$. Let $z_1, z_2 \in S$ be such that $|z_1| = \max_{z \in S}|z|$ and $|z_2| = \min_{z \in S}|z|$. Then $|\sqrt{2}z_1 - z_2|^2$ equals:

Let $\alpha,\beta$ be the roots of the equation $x^2-x+2=0$ with $\text{Im}(\alpha)>\text{Im}(\beta)$. Then $\alpha^6+\alpha^4+\beta^4-5\alpha^2$ is equal to

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left(\frac{1-\sqrt{\pi}i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}i}\right)$, $i=\sqrt{-1}$, then the distance of the point $(\alpha,\beta)$ from the line $4x-3y=7$ is ______.

If $z=\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}$, $i=\sqrt{-1}$, then $(z^{201}-i)^8$ is equal to

Let $S_1=\{z\in\mathbb{C}:|z|\leq5\}$, $S_2=\left\{z\in\mathbb{C}:\text{Im}\left(\dfrac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\geq0\right\}$ and $S_3=\{z\in\mathbb{C}:\text{Re}(z)\geq0\}$. Then the area of the region $S_1\cap S_2\cap S_3$ is:

The area (in sq. units) of the region $S=\{z\in\mathbb{C}:|z-1|\leq2;\,(z+\bar{z})+i(z-\bar{z})\leq2,\,\text{Im}(z)\geq0\}$ is:

Let $S=\{z\in\mathbb{C}:4z^2+\bar{z}=0\}$. Then $\displaystyle\sum_{z\in S}|z|^2$ is equal to:

Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6i|=5$. Then the value of $z^3+3z^2-15z+141$ is equal to

If |z - 2 - i| = |z| sin$\left(\frac{\pi}{4} - \arg z\right)$, where i = √−1, then locus of z, is

If $ z = \dfrac{3+i}{3-i}+\dfrac{3-i}{3+i} $, then $\text{Re}(z)$ equals:

Complex Numbers Questions (654)