Complex Numbers Questions - JEE

$z_1,z_2\in\mathbb{C}$ with $|z_1|=|z_2|=1$, $z_1^2+z_2^2=1$. Then locus of $z_1+z_2$ is

Number of points of intersection of $\arg(z-2-7i)=\cot^{-1}2$ and $\arg\!\left(\dfrac{z-5i}{z+2-i}\right)=\pm\dfrac{\pi}{2}$

Find the number of complex numbers $z$ satisfying both $|z + \bar{z}| + |z - \bar{z}| = 2$ and $|z + i| + |z - i| = 2$.

If z is a complex number of unit modulus and argument θ, then arg (1 + z)/(1 + z̄) equals to

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 $\arg\!\left(\dfrac{z-(10+6i)}{z-(4+2i)}\right)=\dfrac{\pi}{4}$, then the perimeter of locus of $z$ is

If $Z$ is a complex number such that $\arg\left(z(1+\bar{z})\right) + \arg\left(\dfrac{|z|^2}{z - |z|^2 i}\right) = 0$, then

If all roots of $z^3+az^2+bz+c=0$ are of unit modulus, then

If the equation $ z^3 + (3+i)z^2 - 3z - (m+i) = 0 $, where $ m \in \mathbb{R} $, has at least one real root, then the value of $ m $ 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

Number of points of intersection of $\arg(z-2-7i)=\cot^{-1}2$ and $\arg\!\left(\dfrac{z-5i}{z+2-i}\right)=\pm\dfrac{\pi}{2}$

Find the number of complex numbers $z$ satisfying both $|z + \bar{z}| + |z - \bar{z}| = 2$ and $|z + i| + |z - i| = 2$.

If $\omega$ is an imaginary cube root of unity, then $(1 + \omega - \omega^2)^7$ equals

If z is any complex number satisfying |z - 3 - 2i| ≤ 2, where i = √−1, then the minimum value of |2z - 6 + 5i|, is

If $|z-3+2i| \leq 4$, then the difference between the greatest value and the least value of $|z|$ is

If $\arg\!\left(\dfrac{z-(10+6i)}{z-(4+2i)}\right)=\dfrac{\pi}{4}$, then the perimeter of locus of $z$ is

If $\omega = \frac{z}{z - i}$ and $|\omega| = 1$, where $i = \sqrt{-1}$, then z lies on

Express the following in a + ib form:(c) $\dfrac{(\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta)}{(\cos\gamma + i\sin\gamma)(\cos\delta + i\sin\delta)}$

Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^2 + \alpha z + \beta = 0$ has two distinct roots on the line $\text{Re}(z) = 1$, then it is necessary that

If $1,z_1,z_2,\ldots,z_{n-1}$ are $n$-th roots of unity, then $(1-z_1)(1-z_2)\cdots(1-z_{n-1})$ equals

Let $\alpha\in\mathbb{R}$, $z_1,z_2,z_3$ be three distinct complex numbers such that $|z_1|=|z_2|=|z_3|=3$ and $|(kz_1+z_2)-(kz_2+z_3)|_{\min}=\alpha|z_3-z_2||z_3-z_1|$, $\forall k\in\mathbb{R}-\{0\}$. Then $36\alpha=$

Let $\frac{1}{|z_2 - z_3|} = \frac{2}{|z_3 - z_1|} = \frac{3}{|z_1 - z_2|} = \lambda$ (say). Then which of the following is correct?

If $Z$ is a complex number such that $\arg\left(z(1+\bar{z})\right) + \arg\left(\dfrac{|z|^2}{z - |z|^2 i}\right) = 0$, then

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

Let $z_1$ and $z_2$ be two complex numbers such that $|z_1|=1$ and $|z_2|=10$. If $\theta=\arg\left(\dfrac{z_1-z_2}{z_2}\right)$, then maximum value of $\tan^2\theta$ is

Let $z$ be a complex number such that $|z+2|=1$ and $\text{Im}\left(\dfrac{z+1}{z+2}\right)=\dfrac{1}{5}$. Then the value of $|\text{Re}(z+2)|$ is:

Which must be true (T/F in order): I) $\arg(\bar{z})=-\arg(z)$ for any complex number $z$. II) If $|z-z_1|-|z-z_2|=k$ where $k<|z_1-z_2|$, then locus of $z$ is a hyperbola. III) $z_1,z_2,\ldots,z_n$ represent vertices of $n$-sided regular polygon with centre $z_0$, then $\sum z_i^2=nz_0^2$. IV) If $z_1^2+z_2^2+z_1z_2=0$, then $z_1,z_2$ and origin form equilateral triangle.

If $(a + b) - i(3a + 2b) = 5 + 2i$, then find $a$ and $b$.

Which of the following is equal to $\displaystyle\sum_{k=1}^{n}\left(\sin\frac{2\pi k}{n} - i\cos\frac{2\pi k}{n}\right)$, where $i = \sqrt{-1}$?

If $z_1, z_2$ and $z_3, z_4$ are two pairs of conjugate complex numbers, then find the value of $\arg(z_1/z_4) + \arg(z_2/z_3)$.

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

If a = cos(2π/7) + i sin(2π/7), then find the quadratic equation whose roots are α = a + a2 + a4 and β = a3 + a5 + a6.

Find the least positive integer $n$ such that $\left(\dfrac{2i}{1+i}\right)^n$ is a positive integer.

If $x$ and $y$ are complex numbers, then the system of equations $(1+i)x + (1-i)y = 1$, $2ix + 2y = 1+i$ has

Let $z_1$ and $z_2$ be complex numbers such that $z_1 \neq z_2$ and $|z_1| = |z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then $\dfrac{z_1 + z_2}{z_1 - z_2}$ may be

If $|z_1 - 1| \leq 1$, $|z_2 - 2| \leq 2$, $|z_3 - 3| \leq 3$, then find the greatest value of $|z_1 + z_2 + z_3|$.

If complex number $z$ lies on the curve $|z + 1 - i| = 1$, then find the locus of the complex number $w = \dfrac{z + i}{1 - i}$, where $i = \sqrt{-1}$.

Let $\omega \neq 1$ be a complex cube root of unity. If $(4+5\omega+6\omega^2)^{n^2+2} + (6+5\omega^2+4\omega)^{n^2+2} + (5+6\omega+4\omega^2)^{n^2+2} = 0$, and $n \in \mathbb{N}$; where $n \in [1, 100]$, then number of values of n is ___.

The roots of the cubic equation $(z + ab)^3 = a^3$, $a \neq 0$, represent the vertices of a triangle of sides of length

If $p$ and $q$ are distinct prime numbers, then the number of distinct imaginary numbers which are $p$th as well as $q$th roots of unity are

If the six solutions of $x^6 = -64$ are written in the form $a + bi$, where $a$ and $b$ are real, then find the product of those solutions with $a > 0$.

For Problems 17–19: Suppose $z$ and $\omega$ are two complex numbers such that $|z| \leq 1$, $|\omega| \leq 1$, and $|z + i\omega| = |z - i\bar{\omega}| = 2$.Which of the following is true about $|z|$ and $|\omega|$?

Let $\omega$ be the complex number $\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}$. Then the number of distinct complex numbers z satisfying $\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix} = 0$ is equal to _______.(IIT-JEE, 2010)

Consider the equation $10z^2 - 3iz - k = 0$, where $z$ is a complex variable and $i^2 = -1$. Which of the following statements is true?

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

If $a^2 + b^2 = 1$, then $\dfrac{1 + b + ia}{1 + b - ia} =$

If ω is an imaginary fifth root of unity, then find the value of $\log_2 |1 + ω + ω^2 + ω^3 - 1/ω|$.

The value of $i^{1+3+5+\cdots+(2n+1)}$ is ________.

If $2z(1+a) = b + ic$ and $a^2 + b^2 + c^2 = 1$, then $[(1+iz)/(1-iz)] =$

Complex Numbers Questions (654)