Complex Numbers Questions (654)

Let $\left|\dfrac{z-i}{2z+i}\right|=\dfrac{1}{3}$, $z\in\mathbb{C}$, be the equation of a circle with centre $C$. If the area of the triangle whose vertices are at $(0,0)$, $C$, and $(\alpha,0)$ is 11 square units, then $\alpha^2$ equals

Let $z_1$, $z_2$, $z_3$ be three complex numbers lying on the circle $|z|=1$ with $\arg(z_1)=-\pi/4$, $\arg(z_2)=0$, $\arg(z_3)=\pi/4$. If $|z_1\bar{z}_2+z_2\bar{z}_3+z_3\bar{z}_1|^2 = \alpha+\beta\sqrt{2}$ where $\alpha,\beta\in\mathbb{Z}$, then $\alpha^2+\beta^2$ equals

If \(\left|\dfrac{z_1 - 2z_2}{2 - z_1\bar{z}_2}\right| = 1\) and \(|z_2| \neq 1\), then \(|z_1|\) equals:

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\left(2\tan\frac{5\pi}{8}\right)$, then $(r,\theta)$ is equal to:

Find the principal argument of (b) \(\dfrac{1+\sqrt{3}i}{\sqrt{3}+i}\)

The sum of maximum and minimum modulus of a complex number z satisfying |z - 25i| ≤ 15, where i = √(-1) is S. Then S/10 is:

The number of complex numbers $z$ such that $|z|=1$ and $\left|\dfrac{z}{\bar{z}}+\dfrac{\bar{z}}{z}\right|=1$ is

Let $S=\{z\in\mathbb{C}:\ \bar{z}=i(z^2+\text{Re}(\bar{z}))\}$. Then $\displaystyle\sum_{z\in S}|z|^2$ is equal to

Find the number of roots of the equation \(z^{15} = 1\) satisfying \(|\arg z| .

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to

If \(|z + 4| \leq 3\), then the maximum value of \(|z + 1|\) is

Let complex number z satisfy the inequality 2 ≤ |z| ≤ 4. A point P is selected in this region at random. The probability that argument of P lies in the interval [-π/4, π/4] is 1/K, then K = ?

If $\alpha$ and $\beta$ are the roots of $2z^2-3z-2i=0$, where $i=\sqrt{-1}$, then $16\cdot\operatorname{Re}\!\left(\dfrac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)\cdot\operatorname{Im}\!\left(\dfrac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$ is equal to

If \ \(\left(\dfrac{1+i}{1-i}\right)^m = 1\), then find the least positive integral value of \ \(m\).

The value of $\left(\sum_{k=1}^{n}\left(\sin\frac{2k\pi}{n} - i\cos\frac{2k\pi}{n}\right)\right)$ is (where $i$ is iota)

If \(n \geq 3\) and \(1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}\) are \(n\)th roots of unity, then find the sum \(\displaystyle\sum_{1 \leq i

Let $z = re^{i\theta}$ with $r^2e^{i\theta} + 3re^{-i\theta} = 0$. Find the number of solutions.

If $\alpha+i\beta$ and $\gamma+i\delta$ are the roots of $x^2-(3-2i)x-(2i-2)=0$, where $\alpha,\beta,\gamma,\delta$ are real numbers, then $\alpha\gamma+\beta\delta$ equals

Let $S=\left\{z\in\mathbb{C}\setminus\{i,2i\}:\ \dfrac{z^2+8iz-15}{z^2-3iz-2}\in\mathbb{R}\right\}$. If $\alpha-\dfrac{13i}{11}\in S,\ \alpha\in\mathbb{R}\setminus\{0\}$, then $242\alpha^2$ is equal to

Let $z$ be the complex number satisfying $|z-5|\leq3$ and having maximum positive principal argument. Then $34\left|\dfrac{5z-12}{5iz+16}\right|^2$ is equal to:

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha}}$ lie on the circles $|z-z_0|^2=4$ and $|z-z_0|^2=16$ respectively, where $z_0=1+i$. Then the value of $100|\alpha|^2$ is

Let $P=\{z\in\mathbb{C}:|z+2-3i|\le 1\}$ and $Q=\{z\in\mathbb{C}: z(1+i)+\bar{z}(1-i)\le -8\}$. Let in $P\cap Q$, $|z-3+2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2+2|z_2|^2=\alpha+\beta\sqrt{2}$, where $\alpha,\beta$ are integers, then $\alpha+\beta$ equals

We have \(|z| = 1\). Then \(\dfrac{1+z}{1+\bar{z}}\) equals:

For the complex number $z$, the minimum value of $|z| + |z - \cos\alpha - i\sin\alpha|$ is:

All the points in the set $S = \left\{\dfrac{\alpha + i}{\alpha - i} : \alpha \in \mathbb{R}\right\}$ $(i = \sqrt{-1})$ lie on a

For Problems 26–28: Complex numbers \(z\) satisfy the equation \(|z - (4/z)| = 2\).The value of \(\arg(z_1/z_2)\), where \(z_1\) and \(z_2\) are complex numbers with the greatest and the least moduli, can be

If z = (3 + 7i)(λ + iμ), when λ, μ ∈ ℤ \ {0} and i = √−1, is purely imaginary then minimum value of |z|2 is

Let ω be the complex number \(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\), where i = √−1, 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

If $\dfrac{z - \alpha}{z + \alpha}$ ($\alpha \in \mathbb{R}$) is a purely imaginary number and $|z| = 2$, then a value of $\alpha$ is

If $z$ is a complex number such that $|z|\geq 2$, then the minimum value of $\left|z+\dfrac{1}{2}\right|$

If $z$ and $w$ are two complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \dfrac{\pi}{2}$, then

Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\dfrac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\dfrac{1+[a]}{4b}$, where $[t]$ is greatest integer function, is:

Let $\left(-2 - \dfrac{1}{3}i\right)^3 = \dfrac{x + iy}{27}$ $(i = \sqrt{-1})$, where $x$ and $y$ are real numbers, then $y - x$ equals

The maximum area of the triangle formed by the complex coordinates \(z, z_1, z_2\) which satisfy the relations \(|z - z_1| = |z - z_2|\) and \(|z - (z_1 + z_2)/2| \leq r\), where \(r > |z_1 - z_2|\), is

If and only two real numbers lying between $0$ and $1$, such that $Z_1 = a + i$, $Z_2 = 1 + b$ and $Z_3 = 0$ form an equilateral triangle, then

Let $z_0$ be a root of the quadratic equation $x^2 + x + 1 = 0$. If $z = 3 + 6iz_0^{81} - 3iz_0^{93}$, then $\arg z$ is equal to

A square is drawn in the complex plane with one vertex at the origin, one side along a ray at 45° and the opposite vertex at distance \(2\sqrt{2}\). The sum of the \(x\)-coordinates of all four vertices of the square is:

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\!\left(\dfrac{1+z}{1+\bar{z}}\right)$ is equal to

Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to

Let a and b be two fixed non-zero complex numbers and z is a variable complex number. If the lines az + \bar{a}\bar{z} + 1 = 0 and bz + \bar{b}\bar{z} - 1 = 0 are mutually perpendicular, then

Let $A = \left\{\theta \in \left(-\dfrac{\pi}{2}, \pi\right) : \dfrac{3 + 2i\sin\theta}{1 - 2i\sin\theta} \text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is

If \(z_1, z_2 \in \mathbb{C}\), \(z_1^2 + z_2^2 \in \mathbb{R}\), \(z_1(z_1^2 - 3z_2^2) = 2\) and \(z_2(3z_1^2 - z_2^2) = 11\), then the value of \(z_1^2 + z_2^2\) is

If \(\omega\) is a complex nth root of unity, then \(\sum_{r=1}^{n}(ar+b)\omega^{r-1}\) is equal to

A complex number $z$ is said to be unimodular if $|z|=1$. If $z_1$ and $z_2$ are complex numbers such that $\dfrac{z_1-2z_2}{2-z_1\bar{z}_2}$ is unimodular and $z_2$ is not unimodular, then the point $z_1$ lies on a

If \(x = 9^{1/3} \times 9^{1/9} \times 9^{1/27} \times \cdots\); \(y = 4^{1/3} \times 4^{-1/9} \times 4^{1/27} \times \cdots\); and \(z = \sum_{r=1}^{\infty}(1+i)^{-r}\), then \(\arg(x + yz)\) is equal to

If cos(1 - i) = a + ib, where a, b ∈ ℝ and i = √-1, then

Let \(w = \dfrac{\sqrt{3} + i}{2}\) and \(P = \{w^n : n = 1, 2, 3, \ldots\}\). Further \(H_1 = \left\{z \in C : \text{Re}\, z > \dfrac{1}{2}\right\}\) and \(H_2 = \left\{z \in C : \text{Re}\, z

Let the equation of a ray be |z − 2| − |z − 1 − i| = √2. If it strikes the y-axis, then the equation of reflected ray (including or excluding the point of incidence) is

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \dfrac{3z_1}{2z_2} + \dfrac{2z_2}{3z_1}$, then

For Problems 29–31: In an Argand plane \(z_1, z_2\), and \(z_3\) are, respectively, the vertices of an isosceles triangle \(ABC\) with \(AC = BC\) and \(\angle CAB = \theta\). If \(z_4\) is incenter of triangle, then the value of \(AB \times AC/(IA)^2\) is