Complex Numbers Questions (654)

The roots of the equation \(t^3 + 3at^2 + 3bt + c = 0\) are \(z_1, z_2, z_3\) which represent the vertices of an equilateral triangle. Then

Statement-1: Locus of \(z\) satisfying the equation \(|z - 1| + |z - 8| = 5\) is an ellipse.Statement-2: Sum of focal distances of any point on ellipse is constant for an ellipse.

The set \(\left\{\text{Re}\left(\frac{2iz}{1-z^2}\right) : z \text{ is a complex number, } |z| = 1, z \neq \pm 1\right\}\) is

Given that \(\bar{z} + i\bar{\omega} = 0\) and \(\arg(z\omega) = \pi\), then \(\arg(z)\) equals:

If \(|w| = 1\), then \(\left|\dfrac{z}{z - \frac{i}{3}}\right| = 1\) implies that \(z\) lies on:

Let z be a complex number and a a real parameter such that z^2 + ax + a^2 = 0, then

The locus of the points z which satisfy the condition \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3} is

If \(k + |k + z^2| = |z|^2\) (\(k \in R^-\)), then possible argument of \(z\) is

Consider the region \(S\) of complex numbers \(a\) such that \(|z^2 - az + 1| = 1\), where \(|z| = 1\). Then area of \(S\) in the Argand plane is

If z^2 + z + 1 = 0, where z is a complex number, then find the value of \left(z + \frac{1}{z}\right)^2 + \left(z^2 + \frac{1}{z^2}\right)^2 + \left(z^3 + \frac{1}{z^3}\right)^2 + \cdots + \left(z^6 + \frac{1}{z^6}\right)^2

For Problems 14–16: Consider a quadratic equation \(az^2 + bz + c = 0\), where \(a, b, c\) are complex numbers.The condition that the equation has one purely real root is

Let w = eiπ/3, where i = √−1 and a, b, c, x, y and z be non-zero complex numbers such thata + b + c = xa + bω + cω² = ya + bω² + cω = zThe value of \(\frac{|x|^2 + |y|^2 + |z|^2}{|a|^2 + |b|^2 + |c|^2}\) is

The complex number satisfying $\arg\left(z+i\right)=\frac{\pi}{4}$ and $\arg\left(2z+3-2i\right)=\frac{3\pi}{4}$ simultaneously, is :

Let $O$ be the origin, $A = z_1 = \sqrt{3}+2\sqrt{2}\,i$, and $B = z_2$ such that $\sqrt{3}|z_2|=|z_1|$ and $\arg(z_2)=\arg(z_1)+\dfrac{\pi}{6}$. Which of the following is true?

If the centre and radius of the circle $\left|\dfrac{z-2}{z-3}\right|=2$ are respectively $(\alpha,\beta)$ and $\gamma$, then $3(\alpha+\beta+\gamma)$ is equal to:

Let $z_{1},z_{2}$ and $z_{3}$ be three complex numbers on the circle $|z|=1$ with $\arg(z_{1})=-\dfrac{\pi}{4}$, $\arg(z_{2})=0$ and $\arg(z_{3})=\dfrac{\pi}{4}$. If $\bigl|z_{1}\overline{z_{2}}+z_{2}\overline{z_{3}}+z_{3}\overline{z_{1}}\bigr|^{2}=\alpha+\beta\sqrt{2}$, $\alpha,\beta\in\mathbb{Z}$, then the value of $\alpha^{2}+\beta^{2}$ is:

Let $A=\left\{\theta\in(0,2\pi):\ \dfrac{1+2i\sin\theta}{1-i\sin\theta}\text{ is purely imaginary}\right\}$. Then the sum of the elements in $A$ is

If \(|Z - 4| + |Z + 4| = 10\), then the difference between the maximum and the minimum values of \(|Z|\) is:

If \(x^2 - x + 1 = 0\) has roots \(\alpha\) and \(\beta\), then the value of \(\alpha^{2009} + \beta^{2009}\) is:

If \(\frac{3 + i \sin \theta}{4 - i \cos \theta}\), where \(\theta \in [0, 2\pi]\), is a real number, then an argument of \(\sin \theta + i \cos \theta\) is

The system of equations |z + 1 - i| = 2 and |z| = 3 has (where i = √−1)

If $S=\{z\in\mathbb{C}: |z-i|=|z+i|=|z-1|\}$, then $n(S)$ is:

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

Let $\alpha,\beta$ be the roots of the equation $x^{2}-ax-b=0$ with $\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $P_{n}=\alpha^{n}-\beta^{n}$. If $P_{3}=-5\sqrt{7}\,i$, $P_{4}=-3\sqrt{7}\,i$, $P_{5}=11\sqrt{7}\,i$ and $P_{6}=45\sqrt{7}\,i$, then $\bigl|\alpha^{4}+\beta^{4}\bigr|$ is equal to \rule{2cm}{0.4pt}\,.

Let $O$ be the origin, the point $A$ be $z_{1}=\sqrt{3}+2\sqrt{2}\,i$, the point $B(z_{2})$ be such that $\sqrt{3}|z_{2}|=|z_{1}|$ and $\arg(z_{2})=\arg(z_{1})+\dfrac{\pi}{6}$. Then:

If $|Z - 2| = 2|Z - 1|$, then the value of $\frac{|Re(Z)|}{|a|}$ is (where $Z$ is a complex number and $Re(Z)$ represents the real part of $Z$)

Let $S=\{z:3\leq|2z-3(1+i)|\leq7\}$ be a set of complex numbers. Then $\min_{z\in S}\left|z+\dfrac{1}{2}(5+3i)\right|$ is equal to:

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:

For a complex number $Z$, if $|Z - 1 + i| + |Z + i| = 1$, then the range of the principle argument of $Z$ is (where principle arg$(Z) ∈ (-π, π]$)

Equation of tangent drawn to the circle $|z|=r$ at the point $A\left(z_0\right)$, is :

Let integers $a, b \in [-3,3]$ be such that $a+b\neq0$. Then the number of all possible ordered pairs $(a,b)$ for which $\left|\dfrac{z-a}{z+b}\right|=1$ and $\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix}=1$, $z\in\mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$, is equal to

Let $|z_{1}-8-2i|\le 1$ and $|z_{2}-2+6i|\le 2$, $z_{1},z_{2}\in\mathbb{C}$. Then the minimum value of $|z_{1}-z_{2}|$ is:

Let the curve $z(1+i)+\bar{z}(1-i)=4$, $z\in\mathbb{C}$, divide the region $|z-3|\leq1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals

For $\alpha,\beta,z\in\mathbb{C}$ and $\lambda>1$, if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^2+|z-\beta|^2=2\lambda$, then $|\alpha-\beta|$ is equal to _____.

Let $a\neq b$ be two non-zero real numbers. Then the number of elements in the set $X=\{z\in\mathbb{C}:\ \text{Re}(az^2+bz)=a\ \text{and}\ \text{Re}(bz^2+az)=b\}$ is equal to

Suppose that z is a complex number that satisfies \(|z - 2 - 2i| \leq 1\). The maximum value of \(|2iz + 4|\) is equal to ___.

Let $z_1$ and $z_2$ be two complex numbers such that $|z_1-8-2i|\leq1$ and $|z_2-2+6i|\leq2$. Then the minimum value of $|z_1-z_2|$ is

Let $\alpha=\dfrac{-1+i\sqrt{3}}{2}$ and $\beta=\dfrac{-1-i\sqrt{3}}{2}$, $i=\sqrt{-1}$. If $(7-7\alpha+9\beta)^{20}+(9+7\alpha-7\beta)^{20}+(-7+9\alpha+7\beta)^{20}+(14+7\alpha+7\beta)^{20}=m^{10}$, then $m$ is _____

If $\alpha+i\beta$ and $\gamma+i\delta$ are the roots of $x^{2}-(3-2i)x-(2i-2)=0,\ i=\sqrt{-1}$, then $\alpha\gamma+\beta\delta$ is equal to:

Let $\alpha$ and $\beta$ be the roots of $x^2-ax-b=0$ with $\text{Im}(\alpha)<\text{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5\sqrt{7}\,i$, $P_4=-3\sqrt{7}\,i$, $P_5=11\sqrt{7}\,i$, and $P_6=45\sqrt{7}\,i$, then $|\alpha^4+\beta^4|$ is equal to

Let the curve $z(1+i)+\bar z(1-i)=4$, $z\in\mathbb{C}$, divide the region $|z-3|\le 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals:

Complex numbers z₁ and z₂ satisfy z + z̄ = 2|z - 1| and arg(z₁ - z₂) = π/4. Then the value of Im(z₁ + z₂) is:

For $a\in\mathbb{C}$, let $A=\{z\in\mathbb{C}:\ \text{Re}(a+\bar{z})>\text{Im}(\bar{a}+z)\}$ and $B=\{z\in\mathbb{C}:\ \text{Re}(a+\bar{z})<\text{Im}(\bar{a}+z)\}$. Then among the two statements: (S1): If $\text{Re}(a),\text{Im}(a)>0$, then $A$ contains all real numbers. (S2): If $\text{Re}(a),\text{Im}(a)<0$, then $B$ contains all real numbers.

If |z| = 2, the points representing the complex numbers -1 + 5z will lie on

Let integers $a,b\in[-3,3]$ be such that $a+b\ne 0$. Then the number of all possible ordered pairs $(a,b)$, for which $\left|\dfrac{z-a}{z+b}\right|=1$ and $\left|\,\begin{matrix} z+1 & \omega & \omega^{2}\\ \omega & z+\omega^{2} & 1\\ \omega^{2} & 1 & z+\omega\end{matrix}\,\right|=1$, where $\omega$ and $\omega^{2}$ are the complex cube roots of unity, is equal to \rule{2cm}{0.4pt}.

Let $w=z\bar{z}+k_1z+k_2iz+\lambda(1+i),\ k_1,k_2\in\mathbb{R}$. Let $\text{Re}(w)=0$ be the circle $C$ of radius 1 in the first quadrant touching the line $y=1$ and the $y$-axis. If the curve $\text{Im}(w)=0$ intersects $C$ at $A$ and $B$, then $30(AB)^2$ is equal to _______.

Let $z=(1+i)(1+2i)(1+3i)\cdots(1+ni)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to _____

Let $A=\{z\in\mathbb{C}:|z-2|\leq4\}$ and $B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}$. Then the $\max\{|z_1-z_2|:z_1\in A\text{ and }z_2\in B\}$ is:

Let $S=\left\{z=x+iy:\ \dfrac{2z-3i}{4z+2i}\text{ is a real number}\right\}$. Then which of the following is NOT correct?

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