List of top Mathematics Questions on Complex numbers

If one of the values of \[ \sqrt{-1-\sqrt{3}i} \] is \(\alpha+i\beta\), where \(\alpha<0\) and \(\beta>0\), then \(\alpha=\)

$m,n$ and $k$ are integers and $9.5\le n\le12$. If \[ \frac{(\cos\theta+i\sin\theta)^m} {(\sin\theta+i\cos\theta)^n} = k(\sin17\theta-i\cos17\theta), \] then $n-m-k=$

If \(\omega\) is a complex cube root of unity, then \[ (1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^5)(1+\omega^7)(1+\omega^8)\cdots \] (2n factors) is equal to

If \[ A=\{z=x+iy:|z-4||z-3|\}, \] \[ B=\{z=x+iy:-3\le y\le3,\;x\in N,\;y\in N\}, \] and \(C=A\cap B\), then \(n(C)=\)

If \(z_1\) and \(z_2\) are two complex numbers such that \[ |z_1-a|=|z_2-a| \] for \(a\in\mathbb R\), and \[ Arg(z_1-a)+Arg(z_2-a)=\frac{\pi}{2}, \] then \[ \frac{z_1-a}{z_2-a}= \]

\[ \frac{\left(1+\sin\frac{4\pi}{9}-i\cos\frac{4\pi}{9}\right)^6} {\left(1+\sin\frac{4\pi}{9}+i\cos\frac{4\pi}{9}\right)^6} = \ ? \]

If $\omega$ is a complex cube root of unity, evaluate $\left(\frac{\sqrt{3}-i}{-\sqrt{2}+\sqrt{2}i}\right)^{20}$.}

Which of the following is not a possible value of \[ \sqrt{i}+\sqrt{-i}? \]

If \[ \sum_{n=1}^{10}(i^n+i^{n+1}+i^{n+2}) = x+iy, \] then $2x+3y=$

If \(x^2+x+1=0\), then the value of \(\left(x + \frac{1}{x}\right)^2 + \left(x^2 + \frac{1}{x^2}\right)^2 + \left(x^3 + \frac{1}{x^3}\right)^2 + \dots + \left(x^{25} + \frac{1}{x^{25}}\right)^2\) is:

सम्मिश्र संख्याओं के तल में निम्न समुच्चयों पर विचार कीजिए:
\[ A = \left\{ \cos \left(\frac{2n\pi}{5}\right) + i\sin \left(\frac{2n\pi}{5}\right) : n \in \mathbb{Z} \right\} \] तथा
\[ B = \left\{ \cos \left(\frac{2n}{5}\right) + i\sin \left(\frac{2n}{5}\right) : n \in \mathbb{Z} \right\} \].
निम्न कथनों में से कौन सा कथन सत्य है?

Consider the following sets of points in the complex plane \[ A = \left\{ \cos \left( \frac{2n\pi}{5} \right) + i \sin \left( \frac{2n\pi}{5} \right) : n \in \mathbb{Z} \right\} \text{ and} \] \[ B = \left\{ \cos \left( \frac{2n}{5} \right) + i \sin \left( \frac{2n}{5} \right) : n \in \mathbb{Z} \right\} . \] Which of the following statements is TRUE?

Let \( z \) be a complex number such that \( |z-6|=5 \) and \( |z+2-6i|=5 \). Then the value of \( z^3+3z^2-15z+14 \) is equal to:

Let \(\mathbb{R}\) denote the set of all real numbers and let \(i = \sqrt{-1}\). Consider the matrices \[ S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \quad \text{and} \quad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. \] Let \(a, b, c, d\) be real numbers such that \[ ST = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. \] Let \[ H = \{x + iy : x, y \in \mathbb{R} \text{ and } y>0\}. \] Then which of the following statements is (are) TRUE?

The value of $\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^{3}+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^{3}$ is equal to

The imaginary part of $\frac{\cos 50^{\circ}+i\sin 50^{\circ}}{\cos 50^{\circ}-i\sin 50^{\circ}}$ is equal to

The value of $\dfrac{(1+i)^n}{(1-i)^{n-4}}$, where $i=\sqrt{-1}$ and $n$ is an integer, is:

Let $x$ and $y$ be real numbers. If $(3+i)x + y + (1-i)y + 3i - 4 = (2x+1)i + (x-y+2)i$, where $i=\sqrt{-1}$, then the pair $(x,y)$ is equal to:

Let $z_1 = \dfrac{5+7i}{7-5i}, \, z_2 = \dfrac{3+2i}{3-2i}$ and $z_3 = \dfrac{1+11i}{11-i}$. Then $z_1\overline{z_1} + z_2\overline{z_2} + z_3\overline{z_3}$ is equal to:

The number of values of \(z \in \mathbb{C}\) satisfying \(|z-4-8i| = \sqrt{10}\) and \(|z-3-5i| + |z-5-11i| = 4\sqrt{5}\) is

Let the set of all values of $k \in \mathbb{R}$ such that the equation
$z(\bar{z} + 2 + i) + k(2 + 3i) = 0, z \in \mathbb{C}$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha + \beta)$ is equal to:

If \( |z_1| = |z_2| = |z_3| = 1 \) and \( z_1 + z_2 + z_3 = \sqrt{2} + i \), then the number \( z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1 \) is:

Let \[ S=\{z: z^2+4z+16=0,\; z\in\mathbb{C}\} \] then the value of \[ \sum_{z\in S}|z+\sqrt{3}i|^2 \] is

If \( Z_1 \) and \( Z_2 \) are roots of equation \( Z^2 + 4Z - (1 + 12i) = 0 \), where \( Z \) is complex number, then the value of \( |Z_1|^2 + |Z_2|^2 \) is:

If \( S = \{\theta : \theta \in [-\pi, \pi], \cos\theta \cos(50^\circ/2) - \cos 70^\circ \cos(70^\circ/2) = 0\ \), then \( n(S) \) is equal to: