List of top Mathematics Questions on Complex numbers asked in TS EAMCET

If the eight vertices of a regular octagon are given by the complex numbers $\frac{1}{x_j-2i}$ ($j=1,2,3,4,5,6,7,8$), then the radius of the circumcircle of the octagon is
Number of real values of $(-1-\sqrt{3}i)^{3/4}$ is
$(\sqrt{3}+i)^{10} + (\sqrt{3}-i)^{10} =$
If a complex number $z = x+iy$ represents a point $P(x, y)$ in the Argand plane and z satisfies the condition that the imaginary part of $\frac{z-3}{z+3i}$ is zero, then the locus of the point P is
The amplitude of the complex number is:
\[\frac{(\sqrt{3}+i)(1-\sqrt{3}i)}{(-1+i)(-1-i)}\]
If \( n, K \in \mathbb{N} \) such that \( n \neq 3K \), then \( (\sqrt{3}+i)^{2n} + (\sqrt{3}-i)^{2n} = \)
If \( |Z|=2 \), \( Z_1 = \frac{Z}{2}e^{i\alpha} \) and \( \theta \) is the amp(Z), then \( \frac{Z_1^n - Z_1^{-n}}{Z_1^n + Z_1^{-n}} = \)
\( \omega \) is a complex cube root of unity and \( Z \) is a complex number satisfying \( |Z-1| \le 2 \). The possible values of \( r \) such that \( |Z-1| \le 2 \) and \( |\omega Z - 1 - \omega^2| = r \) have no common solution are
If $\alpha$ is a root of the equation $x^2-x+1=0$ then $(\alpha + \frac{1}{\alpha}) + (\alpha^2 + \frac{1}{\alpha^2}) + (\alpha^3 + \frac{1}{\alpha^3}) + \dots$ to 12 terms =
If $\cos\alpha+\cos\beta+\cos\gamma = 0 = \sin\alpha+\sin\beta+\sin\gamma$, then $\sin 2\alpha + \sin 2\beta + \sin 2\gamma =$
The set of all values of $\theta$ such that $\frac{1-i\cos\theta}{1+2i\sin\theta}$ is purely imaginary is
One of the values of $\sqrt{24-70i} + \sqrt{-24+70i}$ is
The point P denotes the complex number \(z = x + iy\) in the Argand plane. If \( \frac{2z - i}{z - 2} \) is purely real number, then the equation of the locus of P is.
x and y are two complex numbers such that \(|x| = |y| = 1\). If \( \text{Arg}(x) = 2\alpha \), \( \text{Arg}(y) = 3\beta \) and \( \alpha + \beta = \frac{\pi}{36} \), then \( x^6 y^4 + \frac{1}{x^6 y^4} \) is:

If \[ z = \frac{(2-i)(1+i)^3}{(1-i)^2} \] then \[ \text{Arg}(z) = ? \]

If \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane. If the amplitude of \( \frac{2z-i}{z+2i} \) is \( \frac{\pi}{4} \), then the equation of the locus of \( P \) is:
If \( \alpha, \beta \) are the roots of the equation \( x^2 + 2x + 4 = 0 \). If the point representing \( \alpha \) in the Argand diagram lies in the 2nd quadrant and \( \alpha^{2024} - \beta^{2024} = ik \), where \( i = \sqrt{-1} \), then \( k = \)

If \( \sqrt{5} - i\sqrt{15} = r(\cos\theta + i\sin\theta), -\pi < \theta < \pi, \) then

\[ r^2(\sec\theta + 3\csc^2\theta) = \]

If Xn = cos \(\frac{ π}{2^n}\) + i sin\(\frac{ π}{2^n}\) , then 

\[\prod_{n=1}^{\infty} x_n =\]

 

If the roots of the equation z2 - i = 0 are α and β, then | Arg β - Arg α | =

The locus of z such that  \(\frac{|z-i|}{|z+i|}\)= 2, where z = x+iy. is