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

One of the values of \[ (\sqrt{3}-i)^{5/3} \] is:

If the curve represented by the locus of a point in the Argand plane corresponding to the complex number \(z\) satisfying the relation \[ \operatorname{Re}\!\left(\frac{z+4}{2z-5}\right) + \operatorname{Re}\!\left(\frac{\bar z+4}{2\bar z-5}\right) =4 \] cuts the \(X\)-axis at two points \(A\) and \(B\), then the sum of the abscissae of those two points is

If a complex number \(\alpha\) is a common root of \[ x^{2026}+x^{1964}+1=0 \] and \[ x^3+2x^2+2x+1=0, \] then the sum of the complex roots of the equation \[ z^3=\alpha^3 \] is

One of the \(5^{\text{th}}\) roots of \(\omega\) is

If \(z\) is a complex number such that \[ \left|9z+\frac{1}{z}\right|=8, \] then the maximum value of \(|z|\) is

If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - 2x + 2 = 0 \), then \( \alpha^{2026} + \beta^{2026} = \)

If \( z = x+iy \) and the point \( P \) denotes \( z \) in the Argand plane, then the locus of \( P \) satisfying the condition \( \text{Im}\left(\frac{z-3i}{z+2}\right) = 1, z \neq -2 \) is:

Number of values of a complex number \( z \) satisfying the condition \( z = z^2 + i\text{Im}(\overline{z}) \) is:

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

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

$m,n$ 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 \[ A=\{z=x+iy:|z-4||z-3|\}, \] \[ B=\{z=x+iy:-3\le y\le3,\;x\in N,\;y\in N\}, \] 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 $\omega \neq 1$ is a cube root of unity, then one root among the $7^{th}$ roots of $(1+\omega)$ is

If $Z=r(\cos\theta+i\sin\theta)$, $(\theta \neq -\pi/2)$ is a solution of $x^3 = i$, then $r^9(\cos(9\theta)+i\sin(9\theta)) =$

If $|Z_1 - 3 - 4i| = 5$ and $|Z_2| = 15$ then the sum of the maximum and minimum values of $|Z_1 - Z_2|$ is

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:
\[\frac{(\sqrt{3}+i)(1-\sqrt{3}i)}{(-1+i)(-1-i)}\]