List of top Mathematics Questions on Complex numbers asked in AP EAPCET

If \(z_1=8+4i,\; z_2=6+4i\) and \[ \operatorname{Arg}\left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi}{4}, \] then \(z\) satisfies

Given that \[ \sum_{k=1}^{n} k(k-1)=\frac{n(n-1)(n+1)}{3} \] and $\omega$ and $\omega^2$ are complex cube roots of unity. If \[ \sum_{k=1}^{2026} \left( k+\frac{1}{\omega} \right) \left( k+\frac{1}{\omega^2} \right) = \frac{2026}{3}(N+3), \] then $N=$

Among the roots of \[ 3\sqrt{3}\,z^3-i=0, \] the sum of the squares of the two roots having non-zero real part is

If \[ \cos\alpha+\cos\beta+\cos\gamma=0 \] and \[ \sin\alpha+\sin\beta+\sin\gamma=0, \] then which of the following is true?

The modulus of the product of all the values of \( (2+3i)^{3/5} \) is:

If \( \text{Arg}(\frac{z-1}{z+5}) = \pi/3 \) and \( |z-1|=|z+5| \), then \( |z|^2 = \)

Among the solutions of the equation \( x^6 = 64 \), the sum of all those solutions whose real part is negative is:

If \( z_{1}=x+iy \), \( z_{2}=a+ib \) and \( x^{2}+y^{2}=a^{2}+b^{2} \), then \( z_{2} = \)

The locus of \( |z-2i|+|z+4i|=10 \) is

If \( z=(1+\sqrt{3}i)^{4/3} \), then the product of all the values of \( z \) is

If \[ \frac{3}{2+\cos\theta+i\sin\theta}=x+iy, \] then \[ (x-1)(x-3)= \]

If $z_1,z_2$ are two roots of the equation \[ z^2+az+b=0 \] and on the Argand plane the points represented by $z_1,z_2$ and the origin form an equilateral triangle, then $a^2=$

If \[ (1+\sin\alpha+i\cos\alpha)^8 = A(\cos\theta+i\sin\theta), \] then $A$ and $\theta$ are respectively

If $\omega$ is a complex cube root of unity, then $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 =$

If $z = \frac{\sqrt{3} + i}{2}$, then $z^{101} + z^{103} =$

If the conjugate of a complex number $z$ is $\frac{1}{z - i}$, then $z$ can be:

If \(\Arg z_1\) and \(\Arg z_2\) are \(\frac{\pi}{3}\) and \(\frac{\pi}{5}\) respectively, then the value of \(\Arg z_1 + \Arg z_2\) is:

Evaluate \[ \left(\frac{1-i}{1+i}\right)^{2022} + \left(\frac{1+i}{1-i}\right)^{2021} \]

If \(1, \omega, \omega^2\) are the cube roots of unity, then the value of \[ (x+y)^2 + (x\omega + y\omega^2)^2 + (x\omega^2 + y\omega)^2 \] is:

Area of the triangle formed by the complex numbers \(z\), \(iz\) and \(z+iz\) in the Argand diagram as vertices is:

Let \(z\) and \(w\) be two distinct non-zero complex numbers. If \[ |z|^2w-|w|^2z=z-w, \] then

If \(1,\omega,\omega^2\) denote the cube roots of unity, then the value of \[ (1-\omega+\omega^2)^5+(1+\omega-\omega^2)^5 \] is

The values of \(\theta\), for which \[ \frac{3+2i\sin\theta}{1-2i\sin\theta} \] is real are

If \[ \frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y}{3-i}=i, \] then \(x+y=\)

If the complex numbers \(z_1\), \(z_2\), \(0\) are vertices of an equilateral triangle, then \[ z_1^2+z_2^2= \] is equal to: