List of top Mathematics Questions on Complex Numbers and Quadratic Equations asked in TS EAMCET

If 2 and 3 are the two roots of the equation \[ 2x^3 + mx^2 - 13x + n = 0, \] then the values of \(m, n\) are respectively
If \( -1 + i \) is a root of the equation \( x^4 + 4x^3 + 5x^2 + 2x - 2 = 0 \), then the real roots of this equation are
One of the roots of the equation \((x + 1)^4 + 81 = 0\) is
If \(z = 1 - \sqrt{3}\,i\), then \(z^3 - 3z^2 + 3z = \;?\)

The product of all the values of \(\bigl(\sqrt{3} - i\bigr)^{25}\) is ?

The number of common roots among the 12th and 30th roots of unity is ?

∫ √(2x2 - 5x + 2) dx = ∫ (41/60) dx,

and

-1/2 > α > 0, then α = ?

If \( x \) and \( y \) are two positive real numbers such that \( x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \), then \( 13y - 26x = \):

If \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane, then the locus of \( z \) satisfying the equation \( |z-1| + |z+i| = 2 \) is:
One of the values of \( (-64i)^{5/6} \) is:}
If \( \alpha, \beta \) are the roots of the equation \( x + \frac{4}{x} = 2\sqrt{3} \), then \( \frac{2}{\sqrt{3}}\left| \alpha^{2024} - \beta^{2024} \right| \) is:
If \( z = x + iy \) satisfies the equation \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}, \] then:
If \( \omega \) is the complex cube root of unity and \[ \left( \frac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2} \right)^k + \left( \frac{a + b\omega + c\omega^2}{b + a\omega^2 + c\omega} \right)^2 = 2, \] then \( 2k + 1 \) is always:
If \( Z_1 = \sqrt{3} + i\sqrt{3} \) and \( Z_2 = \sqrt{3} + i \), and \[ \left( \frac{Z_1}{Z_2} \right)^{50} = x + iy, \] then the point \( (x,y) \) lies in:
If \( Z_1, Z_2, Z_3 \) are three complex numbers with unit modulus such that \[ |Z_1 - Z_2|^2 + |Z_1 - Z_3|^2 = 4 \] then \[ Z_1 Z_2 + \overline{Z_1} Z_2 + Z_1 Z_3 + \overline{Z_1} Z_3 = \]