Let $ \alpha = \dfrac{-1 + i\sqrt{3}}{2} $ and $ \beta = \dfrac{-1 - i\sqrt{3}}{2} $, where $ i = \sqrt{-1} $. If
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of $ m $ is ___________.

Let $z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$ and $|z|^2 = 44200$. Find the value of $n$.

If $ \cos(\theta+\phi)=\frac{3}{5} $ and $ \sin(\theta-\phi)=\frac{5}{13} $, $ 0<\theta,\phi<\frac{\pi}{4} $, then $ \cot(2\theta) $ equals:

If $ |z_1|=|z_2|=|z_3|=1 $ and $ z_1+z_2+z_3=0 $, then the area of the triangle whose vertices are $ z_1,z_2,z_3 $ is:

Let $ \omega (\ne 1) $ be a cube root of unity. Then the minimum value of the set \[ \left\{ |a + b\omega + c\omega^2|^2 : a,b,c \text{ are distinct non-zero integers} \right\} \] equals:

If $ z_1, z_2 $ are complex numbers such that $ \dfrac{2z_1}{3z_2} $ is a purely imaginary number, then the value of \[ \left|\frac{z_1 - z_2}{z_1 + z_2}\right| \] is:

Let $ z = \frac{2 - i}{\alpha + i} $, where $ \alpha $ is a real number. If $ 4\text{Re}(z) = 3\text{Im}(\bar{z}) $, then the value of $ \alpha $ is

If $ z + \bar{z} = 6 $ and $ z - \bar{z} = 4i $, then $ |z|^2 = $

The modulus of the complex number $ (2\sqrt{2} + i2\sqrt{2})^2 $ is equal to

Let $ z = 1 - i $. Then the value of $ z^4 $ is equal to

If $ z = 3 + 4i $, then the modulus of $ z $ is:

Find the roots of the quadratic equation $ 2x^2 - 4x - 6 = 0 $.

Let $ z = x + iy $, where $ y>0 $. If $ z + \overline{z} = 6 $ and $ |z| + | \overline{z} | = 10 $, then $ z = $