Let $\alpha, \beta$ be the roots of the equation $x^2-x+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then $\alpha^6+\alpha^4+\beta^4-5 \alpha^2$ is equal to

If $\cos (u + iv) = \alpha + i\beta ,$ then ${\alpha ^2} + {\beta ^2} + 1$ equals

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i ) \sin \theta+(4+7 i ) \cos \theta$ be $\alpha+ i \beta$,where $i =\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then the value of $p+q$ is equal to

If $z \in \mathbb{C}$,then the minimum value of $|z| + |2z - 3| + |z - 1|$ is

If $\frac{1-10 i \cos \theta}{1-10 \sqrt{3} i \sin \theta}$ is purely real,then one of the values of $\theta$ is

If a complex number $z$ satisfies the equation $z + \sqrt{2} |z + 1| + i = 0$,then $|z|$ is equal to