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 ___________.

The value of ${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$ is :

If [x] be the greatest integer less than or equal to x, then $\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]}$ is…

Let z1 and z2 be two complex numbers such that ${\overline z _1} = i{\overline z _2}$ and $\arg \left( {{{{z_1}} \over {{{\overline z…

The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :

Let f(x) be a polynomial of degree 3 such that $f(k) = - {2 \over k}$ for k = 2, 3, 4, 5. Then the value of 52 $-$ 10f(10) is equal to :