Complex Numbers

Let $\alpha$ and $\beta$ be the roots of $x^2+x+1=0$. The equation whose roots are $\alpha^{19}$ and $\beta^{19}$ is

Roots of $x^{2} - 2x + 4 = 0$ are $\alpha, \beta$. Compute $ \alpha^{6} + \beta^{6} $.

If $\omega$ is a cube root of unity, then find the value of determinant $\begin{vmatrix} 1+\omega &\omega^{2} &-\omega \\ 1+\omega^{2}&\omega &-\omega^{2} \\ \omega^{2}+\omega&\omega &-\omega^{2} \end{vmatrix}$

α, β are the roots of the an equation $x^2- 2x cosθ + 1 = 0$, then the equation having roots αn and βn is

A particle P starts from the point z₀=1+2i, where i=√−1 . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z₁. From z₁ the particle moves √2 units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\dfrac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by

If ${{x}}_k=\cos \Bigg{(}\frac{2\pi k}{n}\Bigg{)}+i\sin \Bigg{(}\frac{2\pi k}{n}\Bigg{)}$ , then $\sum ^n_{k=1}({{x}}_k)=?$