Roots of Polynomials

If $a,b,c$ are the roots of the equation $x^3-3px^2+3qx-1=0$, then the centroid of the triangle with vertices $\left(a,\frac1a\right),\left(b,\frac1b\right),\left(c,\frac1c\right)$ is the point

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

If $a,b$ are the roots of $x^2+px+1=0$ and $c,d$ are the roots of $x^2+qx+1=0$, the value of $E=(a-c)(b-c)(a+d)(b+d)$ is

If $2x^4 + x^3 - 11x^2 + x + 2 = 0$ then the values of $x + \dfrac{1}{x}$ are:

Find $k$ in the equation $x^3 - 6x^2 + kx + 64 = 0$ if roots are in geometric progression.

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

If $\alpha$ and $\beta$ are the roots of the equation $2x^{2}+ 2px + p^{2} = 0$, where $p$ is a non-zero real number, and $\alpha^{4}$ and $\beta^{4}$ are the roots of $x^{2} - rx + s = 0$, then the roots of $2x^{2} - 4p^{2}x + 4p^{4} - 2r = 0$ are:

The value of k for which the equation $(k-2)x^{2}+8x+k+4=0$ has both real, distinct and negative roots is

The equation (x-a)³+(x-b)³+(x-c)³ = 0 has

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