If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-13x^2+kx+189=0$ such that $\beta-\gamma=2$,then $\beta+\gamma: k+\alpha=$

In a triangle $PQR$,$\angle R = \frac{\pi}{2}$. If $\tan(\frac{P}{2})$ and $\tan(\frac{Q}{2})$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then:

If $\alpha, \beta, \gamma$ are the real roots of the equation $18x^3-15x^2-4x+4=0$ such that $\alpha=\beta$ and $\alpha>\gamma$,then $\alpha+\beta^2+\gamma^3=$

The $H.M.$ between the roots of the equation $x^2 - 10x + 11 = 0$ is

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5$ are the roots of $x^5-5 x^4+9 x^3-9 x^2+5 x-1=0$,then $\frac{1}{\alpha_1^2}+\frac{1}{\alpha_2^2}+\frac{1}{\alpha_3^2}+\frac{1}{\alpha_4^2}+\frac{1}{\alpha_5^2}=$

If $\alpha$ and $\beta$ are the roots of $x^2+7x+3=0$ and $\frac{2\alpha}{3-4\alpha}, \frac{2\beta}{3-4\beta}$ are the roots of $ax^2+bx+c=0$ and $GCD(a, b, c) = 1$,then $a+b+c=$