If the roots of the equation $x^2 + px + q = 0$ differ by $1$,then:

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x + 16 = 0$,and if $\alpha^2 + \beta^2$ and $\alpha\beta/2$ are the roots of the equation $x^2 + px + q = 0$,then:

Let $\alpha, \beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is

If the roots of the quadratic equation $x^2-35x+c=0$ are in the ratio $2:3$ and $c=6K$,then $K=$

If $\alpha, \beta, \gamma$ are the real roots of the equation $x^3 - 3px^2 + 3qx - 1 = 0$,then find the centroid of the triangle whose vertices are $(\alpha, \frac{1}{\alpha}), (\beta, \frac{1}{\beta})$ and $(\gamma, \frac{1}{\gamma})$.

If $\alpha$ and $\beta$ are the roots of $x^{2}-ax+b^{2}=0$,then $\alpha^{2}+\beta^{2}$ is equal to