If the difference between the roots of the equation $x^2 + ax + b = 0$ is equal to the difference between the roots of the equation $x^2 + bx + a = 0$ $(a \neq b)$,then:

If $\alpha, \beta$ are the roots of the equation $x^2 - px + r = 0$ and $\alpha/2, 2\beta$ are the roots of the equation $x^2 - qx + r = 0$,then what is the value of $r$?

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 - x - 1 = 0$,then what is the value of $\left( \frac{1 + \alpha}{1 - \alpha} \right) \left( \frac{1 + \beta}{1 - \beta} \right) \left( \frac{1 + \gamma}{1 - \gamma} \right)$?

If $p$ and $q$ are the roots of the equation $x^2 + pq = (p + 1)x$,then $q=$

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$,with $\alpha > \beta$. If $a_n = \alpha^n - \beta^n$ for $n \geq 1$,then the value of $\frac{a_{10}-2a_8}{2a_9}$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 5x^2 - 7x - 1 = 0$,then find the equation whose roots are $\alpha\beta, \beta\gamma, \gamma\alpha$.