If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+r=0$ and $\alpha^3+\beta^3+\gamma^3=5$,then $r=$

For the equation $ax^2 + bx + c = 0$,if $\alpha$ and $\beta$ are its roots,then the equation whose roots are $\alpha + \frac{1}{\beta}$ and $\beta + \frac{1}{\alpha}$ is:

Let $P_n = \alpha^n + \beta^n, n \in N$. If $P_{10} = 123, P_9 = 76, P_8 = 47$ and $P_1 = 1$,then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is:

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

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 2x + 4 = 0$,then $\frac{1}{\alpha^3} + \frac{1}{\beta^3}$ is equal to

Consider the two equations $x^2 - 2ax + b^2 = 0$ and $x^2 - 2bx + a^2 = 0$. The arithmetic mean of the roots of the first equation is equal to what?