If $\frac{\alpha}{\alpha+1}$ and $\frac{\beta}{\beta+1}$ are the roots of the quadratic equation $x^2+7x+3=0$,then the equation having roots $\alpha$ and $\beta$ is

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

If $\alpha$ and $\beta$ are two real numbers satisfying $\alpha^2 + \beta^2 = 5$ and $3(\alpha^5 + \beta^5) = 11(\alpha^3 + \beta^3)$,then the value of $\alpha \beta$ is:

If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0$,$\alpha > \beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $n$,then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{17} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$

If the roots of the equation $2x^2 - 3x + 5 = 0$ are the reciprocals of the roots of the equation $ax^2 + bx + 2 = 0$,then:

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