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

For the equation $3x^2 + px + 3 = 0, p > 0$,if one of the roots is the square of the other,then $p$ is equal to:

If the roots of the equation $lx^2 + nx + n = 0$ are in the ratio $p:q$,then $\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{n}{l}} = $

Suppose that the equation $ax^2+bx+c=0$ has roots $\alpha$ and $\beta$,both of which are different from $\frac{1}{3}$. Then,an equation whose roots are $\frac{1}{3\alpha-1}$ and $\frac{1}{3\beta-1}$ is

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha \beta=-1$. Let $p_{n}=\alpha^{n}+\beta^{n}$,$p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n \geq 1$. Then,the value of $p_{n}^{2}$ is .... .

If $\alpha, \beta$ are the roots of the equation $x^2+3x+k=0$ and $\alpha+\frac{1}{\alpha}, \beta+\frac{1}{\beta}$ are the roots of the equation $4x^2+px+18=0$,then $k$ satisfies the equation: