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 $\alpha+\beta=-2$ and $\alpha^3+\beta^3=-56$,then the quadratic equation whose roots are $\alpha$ and $\beta$ is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$,then the coefficient of $x$ in the cubic equation whose roots are $\alpha(\beta+\gamma), \beta(\gamma+\alpha)$ and $\gamma(\alpha+\beta)$ is

Suppose the quadratic polynomial $p(x)=ax^2+bx+c$ has positive coefficients $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$,then what could be the possible value of $\alpha+\beta+\alpha\beta$ if $0 \leq \alpha+\beta+\alpha\beta \leq 8$?

Let $\alpha$ and $\beta$ be the roots of the equation $5x^{2} + 6x - 2 = 0$. If $S_{n} = \alpha^{n} + \beta^{n}$ for $n = 1, 2, 3, \dots$,then:

If $\frac{x-4}{x^2-5x-2k} = \frac{2}{x-2} - \frac{1}{x+k}$,then $k$ is equal to