If $\alpha, \beta, \gamma$ are the roots of $x^3+2x^2-3x-1=0$,then $\alpha^{-2}+\beta^{-2}+\gamma^{-2}=$

Suppose $\alpha, \beta, \gamma$ are roots of $x^3+x^2+2x+3=0$. If $f(x)=0$ is a cubic polynomial equation whose roots are $\alpha+\beta, \beta+\gamma, \gamma+\alpha$,then $f(x)$ is equal to

Let $\alpha, \beta$ with $\alpha > \beta$ be the roots of the equation $x^2 - \sqrt{2}x - \sqrt{3} = 0$. Let $P_n = \alpha^n - \beta^n$ for $n \in \mathbb{N}$. Then $(11\sqrt{3} - 10\sqrt{2})P_{10} + (11\sqrt{2} + 10)P_{11} - 11P_{12}$ is equal to:

What is the sum of the squares of the roots of the equation $x^2 - 3x + 1 = 0$?

If the sum of the roots of a quadratic equation is $1$ and the sum of the squares of the roots is $13$,then find the equation.

Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$,where $p \neq 0$. If $p, q,$ and $r$ are the consecutive terms of a non-constant $G$.$P$. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$,then the value of $(\alpha-\beta)^2$ is: