If $\alpha, \beta$ are the roots of the equation $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$,then

If $3$ distinct real numbers $a, b, c$ satisfy $a^2(a + p) = b^2(b + p) = c^2(c + p)$ where $p \in \mathbb{R}$,then the value of $bc + ca + ab$ is:

If one of the roots of the equation $x^2+px+q=0$ is equal to the square of the other,then:

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

If $-1$ is a twice repeated root of the equation $ax^3+bx^2+cx+1=0$,then

If $\alpha, \beta$ are the roots of $x^2+ax+2=0$ and $\frac{1}{\alpha}, \frac{1}{\beta}$ are the roots of $x^2-bx+c=0$,then $\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right) = $