Let $f(n)=A(-2)^n+B(-3)^n$ for all $A, B \in \mathbb{R}$ and $n \in \mathbb{N}-\{1, 2\}$. If $f(n)+a f(n-1)+b f(n-2)=0$,then $(a+b)(b-a)=$

If $\alpha, \beta$ are the roots of the equation $x^2-2 \sqrt{3} x+4=0$,then $\alpha^6+\beta^6=$

If the roots of the equation $x^2 + x + 1 = 0$ are $\alpha$ and $\beta$,and the roots of the equation $x^2 + px + q = 0$ are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$,then $p$ is equal to:

The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ assumes the least value is

Let $p(x)$ be a quadratic polynomial with constant term $1$. Suppose $p(x)$,when divided by $x-1$,leaves remainder $2$ and when divided by $x+1$,leaves remainder $4$. Then,the sum of the roots of $p(x)=0$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - a(x + 1) - b = 0$,then $(\alpha + 1)(\beta + 1) = $