Let $\alpha, \beta$ be the roots of the equation $x^2-ax-b=0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5 \sqrt{7} i, P_4=-3 \sqrt{7} i, P_5=11 \sqrt{7} i$ and $P_6=45 \sqrt{7} i$,then $|\alpha^4+\beta^4|$ is equal to . . . . . .

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 $(a+b)(b-a)=$

The equation whose roots are the reciprocals of the roots of the equation $3x^2 - 20x + 17 = 0$ is

Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up with roots $(4, 3)$. Rahul made a mistake in writing down the coefficient of $x$ and ended up with roots $(3, 2)$. The correct roots of the equation are:

Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation $x^2-x-1=0$,where $\alpha \neq \beta$. For $n=0, 1, 2, \ldots$,let $a_n = p \alpha^n + q \beta^n$.
$FACT$: If $a$ and $b$ are rational numbers and $a + b \sqrt{5} = 0$,then $a = 0 = b$.
$(1)$ $a_{12} =$
$[A] a_{11}-a_{10}$ $[B] a_{11}+a_{10}$ $[C] 2a_{11}+a_{10}$ $[D] a_{11}+2a_{10}$
$(2)$ If $a_4 = 28$,then $p+2q =$
$[A] 21$ $[B] 14$ $[C] 7$ $[D] 12$

The harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is