If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x + 4 = 0$,then the equation whose roots are $\alpha^2$ and $\beta^2$ is

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0, (abc \neq 0)$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3 - ax^2 + bx - c = 0$,then $\alpha^{-2} + \beta^{-2} + \gamma^{-2} = $

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

If one of the roots of the equation $x^2 + ax + 3 = 0$ is $3$,and one of the roots of the equation $x^2 + ax + b = 0$ is three times the other root,then the value of $b$ is equal to

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 $p+2q =$
$[A] 21$ $[B] 14$ $[C] 7$ $[D] 12$