The condition to be satisfied in order that one root of $x^3+b x^2+c x+d=0$ is the sum of the other two roots,is

Which equation has roots that are the squares of the roots of the equation $ax^2 + bx + c = 0$?

If the harmonic mean of the roots of the equation $\sqrt{2} x^2 - bx + (8 - 2\sqrt{5}) = 0$ is $4$,then the value of $b$ is

If $x_1$ and $x_2$ are the real roots of the equation $x^2-kx+c=0$,then the distance between the points $A(x_1, 0)$ and $B(x_2, 0)$ is

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

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-4x^2-9x+36=0$ such that $\alpha+\beta=0$,then $\alpha^2+2\beta^2+3\gamma^2=$