The $G.M.$ of the roots of the equation $x^2 - 18x + 9 = 0$ is

Suppose the quadratic polynomial $p(x)=ax^2+bx+c$ has positive coefficients $a, b, c$ such that $b-a=c-b$. If $p(x)=0$ has integer roots $\alpha$ and $\beta$,then what could be the possible value of $\alpha+\beta+\alpha\beta$ if $0 \leq \alpha+\beta+\alpha\beta \leq 8$?

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 quadratic equation whose sum of the roots is $11$ and sum of squares of the roots is $61$ is

If $\alpha$ and $\beta$ are the roots of $6x^2 - 6x + 1 = 0$,then the value of $\frac{1}{2}[a + b\alpha + c\alpha^2 + d\alpha^3] + \frac{1}{2}[a + b\beta + c\beta^2 + d\beta^3]$ is

The expression $x^3 - 3x^2 - 9x + c$ can be written in the form $(x - a)^2 (x - b)$ if the value of $c$ is