If $\alpha, \beta, \gamma, \delta$ are the roots of $x^4 - 100x^3 + 2x^2 + 4x + 10 = 0$,then $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}$ is equal to:

The difference between two roots of the equation $x^3-13x^2+15x+189=0$ is $2$. Then the roots of the equation are:

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

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 $x^3-2x^2+3x-4=0$,then find $\sum \alpha \beta(\alpha+\beta)$.

If the roots of the equation $ax^2 + bx + c = 0$ are $\alpha$ and $\beta$,then the roots of the equation $cx^2 + bx + a = 0$ are