The condition that the roots of $x^3-b x^2+c x-d=0$ are in geometric progression is

$\alpha, \beta, \gamma$ are the roots of the equation $x^3-10x^2+7x+8=0$. Match the following and choose the correct answer.

Column-$I$	Column-$II$
$A$. $\alpha+\beta+\gamma$	$(1)$ $-\frac{43}{4}$
$B$. $\alpha^2+\beta^2+\gamma^2$	$(2)$ $-\frac{7}{8}$
$C$. $\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}$	$(3)$ $86$
$D$. $\frac{\alpha}{\beta\gamma}+\frac{\beta}{\gamma\alpha}+\frac{\gamma}{\alpha\beta}$	$(4)$ $0$
	$(5)$ $10$

The values of $a$ and $b$ for which the equation $x^4 - 4x^3 + ax^2 + bx + 1 = 0$ has four real roots are:

If the difference between the roots of the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ is the same,and $a \neq b$,then:

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$,with $\alpha>\beta$. For all positive integers $n$,define $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$ and $b_1=1$ and $b_n=a_{n-1}+a_{n+1}, n \geq 2$. Then which of the following options is/are correct?
$(1)$ $\sum_{i=1}^{n} a_i = a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
$(4)$ $b_n = \alpha^n+\beta^n$ for all $n \geq 1$

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 6x + a = 0$ and satisfy the relation $3\alpha + 2\beta = 16$,then the value of $a$ is