Question

If $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-8x+8=0$, then $\sum {\alpha }^2$ and $\sum \frac{1}{\alpha \beta }$ are respectively

• A. $0$ and $-16$
• B. $16$ and $18$
• C. $-16$ and $0$
• D. $16$ and $0$

Answer 1

The correct option is D

$16$ and $0$

Explanation for the correct option:

Find the value of $\sum {\alpha }^2$ and $\sum \frac{1}{\alpha \beta }$:

Given, $\alpha ,\beta$ and $\gamma$ are the roots of the equation $x^3-8x+8=0$

$\Rightarrow$$\sum \alpha =0,\sum ɑ\beta =–8,ɑ\beta ɣ=–8$

As we know,

$\begin{array}{rcl}\sum {ɑ}^2& =& {(\alpha +\beta +\gamma )}^2–2\sum ɑ\beta \\ & =& 0–2(-8)\\ & =& 16\end{array}$

and

$\begin{array}{rcl}\sum \frac{1}{ɑ\beta }& =& \sum \frac{ɑ}{ɑ\beta ɣ}\\ & =& 0\end{array}$ $\left[\because \sum \alpha =0\right]$

$\therefore$The required values are $\mathbf{16}$ and $\mathbf{0}$

Hence, Option ‘D’ is Correct.