Question

If $\alpha ,\beta ,\gamma$ are roots of equation $x^3-x-1=0$, then the equation whose roots are $\frac{1}{\beta +\gamma },\frac{1}{\gamma +\alpha },\frac{1}{\alpha +\beta }$ is -

• A. $x^3+x-1=0$
• B. $x^3-x^2+1=0$
• C. $x^3+x^2-1=0$
• D. $x^3-x+1=0$

Answer 1

The correct option is B $x^3-x^2+1=0$

$\because \alpha ,\beta ,\gamma \text{are roots of the given equation}{x}^3-x-1=0\text{and coefficient of}{x}^{2}is0.$
$\therefore \alpha +\beta +\gamma =0$
So roots of required equation are $-\frac{1}{\alpha },-\frac{1}{\beta },-\frac{1}{\gamma }$
So replace $x$ by $-\frac{1}{x}$ in the given equation to get the required equation
$\frac{-1}{x^3}-\left(\frac{-1}{x}\right)-1=0$
$\Rightarrow x^3-x^2+1=0$