Question

If $x^3-x^2+5x-1=0$ has roots $\alpha ,\beta ,\gamma$ and $x^3+a{x}^2+bx+c=0$ has roots $\alpha \beta ,\beta \gamma ,\gamma \alpha$ then $(a+b+c)$ is:

• A. -1
• B. 5
• C. 3
• D. 7

Answer 1

The correct option is B 5

Let $x^3-x^2+5x-1=0$ ..(i)
$\Rightarrow \alpha \beta \gamma =1$

$d\Rightarrow \alpha \beta =\frac{1}{\gamma };\beta \gamma =\frac{1}{\alpha },\alpha \gamma =\frac{1}{\beta }$
We have to determine the equation whose roots are
$\frac{1}{\alpha },\frac{1}{\beta },\frac{1}{\gamma }$

So, replace $x\to \frac{1}{x}$ in equation (i)
$\Rightarrow \frac{1}{x^3}-\frac{1}{x^2}+\frac{5}{x}-1=0\Rightarrow x^3+5x^2-x+1=0$
$\Rightarrow a=5,b=-1,c=1$