Question

Two roots of $9x^3-30x^2+31x-10=0$ are in the ratio $2:5$, then the roots are

• A. $1,-\frac{2}{3},-\frac{5}{3}$
• B. $-1,\frac{2}{3},\frac{5}{3}$
• C. $1,\frac{2}{3},\frac{5}{3}$
• D. $1,2,5$

Answer 1

Answer: (C)

$1,\frac{2}{3},\frac{5}{3}$

Let $\alpha ,\beta ,\gamma$ be the roots of $9x^3-30x^2+31x-5=0$
$s_1=\alpha +\beta +\gamma =\frac{30}{9}=\frac{10}{3}$
Now observing options
A) $\alpha +\beta +\gamma =1-\frac{2}{3}-\frac{5}{3}=-\frac{4}{3}$
B) $\alpha +\beta +\gamma =-1+\frac{2}{3}+\frac{5}{3}=\frac{4}{3}$
C) $\alpha +\beta +\gamma =1+\frac{2}{3}+\frac{5}{3}=\frac{10}{3}$
D) $\alpha +\beta +\gamma =1+2+5=8$
Hence, (C) is correct answer as $\frac{2}{3}:\frac{5}{3}\Rightarrow 2:5$