Q53 / 160 · TS EAMCET 2013 · Mathematics
If the roots of $x^3-42 x^2+336 x-512=0$, are in increasing geometric progression, then its common ratio is
- A$2: 1$
- B$3: 1$
- Correct answerC$4: 1$
- D$6: 1$
Solution
Given, cubic equation is
$$
\begin{aligned}
& x^3-42 x^2+336 x-512=0 \\
& \Rightarrow x^2(x-2)-40 x(x-2)+256(x-2)=0 \\
& \Rightarrow \quad(x-2)\left(x^2-40 x+256\right)=0 \\
&
\end{aligned}
$$
$$
\begin{array}{rlrl}
\Rightarrow & (x-2)\left\{x^2-32 x-8 x+256\right\} & =0 \\
\Rightarrow & & (x-2)\{x(x-32)-8(x-32)\} & =0 \\
\Rightarrow & & (x-2)(x-32)(x-8) & =0 \\
\Rightarrow & & (x-2)(x-8)(x-32) & =0 \\
\Rightarrow & & x=2,8,32
\end{array}
$$
Which represents a geometric progression in increasing order.
Common ratio $=\frac{8}{2}=4: 1$
Hence, the correct answer is option: 'C'.