Question

If the roots of the equation $72 x^{3}-108 x^{2}+46 x-$ $5=0$ are in arithmetic progression then the difference between largest and smallest root lies in the interval

• A. $\left(0, \frac{1}{2}\right)$
• B. $\left(\frac{1}{2}, \frac{1}{\sqrt{2}}\right)$
• C. $\left(\frac{1}{\sqrt{2}}, 1\right)$
• D. $(1,2)$

Answer 1

Answer: (B)

$72 x^{3}-108 x^{2}+46 x-5=0$
Let roots are $a-d, a, a+d$
$\therefore \quad 3 a =\frac{108}{72}=\frac{3}{2} $
$\Rightarrow a =\frac{1}{2}$
Now $a\left(a^{2}-d^{2}\right)=\frac{5}{72}$
$\frac{1}{4}- d ^{2}=\frac{5}{36} $
$\Rightarrow d ^{2}=\frac{1}{9}$
$\Rightarrow d=\frac{1}{3} $
$\Rightarrow 2 d=\frac{2}{3}$
Hence, difference between largest and smallest root $2 d =\frac{2}{3}$
which lies in the interval $\left(\frac{1}{2}, \frac{1}{\sqrt{2}}\right)$