Question

If the roots of the equation $72x^3-108x^2+46x-$ $5=0$ 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)

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