The square root of the expression 1 a b c (a^2+b^2+c^2 )+2 (1 a +1 b +1 c ) is

The square root of the expression 1 a b c (a^2+b^2+c^2 )+2 (1 a +…

MCQ

The square root of the expression $\frac{1}{a b c}\left(a^2+b^2+c^2\right)+2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$ is

A
$\frac{a+b+c}{a b c}$
B
$\sqrt{a}+\sqrt{b}+\sqrt{c}$
C
$\sqrt{\frac{b c}{a}}+\sqrt{\frac{c a}{b}}+\sqrt{\frac{a b}{c}}$
✓
$\sqrt{\frac{a}{b c}}+\sqrt{\frac{b}{c a}}+\sqrt{\frac{c}{a b}}$

✓

Answer

Correct option: D.

$\sqrt{\frac{a}{b c}}+\sqrt{\frac{b}{c a}}+\sqrt{\frac{c}{a b}}$

(d)
We have
$\frac{1}{a b c}\left(a^2+b^2+c^2\right)+2\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1}{a b c}\left(a^2+b^2+c^2\right)+2\left(\frac{a b+b c+c a}{a b c}\right)$
$=\frac{1}{a b c}\left(a^2+b^2+c^2+2 a b+2 b c+2 c a\right)=\frac{1}{a b c}(a+b+c)^2=\left(\frac{a+b+c}{\sqrt{a b c}}\right)^2=\left(\sqrt{\frac{a}{b c}}+\sqrt{\frac{b}{c a}}+\sqrt{\frac{c}{a b}}\right)^2$
So, the square root of the given expression is $\sqrt{\frac{a}{b c}}+\sqrt{\frac{b}{c a}}+\sqrt{\frac{c}{a b}}$.