Question

If $a,b,c$ be +ve and unequal, then

• A. $>1$
• B. $<1$
• C. $>6$
• D. $<6$

Answer 1

The correct option is C

$>6$

Explanation for the correct option:

Step 1. Given that, $a,b,c$ be +ve and unequal

$\because AM>GM$

$\therefore \frac{(a/b)+(b/a)}{2}>\sqrt{\left(\frac{a}{b}\right)\left(\frac{b}{a}\right)}=1$

$\Rightarrow$ $\frac{a}{b}+\frac{b}{a}>2$ …(1)

Similarly,

$\frac{b}{c}+\frac{c}{b}>2$ …(2)

Step 2. By adding equation (1) and (2), we get

$\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}>6$

$\Rightarrow$ $\left(\frac{b}{a}+\frac{c}{a}\right)+\left(\frac{c}{b}+\frac{a}{b}\right)+\left(\frac{a}{c}+\frac{b}{c}\right)>6$

$\Rightarrow$ $\left(\frac{b+c}{a}\right)\mathbf{}\mathbf{+}\mathbf{}\left(\frac{c+a}{b}\right)\mathbf{}\mathbf{+}\mathbf{}\left(\frac{a+b}{c}\right)\mathbf{}\mathbf{>}\mathbf{}\mathbf{6}$

Hence, Option ‘C’ is Correct.