GATE Papers >> Mechanical >> 2018 >> Question No 6

Question No. 6

For integers $a,\;b$ and $c$ what would be the minimum and maximum values respectively of $a+\;b+\;c$ if $\log\left|a\right|+\log\left|b\right|+\log\left|c\right|=0?$

Answer : (A) -3 and 3

Solution of Question No 6 of GATE 2018 Mechanical Paper

$\log\left|\mathrm a\right|+\log\left|\mathrm b\right|+\log\left|\mathrm c\right|=0$

We know that

$\begin{array}{l}\log\left(\mathrm{mn}\right)=\mathrm{logm}+\mathrm{logn}\\\mathrm{So},\\\log(\mathrm a)+\log(\mathrm b)+\log(\mathrm c)=\log1\\\log\left|\mathrm a\right|.\left|\mathrm b\right|.\left|\mathrm c\right|=\log1\\\left|\mathrm a\right|.\left|\mathrm b\right|.\left|\mathrm c\right|=1\end{array}$

This is only possible when

$\begin{array}{l}a=\pm1,\;b=\pm1,\;C=\pm1\\\end{array}$

So,

Maximum value of $(\mathrm a+\mathrm b+\mathrm c),=3$

When $\mathrm a=1,\mathrm b=1,\mathrm c=1$

And minimum value of $(\mathrm a+\mathrm b+\mathrm c)=-3$

When, $\mathrm a=-1,\mathrm b=-1,\mathrm c=-1$

So, the answer is -3 & 3