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

Question No. 6 Mechanical | GATE 2018

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

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