GATE Mechanical 2018 Set 1 | GA | Question: 6

Question

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

• A. $\text{-3 and 3}$
• B. $\text{-1 and 1}$
• C. $\text{-1 and 3}$
• D. $\text{1 and 3}$

Comments

Similar question asked in https://gateoverflow.in/313388/gate2018-ce-2-ga-5

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Answer here!   

Answer 1

Given that $a,b$ and $c$ are integers.

and $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =0$

we know that $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =\log(\mid a \mid\cdot\mid b\mid\cdot\mid c\mid)$

$\implies\log(\mid a \mid\cdot\mid b\mid\cdot\mid c\mid)= 0$

$\implies \mid a \mid\cdot\mid b\mid\cdot\mid c\mid={10}^{0}$

$\implies \mid a \mid\cdot\mid b\mid\cdot\mid c\mid=1$

For minimum value of $a+b+c$ we can put $a=b=c=-1$ and get
$(a+b+c)_{\min}=-1-1-1=-3$

For maximum value of $a+b+c$ we can put $a=b=c=1$ and get
$(a+b+c)_{\max}=1+1+1=3$

So, (A) is the correct answer.

Comments

If someone has doubt similar to this:

$log|a| + log|b| + log|c| = 0$

so, if we have something like this:

$\dfrac{1}{2} + (-\dfrac{1}{4}) + (-\dfrac{1}{4}) = 0$

but for this type of combination of values for $log|a|, \, log|b|, \, log|c|$ atleast one of $a, b, c$ needs to be $<1$. but it is given $a, b, c$ are integers so that is not possible. they can be $-ve$ but they can't be fractions.

graph of $log(x)$:

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But when we are taking a=b=c=-1 then a*b*c is becoming -1 ,can anyone please clarify?

@Lakshman Bhaiya

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Please see the question carefully, mod is given. So ABC = 1 only. @Ujjal Das