If a^4b^3c^7 > 0, then which of the following must be true?

Official Solution:






If \(a^4b^3c^7 > 0\), then which of the following must be true?

I. \(\frac{b}{c} > 0\)

II. \(ab > 0\)

III. \(abc > 0\)


A. I only
B. II only
C. III only
D. I and II only
E. I and III only


From \(a^4b^3c^7 > 0\) we can deduce the following:

\(•\) None of the unknowns is 0.

\(•\) \(b\) and \(c\) have the same sign (\(bc > 0\)).

I. \(\frac{b}{c} > 0\). Since \(b\) and \(c\) have the same sign, then this must be true.

II. \(ab > 0\). We don't know anything about relationship between signs of \(a\) and \(b\). So, this option is not necessarily true.

III. \(abc > 0\). We know that \(bc > 0\) but the only thing we know about \(a\) that it's non-zero: it can be positive as well as negative. So, this option is not necessarily true.


Answer: A
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\(a^4\) will always be positive.
The question boils down to \(b^3c^7 > 0\), means both b and c are both positive or both negative.

Statement 1:

As b and c should both be +ve or -ve at the same time.
b/c > 0.
Always true.

Statement 2:

ab > 0
Is a is +ve and b is +ve, then ab>0
Or If both a and b are -ve, then ab>0
If either one is -ve, then ab<0
May or may not be true.

Statement 3:

Same again. If bc is +ve and a is +ve, then abc > 0.
If bc is +ve and a is -ve, then abc < 0.
Not always true.

A is the answer.

Given

\(a^4b^3c^7 > 0\)

\(a^4\) is always positive but we don't know whether a is positive or negative.

\(b^3\)could be positive or negative.

\(c^7\) could be positive or negative.

\(a^4b^3c^7>0.\)

From this information , it's clear that either b and c will both be negative or both be positive.

I. \(\frac{b}{c} > 0\). Always true. b and c will have the same sign. Result must be greater than 0.

II. \(ab > 0\). could be true. we don't know the sign of a.

III. \(abc > 0\). bc will always be positive but we don't know about a. could be . MUST not be.

A is the correct answer.

\(a^4b^3c^7 > 0\)

for it to hold true
a can be either +/- and b & c have to be of same sign

so
out of given options
only \(\frac{b}{c} > 0\)
would be valid
IMO A

IMO A

\(a^4b^3c^7 > 0\), has to be true, then

a can be +ive or -ive
b has to be -ive, since it is having an odd power
c has to be -ive, since it is having an odd power

\(\frac{b}{c} > 0\), This will always hold good

II. \(ab > 0\), can be Yes when a -ive and b -ive and a No when a +ive and b -ive

III. \(abc > 0\), can be Yes when a +ive and b -ive and a No when a -ive and b -ive
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bc > 0
b/c > 0

IMO A

I. \(\frac{b}{c} > 0\)
b<0, c<0 , yes definitely
II. \(ab > 0\)
b can be <0 then the eqn is not satisfied and a can be >0 and <0
therefore cannot be true always

III. \(abc > 0\)
b,c<0 and a>0 satisfied however if a<0 it cannot be satisfied

Therefore IMO A

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