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

Question

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

M37-14

Bunuel · Accepted Answer

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

Chethan92 · Answer

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.

KSBGC · 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.

Archit3110 · 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

KanishkM · Answer

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

Kinshook · Answer

bc > 0
b/c > 0

IMO A

Crytiocanalyst · Answer

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

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