If a^5 ≤ a, which of the following must be true?

Question

I. –1 ≤ a ≤ 0
II. a = 0
III. 0 ≤ a ≤ 1

A. None of the above
B. I only
C. II only
D. III only
E. I and III only

astiles67 · Accepted Answer

Hi - this problem seems to have a language error, please confirm logic:

"which of of the following must be true"

e.g.
Stem: a^5 ≤ a
Condition: a = 0
Solution o^5 ≤ a
Condition is DEFINITELY TRUE. Therefore, II satisfies question completely.

I don't see how a II and III are void. They both work for all a values stated and therefore 'must' be true.

Bunuel · Answer

a^5\leq{a} ;

a\leq{-1}  or  0\leq{a}\leq{1} .

If a = -2, then none of the options MUST be true.

Answer: A.

Kinshook · Answer

Asked: If a^5 ≤ a, which of the following must be true?

a^5 - a <=0 
a(a^4-1) <=0
a(a-1)(a+1)(a^2+1) <=0
Since (a^2 + 1) > 0
Using wavy line method,
a<=-1 or 0<=a<=1

I. –1 ≤ a ≤ 0: CAN NOT BE TRUE
II. a = 0; MAY BE TRUE
III. 0 ≤ a ≤ 1; MAY BE TRUE

A. None of the above
B. I only
C. II only
D. III only
E. I and III only

IMO A

Simerpreet19 · Answer

Can someone please explain why II is wrong ?

Bunuel · Answer

The question asks which of the following MUST be true, not which COULD be true. II is wrong because a = 0 is just one possible case; the inequality holds for other values too, so it’s not always true.

Nikkz99 · Answer

Bunuel   Krunaal  how to deal with these kind of questions?
I often mess up with these kind of questions (for instance, here, I was confused like, either II or III seems feasible, if there was a option Both II and III , I would have probably chosen that as my option for this quesiton)

Krunaal · Answer

When asked to check if something  must  be true, one strategy you can follow is to  find a case which doesn't adhere to the statement that must be true but still satisfies the criteria.

For example,

a^5 ≤ a = a^5 - a ≤ 0

=>  a*(a^4 - 1) ≤ 0

From here we know that  a ≤ -1  or  0 ≤ a ≤ 1

Now we are asked which of the below statements  must  be true,

I. -1 ≤ a ≤ 0

a  can be -1, so this statement  can  be true, but  a  can also be something from 0 to 1, in which case  this statement necessarily need not be true . So it's  not necessary  that this statement  must  be true.

II. a = 0

a  can be 0, so a = 0  can  be true, but  a  can also be something less or equal to than -1, in which case  this statement necessarily need not be true . So it's  not necessary  that this statement  must  be true.

III. 0 ≤ a ≤ 1

Again, while  a  can be something from 0 to 1, but it  can also be something less than equal to -1 , so  this statement necessarily need not be true .

If you observed above, we tried to find a case that proves that the statement while  can be true ,  but must necessarily need not be true.

If there was an option both II and III then you would've asked yourself that  a  can also be less than or equal to -1.

Hope it helps.

If a^5 ≤ a, which of the following must be true?

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GMAT Problem Solving (PS) Questions

If a^5 ≤ a, which of the following must be true?

Prep Toolkit

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