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

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If a^5 ≤ a, which of the following must be true?  [#permalink]

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Updated on: 15 Sep 2013, 12:08
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Question Stats:

35% (01:04) correct 65% (01:05) wrong based on 435 sessions

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

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

Originally posted by nroy347 on 15 Sep 2013, 12:04.
Last edited by Bunuel on 15 Sep 2013, 12:08, edited 1 time in total.
Edited the question.
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Posts: 48044
Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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15 Sep 2013, 12:12
2
1
nroy347 wrote:
If a^5 ≤ a, which of the following must be true?

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

$$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

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

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 07:27
can u plz elaborate y D is nt d correct answer
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Posts: 48044
Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 07:45
fnumiamisburg wrote:
can u plz elaborate y D is nt d correct answer

The question asks: which of the following MUST be true.

Now, 0 ≤ a ≤ 1 is NOT always true, because, a can be less than or equal to -1, say -2 ((-2)^5<-2), and in this case this option is not true.

Does this make sense?
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 07:48
fnumiamisburg wrote:
can u plz elaborate y D is nt d correct answer

-1<=X=<1 so none are true
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 07:51
Valerun wrote:
fnumiamisburg wrote:
can u plz elaborate y D is nt d correct answer

-1<=X=<1 so none are true

The range is not correct.

$$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

Check here: if-a-5-a-which-of-the-following-must-be-true-159797.html#p1267231
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 08:01
thanks Brunel... I got it now..
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 09:52
Just put (-2) as a possible "a" and do the work, it took me 30 sec to get to answer "A"
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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18 Sep 2013, 10:00
1
1
Predecessor wrote:
Just put (-2) as a possible "a" and do the work, it took me 30 sec to get to answer "A"

That's a valid approach.

"MUST BE TRUE" questions:
These questions ask which of the following MUST be true, or which of the following is ALWAYS true for ALL valid sets of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular valid set of numbers, it will mean that this statement is not always true and hence not a correct answer.

So, for "MUST BE TRUE" questions plug-in method is good to discard an option but not 100% sure thing to prove that an option is ALWAYS true.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

So, for "COULD BE TRUE" questions plug-in method is fine to prove that an option could be true. But here, if for some set of numbers you'll see that an option is not true, it won't mean that there does not exist some other set which will make this option true.
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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20 Nov 2013, 07:21
Bunuel wrote:
nroy347 wrote:
If a^5 ≤ a, which of the following must be true?

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

$$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

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

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

Hi Bunuel, I don't get it. How come do you get two ranges.

It isn't just a^5<=a
Then solve this inequality with key points after factorizing that gives 0<=a<=1?

Where did you get the other range from?

Cheers
J
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Joined: 02 Sep 2009
Posts: 48044
Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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20 Nov 2013, 07:32
jlgdr wrote:
Bunuel wrote:
nroy347 wrote:
If a^5 ≤ a, which of the following must be true?

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

$$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

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

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

Hi Bunuel, I don't get it. How come do you get two ranges.

It isn't just a^5<=a
Then solve this inequality with key points after factorizing that gives 0<=a<=1?

Where did you get the other range from?

Cheers
J

$$a^5\leq{a}$$ --> $$a(a-1)(a+1) (a^2+1)\leq{0}$$ --> reduce by a^2+1 since it's always positive: $$a(a-1)(a+1)\leq{0}$$. Now, solve it with key points approach.
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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Updated on: 07 Feb 2014, 05:44
Bunuel wrote:
fnumiamisburg wrote:
can u plz elaborate y D is nt d correct answer

The question asks: which of the following MUST be true.

Now, 0 ≤ a ≤ 1 is NOT always true, because, a can be less than or equal to -1, say -2 ((-2)^5<-2), and in this case this option is not true.

Does this make sense?

Now I get it we need to factorize everything and get two ranges
Therefore, none of the statement will HAVE to be true

Thanks all
J

Originally posted by jlgdr on 01 Jan 2014, 07:43.
Last edited by jlgdr on 07 Feb 2014, 05:44, edited 1 time in total.
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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01 Jan 2014, 10:28
jlgdr wrote:
Bunuel wrote:
fnumiamisburg wrote:
can u plz elaborate y D is nt d correct answer

The question asks: which of the following MUST be true.

Now, 0 ≤ a ≤ 1 is NOT always true, because, a can be less than or equal to -1, say -2 ((-2)^5<-2), and in this case this option is not true.

Does this make sense?

I don't quite get it Bunuel. I mean it looks fine that when you pick -2 it satisfies the inequality on question stem and is not good for any of the statements
BUT, when one tries to find the range for a^5<=a, one ends up with 0<=a<=1, which is the exact same as option III

I'm pretty sure I'm getting the reasoning incorrectly

Cheers!
J

The trick here is more on the answer choices.

Look at I II and III. all are right (pick out some numbers).

Therefore I was totally confused because non of the answers had the right answer (I II and III)

I finally chose D, but hesitate between A and D quite a long time...

This is a tricky tricky question, since it takes into consideration that the answers are correct but they do not cover all the POSSIBLE answers.

Therefore it is A.

Confusing...
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If a^5 ≤ a, which of the following must be true?  [#permalink]

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06 Jun 2014, 08:26
If a^5 ≤ a, which of the following must be true?

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
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Posts: 48044
Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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06 Jun 2014, 08:29
goodyear2013 wrote:
If a^5 ≤ a, which of the following must be true?

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

Merging similar topics. Please refer to the discussion above.
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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07 Jun 2014, 04:48
Hi,

What is the difference between these 2 equations?
I am trying to apply this inequality trick. http://gmatclub.com/forum/inequalities-trick-91482.html

http://gmatclub.com/forum/which-of-the-following-represents-the-complete-range-of-x-108884.html
$$x^3-4x^5<0$$ --> $$x^3(1-4x^2)<0$$ --> $$(1+2x)(x^3)(1-2x)<0$$ --> roots are -1/2, 0, and 1/2 --> $$-\frac{1}{2}<x<0$$ or $$x>\frac{1}{2}$$.

$$a^5\leq{a} --> a(a-1)(a+1) (a^2+1)\leq{0}$$ --> reduce by $$(a^2+1)$$ since it's always positive: $$a(a-1)(a+1)\leq{0}$$ --> $$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.
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Posts: 48044
Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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08 Jun 2014, 04:55
1
goodyear2013 wrote:
Hi,

What is the difference between these 2 equations?
I am trying to apply this inequality trick. http://gmatclub.com/forum/inequalities-trick-91482.html

http://gmatclub.com/forum/which-of-the-following-represents-the-complete-range-of-x-108884.html
$$x^3-4x^5<0$$ --> $$x^3(1-4x^2)<0$$ --> $$(1+2x)(x^3)(1-2x)<0$$ --> roots are -1/2, 0, and 1/2 --> $$-\frac{1}{2}<x<0$$ or $$x>\frac{1}{2}$$.

$$a^5\leq{a} --> a(a-1)(a+1) (a^2+1)\leq{0}$$ --> reduce by $$(a^2+1)$$ since it's always positive: $$a(a-1)(a+1)\leq{0}$$ --> $$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

The first one is solved with slightly different approach, check here: which-of-the-following-represents-the-complete-range-of-x-108884.html#p868863

If you want to solve it with the approach given here: which-of-the-following-represents-the-complete-range-of-x-108884.html, then ensure that the factors are of the form (x - a)(x - b)(x - c)... So, it would be (x+1/2)(x^3)(x-1/2)>0.
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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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15 Jul 2014, 09:45
Bunuel wrote:
nroy347 wrote:
If a^5 ≤ a, which of the following must be true?

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

$$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

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

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

Hi

How can 2 comes in the range?? 0<=a<=1..
I dont understand.. Can you pls explain....
I checked all threads in the discussion still did not get it.

Thanks
Math Expert
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Posts: 48044
Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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15 Jul 2014, 10:12
GGMAT760 wrote:
Bunuel wrote:
nroy347 wrote:
If a^5 ≤ a, which of the following must be true?

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

$$a^5\leq{a}$$ --> $$a\leq{-1}$$ or $$0\leq{a}\leq{1}$$.

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

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

Hi

How can 2 comes in the range?? 0<=a<=1..
I dont understand.. Can you pls explain....
I checked all threads in the discussion still did not get it.

Thanks

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Re: If a^5 ≤ a, which of the following must be true?  [#permalink]

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21 Feb 2017, 22:09
Hi,
I have understood the whole solution except for the part where a^5 reduces to a(a-1)(a+1)(a^2 -1).can anyone guide me if I am doing it correctly?.
a^5 <a
a^5 -a <0
a (a^4 -1)<0
a(a^2-1)(a^2+1)<0
a(a+1)(a-1)(a^2+1)<0
a<0 a<1 a<-1
Re: If a^5 ≤ a, which of the following must be true? &nbs [#permalink] 21 Feb 2017, 22:09

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