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# Is a positive? (1) a^6 < a (2) a^5 < a

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Joined: 02 Sep 2009
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Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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13 Aug 2018, 03:33
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55% (hard)

Question Stats:

49% (01:08) correct 51% (01:14) wrong based on 63 sessions

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Is a positive?

(1) a^6 < a

(2) a^5 < a

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Math Expert
Joined: 02 Aug 2009
Posts: 6797
Re: Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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13 Aug 2018, 03:42
Is a positive?

(1) a^6 < a
$$a-a^6>0.......a(1-a^5)>0$$
So two cases
a) a>0, then 1-a^5>0 or 1>a^5 so 0<a<1
b) a<0, then 1-a^5<0 or a^5>1 or a>1........NOT possible
So a>0 as it cannot be 0 or <0
Sufficient

(2) a^5 < a
$$a-a^5>0.......a(1-a^4)>0$$
So two cases
a) a>0, then 1-a^4>0 or 1>a^4 so 0<a<1
b) a<0, then 1-a^4<0 or a^4>1......$$a^2>1$$.....thus either a>1 or a<-1.......so a<-1
So a>0 or a<0

A
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Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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13 Aug 2018, 05:27
Bunuel wrote:
Is a positive?

(1) a^6 < a

(2) a^5 < a

St1:- $$a^6 < a$$
Or, $$a^6 -a < 0$$
Or, $$a(a^5-1)<0$$
Or, a>0
Sufficient.

St2:- $$a^5 < a$$
Or,$$a^5-a<0$$
Or, $$a\left(a+1\right)\left(a-1\right)\left(a^2+1\right)<0$$
Or, a<-1 or, 0<a<1
Insufficient.
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Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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02 Sep 2018, 11:21
Hello,

I am confused. According to me it should be E. Lets forget the algebra for a while and lets substitute a = 0.5 or a = -0.5

Statement 1

a^6 < a. For either of the values of a (0.5 or -0.5), the answer will be 0.015625 which is lesser than a.

Statement 2

a^5 < a. For 0.5, the answer will be 0.03125 and for -0.5 the answer is -0.03125. The answer is still lesser than a.

If 1 + 2, then also the answer is either +ve or -ve. I am confused as to how the answer is A? Please clarify
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Posts: 6797
Re: Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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02 Sep 2018, 11:26
1
anant327 wrote:
Hello,

I am confused. According to me it should be E. Lets forget the algebra for a while and lets substitute a = 0.5 or a = -0.5

Statement 1

a^6 < a. For either of the values of a (0.5 or -0.5), the answer will be 0.015625 which is lesser than a.

Statement 2

a^5 < a. For 0.5, the answer will be 0.03125 and for -0.5 the answer is -0.03125. The answer is still lesser than a.

If 1 + 2, then also the answer is either +ve or -ve. I am confused as to how the answer is A? Please clarify

In statement I, when a=-0.5
a^6=0.15... But a is -0.5 so a^6 >a
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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02 Sep 2018, 11:29
Ohhhhhh silly mistake. Thank you so much!!!!
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Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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02 Sep 2018, 11:35
1
anant327
Option A alone is sufficient
As per Statement 1
a( a^5 -1 ) <0
Now when you plot on number line marking + and - as alternative signs
The range would come between 0 -1

As per Statement 2
a( a^4-1)<0
a( a^2+1) ( a^2-1) <0
a( a^2+1)(a+1)(a-1)<0
Now when you plot on number line marking + and - as alternative signs
The range would come between 0 -1 and less than -1
So a can be in 0-1 range and can be less than -1 also
Hence B alone is insufficient

Kindly give kudos if my explanation is helpful
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Re: Is a positive? (1) a^6 < a (2) a^5 < a  [#permalink]

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02 Sep 2018, 12:59
Bunuel wrote:
Is A positive?

(1) A^6 < A

(2) A^5 < A

$$A\,\,\mathop > \limits^? \,\,0$$

$$\left( 1 \right)\,\,\underline {{A^6} < A}$$

$$\left. \begin{gathered} A = 0\,\,\,\, \Rightarrow \,\,\,0 = \underline {{A^6} < A} = 0\,\,\,\, \Rightarrow \,\,\,{\text{impossible}}\,\,\,\, \hfill \\ A < 0\,\,\,\, \Rightarrow \,\,\,0 < \underline {{A^6} < A} < 0\,\,\,\, \Rightarrow \,\,\,{\text{impossible}} \hfill \\ \end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle$$

$$\left( 2 \right)\,{A^5} < A\,\,\,\left\{ \begin{gathered} \,Take\,\,A = \frac{1}{2}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\, \hfill \\ Take\,\,A = - 2\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \end{gathered} \right.\,\,$$

The above follows the notations and rationale taught in the GMATH method.
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Re: Is a positive? (1) a^6 < a (2) a^5 < a &nbs [#permalink] 02 Sep 2018, 12:59
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# Is a positive? (1) a^6 < a (2) a^5 < a

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