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# Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​

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Senior Manager
Joined: 27 Feb 2014
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Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​  [#permalink]

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Updated on: 15 Feb 2020, 00:54
1
00:00

Difficulty:

85% (hard)

Question Stats:

37% (02:07) correct 63% (01:53) wrong based on 35 sessions

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Is –1 < a < 0 ?

(1) a^3 > a​

(2) a^2 > a^3​

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Originally posted by kapil1 on 14 Feb 2020, 23:15.
Last edited by Bunuel on 15 Feb 2020, 00:54, edited 1 time in total.
Renamed the topic and edited the question.
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Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​  [#permalink]

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16 Feb 2020, 16:55
1)+2):

a^2 > a^3​ > a
For any positive number .. this can’t be true ...a^2 has to be the middle number (except when a=1,all equal)

So a must be negative

If a<-1 ..then a2>a>a^3 ...wrong

When -1<a<0... a^2 > a^3​ > a ....... perfect

Hence C

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Senior Manager
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Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​  [#permalink]

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18 Feb 2020, 23:39
1
1) $$a^3$$ > a​

Step 1:
Finding critical points $$a^3$$ – a >0 -> a ($$a^2$$-1) >0 -> a (a+1)(a-1)>0​
Critical points are a = -1, 0 , 1​

Step 2: Validating the regions​ on the number line
i) -∞ to –1: when a=-2, inequlaity is invalid​
ii) -1 to 0; when a=-1/2, inequlaity is valid​
iii) 0 to –1; when a=½, it is invalid​
iv) 1 to ∞, when a =2, it is valid​

Step 3: Solution set is –1<a<0 and a>1
No concrete answer to the question

2) $$a^2$$ > $$a^3$$​

Step 1:
Finding critical points
$$a^3 – a^2$$ < 0​
$$a^2$$ (a-1) <0​
Critical points are a = 0, 1​

Step 2:
Validating the regions​
-∞ to 0, when a=-1, valid ​
i) 0 to 1, when a = ½ , valid​
ii)1 to ∞, when a =2, invalid​

Step 3: Solution set -∞ <a<1
No concrete answer to the question​

Combining statements (1) and (2)​
Common solution set is –1<a<0
C is correct​
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Re: Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​  [#permalink]

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19 Feb 2020, 07:57
kapil1 wrote:
Is –1 < a < 0 ?

The answer should be D. because using the two statements we are able to give a "Yes" or "No".

Based on statement 1, since the range asked in the question is a subset of valid range, answer is Yes.
Similarly in statement 2, the range asked is in the question(-1 to 0) is a subset of the valid range(-∞ to 1), answer is YES.
Thus, answer should be D

(1) a^3 > a​

(2) a^2 > a^3​
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Re: Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​  [#permalink]

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19 Feb 2020, 19:21
shanks2020 wrote:
kapil1 wrote:
The answer should be D. because using the two statements we are able to give a "Yes" or "No".

Based on statement 1, since the range asked in the question is a subset of valid range, answer is Yes.
Similarly in statement 2, the range asked is in the question(-1 to 0) is a subset of the valid range(-∞ to 1), answer is YES.
Thus, answer should be D

shanks2020 Are you getting a definite answer (YES/NO) to the question stem from statement (1) and statement (2) alone?
Lets see:

(1) When a = -1/2, answer to the question stem is yes
when a = 2, answer to the question stem is no

Same reasoning is valid for statement (2) alone.
_________________
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Re: Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​   [#permalink] 19 Feb 2020, 19:21
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# Is –1 < a < 0 ? (1) a^3 > a​ (2) a^2 > a^3​

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