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

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Director
Joined: 26 Oct 2016
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Location: United States
Concentration: Marketing, International Business
Schools: HBS '19
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WE: Education (Education)
Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3  [#permalink]

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23 Feb 2017, 18:04
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65% (hard)

Question Stats:

66% (02:23) correct 34% (02:08) wrong based on 89 sessions

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Is a < 0 ?

(1) a^3 < a^2 + 2a
(2) a^2 > a^3

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Thanks & Regards,
Anaira Mitch

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Joined: 11 Sep 2015
Posts: 3350
Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3  [#permalink]

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Updated on: 23 Feb 2017, 22:15
2
Top Contributor
anairamitch1804 wrote:
Is a < 0 ?
(1) a³ < a² + 2a
(2) a² > a³

Target question: Is a < 0 ?

Statement 1: a³ < a² + 2a
Subtract a² and 2a from both sides to get: a³ - a² - 2a < 0
Factor: a(a² - a - 2) < 0
Factor more: a(a - 2)(a + 1) < 0
There are several values of a that satisfy this inequality. Here are two:
Case a: a = 0.5. In this case, a > 0
Case b: a = -10. In this case, a < 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a² > a³
There are several values of a that satisfy this inequality. Here are two:
Case a: a = 0.5. In this case, a > 0
Case b: a = -10. In this case, a < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of a that satisfy BOTH statements Here are two:
Case a: a = 0.5. In this case, a > 0
Case b: a = -10. In this case, a < 0
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Originally posted by GMATPrepNow on 23 Feb 2017, 18:40.
Last edited by GMATPrepNow on 23 Feb 2017, 22:15, edited 1 time in total.
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Posts: 7212
Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3  [#permalink]

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23 Feb 2017, 18:48
1
Is a < 0 ?
(1) a^3 < a^2 + 2a
$$a^3-a^2-2a<0......a( a^2-a-2)<0......a(a-2)(a+1)<0$$
So a <-1 will give ans as YES..
a=0 or 1 will also be true and ans will be NO
Insufficient

(2) a^2 > a^3
a^2-a^3>0......$$a^2(1-a)>0$$..
So 1-a>0..a<1...
So a can be -1 or 0 or 0.5
Insuff..

Combined
Again a as 0, 0.5 or -3 etc still remains..
Insufficient

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

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23 Feb 2017, 19:57
Is a < 0?

Statement 1: a^3 < a^2 +2a --> a(a+1)(a-2) < 0 Therefore, if this equals zero, then a can be -1, 0, or 2.
Try a = 1, the expression is negative. We can fill in the number line with signs because signs will switch back and forth.
Therefore <----(-1)++++(0)-----(2)+++++>. Statement 1 is correct when a is less than -1 or between 0 and 2. Insufficient.

Statement 2: a^2 > a^3 --> a(a)(a-1) > 0 Therefore, if this equals zero, then a can be 0 or 1.
Try a = 2, the expression is positive. We can fill in the number line with signs because signs will switch back and forth.
Therefore <++++(0)------(1)+++++>. Statement 2 is correct when a is less than 0 or greater than 1. Insufficient.

Combined:
From statement 1, a can be less than -1 or between 0 and 2.
From statement 2, a can be less than 0 or greater than 1.
Therefore, a must be less than -1 or between 1 and 2.
Is a < 0? Yes and no. Insufficient.
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Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3  [#permalink]

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25 Mar 2018, 07:49

Solution

We need to find if $$a$$ is less than $$0$$ or not.

Statement-1$$a^3 < a^2 + 2a$$”.

• $$a^3 -a^2 -2a<0$$
• $$a * (a^2 -a -2) < 0$$
• $$a (a+1) (a-2) < 0$$

Thus, the values of a for which inequality, $$a^3 < a^2 + 2a$$, satisfies belongs to $$(-∞, -1)∪ (0,2)$$.

In the above range, the value of $$a$$ can be both positive and negative.
Hence, Statement 1 alone is not sufficient to answer the question.

Statement-2$$a^2 > a^3$$”.

•$$a^2 - a^3> 0$$
•$$a^3- a^2 < 0$$
•$$a^2 (a-1) < 0$$

Since $$a^2$$ is always positive, only $$(a-1)$$will give the negative sign.
• $$(a-1) <0$$

Thus, $$a$$ is less than $$1$$.
•When $$a$$ is less than $$1$$, $$a$$ can be both, positive and negative.

Thus, Statement 2 alone is not sufficient to answer the question.

Combining both the statements:

By combining both the statements, the range of $$a$$ will be $$(-∞, -1)∪ (0,1)$$.

Since the value of $$a$$ can be either positive or negative, Statement (1) and (2) TOGETHER are NOT sufficient.

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Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 &nbs [#permalink] 25 Mar 2018, 07:49
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# Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3

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