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25 Mar 2018, 08:49
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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.
Answer: E
26 Aug 2020, 20:40
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Genoa2000
Hi Genoa2000,
\(a^3 - a^2 - 2a <0\)
\(a^3 - a^2 <0\)
At this 'step' in the process, you know that each of those two TOTALS is less than 0, but that does NOT mean that all of those terms are INDIVIDUALLY less than 0.
For example, if A = 1/2, we have...
(1/8) - (1/4) - 2(1/2) =
(1/8) - (2/8) - (8/8) =
- 9/8... which is less than 0
and
(1/8) - (1/2) = -3/8... which is less than 0
But our value of A was POSITIVE (in this case, positive 1/2).
GMAT assassins aren't born, they're made,
Rich
03 Oct 2024, 23:48
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A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.
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