GMAT Kolaveri wrote:Is p a negative number?
(1) p^3(1 - p^2) < 0
(2) p^2 - 1 < 0
Source: grockit. OA: C
The critical points method is great.
For some test-takers, plugging in numbers might be easier and more efficient.
Typically, we need to consider the following types of numbers:
Zero
One (positive and negative)
Integers (greater than positive 1 and less than negative 1)
Fractions (positive and negative)
Statement 1: p^3(1 - p^2) < 0
p=0 and p=±1 make the left-hand side equal to 0, so none of them works here.
p=-2:
(-2)³(1-(-2)²) < 0
(neg)(neg) < 0.
Doesn't work, implying that p cannot be less than -1.
p=-1/2:
(-1/2)³(1-(-1/2)²) < 0
(neg)(pos) < 0.
This works, implying that p can be a negative fraction.
p=2:
(2)³(1-2²) < 0
(pos)(neg) < 0.
This works, implying that p can be greater than 1.
p=1/2:
(1/2)³(1-(1/2)²) < 0
(pos)(pos) < 0.
Doesn't work, implying that p cannot be a positive fraction.
Since p=-1/2 and p=2 both work, INSUFFICIENT.
Statement 2, rephrased: p²<1.
Thus, p can be a negative fraction, 0, or a positive fraction.
INSUFFICIENT.
Statements 1 and 2 combined:
The only type of value that satisfies both statements is a negative fraction such as p=-1/2.
SUFFICIENT.
The correct answer is
C.
An alternate way to combine the statements:
Since p²-1<0 (statement 2), 1-p²>0.
Thus, statement 1 becomes:
p³(positive) < 0.
p³<0, implying that p is a negative number.
SUFFICIENT.
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