An "Is" question will fetch an "Yes" or a "No" as an answer.
The data provided in the statements will be considered sufficient if the question is answered with a conclusive Yes or a conclusive No.
If x^{3} > x^{2}, the answer to the question is a conclusive Yes.
If x^{3} ≤ x^{2}, the answer to the question is a conclusive No.
Note: When x^{3} = x^{2}, the answer is No.
It pays rich dividend to note down the answer to questions 2 and 3 mentioned above in your scratch paper while solving DS questions.
Now let us evaluate each statement independently.
x^{3} is greater than x^{2} for certain values of x and will not be greater for other values of x.
There are 4 intervals to keep in mind while evaluating the relation between two different exponents of x.
Interval 1: -∞ < x < -1 x^{3} is negative in this interval and x^{2} is positive in this interval. So, x^{3} < x^{2}.
Interval 2: -1 < x < 0 x^{3} is negative in this interval and x^{2} is positive in this interval. So, x^{3} < x^{2}.
Interval 3: 0 < x < 1 x^{3} is positive and so is x^{2}.
Let us substitute a value and check. When x = ½, x^{3} = ⅛ and x^{2} = ¼
⅛ < ¼. So, x^{3} < x^{2} in this interval as well.
Interval 4: 1 < x < ∞ Both x^{3} and x^{2} are positive in this interval.
Let us plug in a value and check. When x = 2, x^{3} = 8 and x^{2} = 4. So, x^{3} > x^{2}.
In addition to these 4 intervals, we also need to check the relation between x^{3} and x^{2} when x = -1, x = 0 and x = 1.
We need to check if we get a conclusive Yes or No using this statement to determine whether statement 1 is sufficient
When x > 0, we need to evaluate the following conditions:
The values that x takes based on statement 1 do not give a conclusive answer to the question.
Hence, statement 1 is NOT Sufficient.
Therefore, we can eliminate choices A and D. We can narrow down the choices to this DS question to B, C, or E.
We need to check if we get a conclusive Yes or No using this statement to determine whether statement 2 alone is sufficient
When x < 0, we need to evaluate the following conditions:
The values that x takes based on statement 2 give a conclusive 'NO' as an answer to the question.
Please bear in mind that what is important is whether we get a uniform YES or a uniform NO as an answer. A NO does not mean that the data is not sufficient. Getting a conclusive NO means we have a definitive answer to the question.
Hence, statement 2 alone is Sufficient.
Statement 2 alone is sufficient to answer the question; statement 1 is not sufficient to answer the question.
Choice B is the correct answer.