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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify the conditions, if necessary.
Condition 1)
x^2 > x|x|
=> |x|^2 > x|x|
=> |x|^2 - x|x| > 0
=> |x|(|x| - x) > 0
=> (|x| - x) > 0
=> |x| > x
=> x < 0
Thus, condition 1) is sufficient since it yields the unique answer, 'no'.
Condition 2)
x|x| > x
Assume x ≥ 0. x|x| > x => x^2 > x => x^2 - x > 0 => x(x-1) > 0 => x < 0 or x > 1.
We must have x > 1 by the assumption.
Assume x < 0. x|x| > x ⇔ -x2 > x ⇔ x2 + x < 0 ⇔ x(x+1) < 0 ⇔ -1 < x < 0.
In this case, -1 < 0 < 1.
So, condition 2) tells us that -1 < x < 0 or x > 1.
In inequality questions, the law "Question is King" tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
The solution set of the question "x > 1" doesn't include the solution set of the condition 2) "-1 < x < 0 or x > 1". Condition 2) is not sufficient.
Therefore, A is the answer.
Answer: A
