Statement 1) \(-x^3 = -x^3\). INSUFFICIENT
Statement 2) \(x^2 = -x^2\)

\(2x^2\) = 0 => x = 0. SUFFICIENT

OPTION: B

Statement 1:

\((-x)^3 = -x^3\)

\(-x^3 = -x^3\)

\(x^3 = x^3\)

x could be 10, o , 100.

NOT sufficient.

Statement 2;

\((-x)^2 = -x^2\)

\(x^2 = -x^2\)

It only possible when x = 0.

Sufficient.

The best answer is B.

selim Could you please elaborate your solution?

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

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I think the trap in this question lies in tricking you to think that statement 2 will also give a similar solution as statement 1 did. This tends to happen in a lot of DS questions with seemingly similar statements.

Lets dive in!

Statement 1: \((-x)^3 = -x^3\)

An odd power does not change the sign of a number.

So, whether x is positive or negative or 0, \((-x)^3\) will always be equal to \(-x^3\). To illustrate this point, lets take examples:

Case 1: x is 1

\((-1)^3 = -1^3\) which is essentially -1

Case 2: x is 0

\((-0)^3 = -0^3\) which is essentially 0

Case 3: x is -1

\((--1)^3 = --1^3\) which is essentially 1

Since we get no unique value for x, this statement is insufficient

Statement 2: Even power of a number always changes the sign of a number to non-negative.

Thus, \((-x)^2\) will be equal to \(x^2\)
Thus, expression translates to: \(x^2 = -x^2\)
Which implies: \(2x^2 = 0\)
Which implies x=0.

Hence, statement 2 is sufficient and answer is B