Asad wrote:
If \(x ≠ 0\), is \(x = 1\)?
(1) \( x^2 =\frac{1}{x^2}\)
(2) \(x^2 =\frac{1}{x}\)
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.
Visit
https://www.mathrevolution.com/gmat/lesson for details.
Since we have 1 variable (\(x\)) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.
Condition 1)
\(x^2 = \frac{1}{x^2}\)
⇔ \(x^4 = 1\) when both sides are multiplied by \(x^2\)
⇔ \(x^4 - 1 = 0\)
⇔ \(( x^2 + 1 )(x^2 - 1) = 0\)
⇔ \(( x^2 + 1 )(x + 1)(x - 1) = 0\)
⇔ \((x + 1)(x - 1) = 0\) since \(x^2 + 1 > 0\)
⇔ \(x = -1\) or \(x = 1\)
Since condition 1) does not yield a unique solution, it is not sufficient.
Condition 2)
\(x^2 = \frac{1}{x}\)
⇔ \(x^3 = 1\) when both sides are multiplied by \(x\)
⇔ \(x^3 - 1 = 0\)
⇔ \((x-1)(x^2+x+1) = 0\)
⇔ \(x - 1 = 0\) since \(x^2+x+1>0\)
⇔ \(x = 1\)
Since condition 2) yields a unique solution, it is sufficient.
Therefore, B is the answer.
If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spareThe one-and-only
World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course""Free Resources-30 day online access & Diagnostic Test""Unlimited Access to over 120 free video lessons - try it yourself"