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If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x

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If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x  [#permalink]

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New post 25 Feb 2020, 13:04
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A
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D
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If \(x ≠ 0\), is \(x = 1\)?

(1) \( x^2 =\frac{1}{x^2}\)
(2) \(x^2 =\frac{1}{x}\)
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Re: If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x  [#permalink]

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New post 25 Feb 2020, 20:14
1) x=1,-1 not sufficient
2)x^3=1. ...x must be 1. Sufficient

B

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Re: If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x  [#permalink]

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New post 25 Feb 2020, 23:30
Asad wrote:
If \(x ≠ 0\), is \(x = 1\)?

(1) \( x^2 =\frac{1}{x^2}\)
(2) \(x^2 =\frac{1}{x}\)


Solution


Step 1: Analyse Question Stem


    • \(x≠ 0\)
We need to find if x is 1

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE


Statement 1: \(x^2 = \frac{1}{x^2}\)
    • \(x^4 = 1\)
      o Since, power of x is even so x can be 1 or -1
Hence, statement 1 is not sufficient and we can eliminate answer options A and D.

Statement 2: \(x^2 = \frac{1}{x}\)
    • \(x^3 = 1\)
      o As the power of x is odd, so only possible value of \(x = 1\)
Hence, statement 2 is sufficient.
Thus, the correct answer is Option B.
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Re: If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x  [#permalink]

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New post 25 Feb 2020, 23:56
Before attempting question first we have to see is there anything mention which states that x has to be only positive or negative integers or fraction.

For option 1- x can be equals to 1,-1. So option D and A is not valid.

Now,

For option 2 - x must be equal to 1 else equation will not be valid. So option B is the correct choice

Correct option- B

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Re: If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x  [#permalink]

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New post 29 Feb 2020, 19:12
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.
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Re: If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x  [#permalink]

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New post 29 Feb 2020, 20:57
Statement 1

x^2 - 1/x^2 = 0
=> x^4-1/x^2 =0
=> (x^2+1)(x^2-1) / x^2 =0
Hence x can be 1 or -1. Not sufficient

Statement 2
×^2-1/x =0
X^3-1/x =0
(X-1)(x^2+x+1)/x =0
This will be only possible when x=1
Hence sufficient.
B should be the correct answer
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Re: If x ≠ 0, is x = 1? (1) x^2 = 1/x^2 (2) x^2 = 1/x   [#permalink] 29 Feb 2020, 20:57
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