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

1) x=1,-1 not sufficient
2)x^3=1. ...x must be 1. Sufficient

B

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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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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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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.

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