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Mathematics

If x = 2−1, then (x−1x)2\sqrt{2} - 1, \text{ then } \Big(x - \dfrac{1}{x}\Big)^2 is :

  1. 222\sqrt{2}

  2. 2

  3. 4

  4. 2−22 - \sqrt{2}

Rational Irrational Nos

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Answer

Given,

x = 2−1\sqrt{2} - 1

1x=12−1\dfrac{1}{x} = \dfrac{1}{\sqrt{2} - 1}

Rationalizing,

⇒1x=12−1×2+12+1=2+1(2)2−(1)2=2+12−1=2+11=2+1.⇒(x−1x)2=[2−1−(2+1)]2=[2−2−1−1]2=[−2]2=4.\Rightarrow \dfrac{1}{x} = \dfrac{1}{\sqrt{2} - 1} \times \dfrac{\sqrt{2} + 1}{\sqrt{2} + 1} \\[1em] = \dfrac{\sqrt{2} + 1}{(\sqrt{2})^2 - (1)^2} \\[1em] = \dfrac{\sqrt{2} + 1}{2 - 1} \\[1em] = \dfrac{\sqrt{2} + 1}{1} \\[1em] = \sqrt{2} + 1. \\[1em] \Rightarrow \Big(x - \dfrac{1}{x}\Big)^2 = [\sqrt{2} - 1 - (\sqrt{2} + 1)]^2 \\[1em] = [\sqrt{2} - \sqrt{2} - 1 - 1]^2 \\[1em] = [-2]^2 \\[1em] = 4.

Hence, Option 3 is the correct option.

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