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Mathematics

Assertion (A): If x = 1x\dfrac{1}{x}, then x = ± 1.

Reason (R): (x−1x)(x+1x)=x2−1x2\Big(x - \dfrac{1}{x}\Big)\Big(x + \dfrac{1}{x}\Big) = x^2 - \dfrac{1}{x^2}

  1. Assertion (A) is true, Reason (R) is false.

  2. Assertion (A) is false, Reason (R) is true.

  3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

  4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Expansions

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Answer

Given,

⇒ x = 1x\dfrac{1}{x}

⇒ x.x = 1

⇒ x2 = 1

⇒ x = 1\sqrt{1}

⇒ x = ± 1

∴ Assertion (A) is true.

Solving,

⇒(x−1x)(x+1x)⇒x(x+1x)−1x(x+1x)⇒x2+xx−xx−1x2⇒x2+1−1−1x2⇒x2−1x2.\Rightarrow\Big(x - \dfrac{1}{x}\Big)\Big(x + \dfrac{1}{x}\Big)\\[1em] \Rightarrow x\Big(x + \dfrac{1}{x}\Big) - \dfrac{1}{x}\Big(x + \dfrac{1}{x}\Big)\\[1em] \Rightarrow x^2 + \dfrac{x}{x} - \dfrac{x}{x} - \dfrac{1}{x^2}\\[1em] \Rightarrow x^2 + 1 - 1 - \dfrac{1}{x^2}\\[1em] \Rightarrow x^2 - \dfrac{1}{x^2}.

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.

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