Statement-1 (A): (a^ -1 +b^ -1 ) (a^ -1 -b^ -1 ) (1 a^ -1 -1 b^ -… - Vidyadip

MCQ

Statement-1 $(A): \left\{\left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right)\right\} \div\left\{\left(\frac{1}{a^{-1}}-\frac{1}{b^{-1}}\right)\left(\frac{1}{a^{-1}}+\frac{1}{b^{-1}}\right)\right\}=1$.
Statement-2 ( $R$ ): For any $a \neq 0, a^{-m}=\frac{1}{a^m}$ and $a^m=\frac{1}{a^{-m}}$.

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-2
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-2
C
Statement-1 is True, Statement-2 is False
✓
Statement-1 is False, Statement-2 is True

✓

Answer

Correct option: D.

Statement-1 is False, Statement-2 is True

(d)
We observe that statement-2 is true.
Now,
\begin{array}{l} \left\{\left(a^{-1}+b^{-1}\right)\left(a^{-1}-b^{-1}\right)\right\} \div\left\{\left(\frac{1}{a^{-1}}-\frac{1}{b^{-1}}\right)\left(\frac{1}{a^{-1}}+\frac{1}{b^{-1}}\right)\right\} \\ =\left\{\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{a}-\frac{1}{b}\right)\right\} \div\{(a-b)(a+b)\} \\ =\left\{\left(\frac{a+b}{a b}\right)\left(\frac{b-a}{a b}\right)\right\} \div\{(a-b)(a+b)\}=-\frac{(a-b)(a+b)}{a^2 b^2} \div(a-b)(a+b)=-\frac{1}{a^2 b^2} \end{array}
So, statement-1 is not true. hence, option (d) is correct.

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