Question

If \({{\rm{q}}^{ - {\rm{a}}}} = \frac{1}{{\rm{r}}}\) and \({{\rm{r}}^{ - {\rm{b}}}} = \frac{1}{{\rm{s}}}\) and \({{\rm{S}}^{ - {\rm{c}}}} = \frac{1}{{\rm{q}}}\), the value of abc is _______
1. (rqs)^-1
2. 0
3. 1
4. r⁺q⁺s⁺

Answer 1

Correct Answer - Option 3 : 1

\({{\rm{q}}^{ - {\rm{a}}}} = {\rm{\;}}\frac{1}{{\rm{r}}}\) ;   \({{\rm{r}}^{ - {\rm{b}}}} = {\rm{\;}}\frac{1}{{\rm{s}}}\)  and \({{\rm{s}}^{ - {\rm{c}}}} = {\rm{\;}}\frac{1}{{\rm{q}}}\)

∴ \({{\rm{q}}^{\rm{a}}} = {\rm{\;r}}\)  ; \({{\rm{r}}^{\rm{b}}} = {\rm{\;s}}\) and \({{\rm{s}}^{\rm{c}}} = {\rm{\;q}}\)

∴ \({\rm{a}}\log {\rm{q}} = \log {\rm{r\;}}\)-------(1)

And   \({\rm{b}}\log {\rm{r}} = \log {\rm{s\;}}\)------(2)

And \({\rm{c}}\log {\rm{s}} = \log {\rm{q\;}}\)-----(3)

Multiplying equation  (1),(2) and (3)

\(\begin{array}{l} {\rm{abc\;}}(\log {\rm{q}}){\rm{\;}}(\log {\rm{r}})(\log {\rm{s}}) = (\log {\rm{r}})(\log {\rm{s}})(\log {\rm{q}})\\ \therefore {\rm{\;abc\;}} = {\rm{\;}}1 \end{array}\)