A) \[\frac{{{3}^{10}}{{.2}^{4}}}{{{7}^{7}}}\]
B) \[\frac{{{3}^{9}}{{.2}^{4}}}{{{7}^{7}}}\]
C) \[\frac{{{3}^{8}}{{.2}^{4}}}{{{7}^{7}}}\]
D) None of these
Correct Answer: A
Solution :
[a] Taking A.M. and G.M. of 7 numbers \[\frac{a}{2},\,\,\frac{a}{2},\,\,\frac{b}{3},\,\,\frac{b}{3},\,\,\frac{b}{3},\,\,\frac{c}{2},\,\,\frac{c}{2},\] we get \[\frac{2.\frac{a}{2}+3.\frac{b}{3}+2.\frac{c}{2}}{7}\ge {{\left\{ {{\left( \frac{a}{2} \right)}^{2}}{{\left( \frac{b}{3} \right)}^{3}}{{\left( \frac{c}{2} \right)}^{2}} \right\}}^{\frac{1}{7}}}\] \[\Rightarrow \,\,\,\frac{{{3}^{7}}}{{{7}^{7}}}\ge \frac{{{a}^{2}}{{b}^{3}}{{c}^{2}}}{{{2}^{2}}{{.3}^{3}}{{.2}^{2}}}\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\,\,\,\,{{a}^{2}}{{b}^{3}}{{c}^{2}}\le \frac{{{3}^{10}}{{.2}^{4}}}{{{7}^{7}}}\] \[\therefore \] greatest value of \[{{a}^{2}}{{b}^{3}}{{c}^{2}}=\frac{{{3}^{10}}{{.2}^{4}}}{{{7}^{7}}}\]
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