A) \[a:b\]
B) \[{{a}^{2}}:{{b}^{2}}\]
C) \[{{a}^{3}}:{{b}^{3}}\]
D) \[{{a}^{3}}:{{c}^{3}}\]
Correct Answer: C
Solution :
By question, b is the mean-proportional of a and c. So, \[\frac{a}{b}=\frac{b}{c}=k\] (let say); \[\therefore \] \[a=bk\] and \[b=ck\] \[\therefore \] \[\frac{{{(a-b)}^{3}}}{{{(b-c)}^{3}}}=\frac{{{(bk-b)}^{3}}}{{{(ck-c)}^{3}}}\] \[=\frac{{{b}^{3}}{{(k-1)}^{3}}}{{{c}^{3}}{{(k-1)}^{3}}}=\frac{{{b}^{3}}}{{{c}^{3}}}=\frac{{{a}^{3}}}{{{b}^{3}}}\]
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