\[(a)\;T^2 \; \alpha \; r^{\large\frac{+7}{2}} \quad (b)\;T^2 \; \alpha \; r^{\large\frac{-7}{2}} \quad (c)\; T^2 \; \alpha \; r^{\large\frac{3}{2}} \quad (d)\;T^2 \; \alpha \; r^{\large\frac{-3}{2}}\]

As the gravitation provides the centripetal force

$\large\frac{mv^2}{r}=\frac{k}{r^{\frac{5}{2}}}$

$v^2=\large\frac{k}{mr^{\Large\frac{3}{2}}}$

$T=\large\frac{2 \pi r}{v}$$=2 \pi r \sqrt {\large\frac{mr^{\Large\frac{3}{2}}}{k}}$

$T^2=\large\frac{4 \pi^2 m}{k} $$r ^{\large\frac{7}{2}}$

$T^2 \; \alpha \; r^{\large\frac{+7}{2}}$

Hence a is the correct answer.

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