\[(a)\;\frac{2}{3} \quad (b)\;\frac{3}{4} \quad (c)\;\frac{1}{3} \quad (d)\;\frac{1}{2} \]

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Let r be the direction between the parts

$F= \large\frac{Gm(M-m)}{r^2}$

$\qquad= \large\frac{G}{r^2} $$(mM-m^2)$

The force will be maximum

$\large\frac{dF}{dm}$$=0$

ie $\large\frac{d}{dm} \bigg[\large\frac{G}{r^2} $$(mM-m^2)\bigg]=0$

$\large\frac{G}{r^2}$$(M-2m)=0$

$M-2m=0$

$M=2m$

$\large\frac{M}{m}$$=2$

or $\large\frac{m}{M}=\large\frac{1}{2}$

Hence d is the correct answer.

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