A uniform solid sphere of mass M and radius R has its Centre at the origin O of the coordinate system. Two spherical cavities A and B, each of radius $r= R/4$ are made such that their coordinates are$(2r,0,0) and (- 2r,0,0)$ respectively as shown in fig. The gravitational field at O due to the remaining part of the sphere is 

Mass of complete sphere is $M=\frac{4\pi }{3}{R}^3\rho$. Mass of each spherical cavity is
$m=\frac{4\pi }{3}{\left(\frac{R}{4}\right)}^3\rho =\frac{M}{64}$
Gravitational at O due to cavity A is
$I_A=\frac{GM}{(2r{)}^2}$
$=\frac{GM\times 16}{64\times 4\times R^2}=\frac{GM}{16R^2}$

directed to the left.
Gravitational field at O due to cavity B is
$I_B=\frac{GM}{16R^2}$directed to the right.
If I is gravitational field at O due to the complete sphere, then the gravitational field at O due to the remaining part is
$I_r=I-I_A+I_B$
Now$I_A=I_B$ and $I=0$ (because $x=0$), $I_r=0$. So the correct choice is (a)