A ring of mass M and radius R lies in x-y plane with its centre at origin as shown. The mass distribution of ring is non uniform such that, at any point P on the ring, the mass per unit length is given by $\mathrm{\lambda } = {\mathrm{\lambda }}_0{\cos }^2\mathrm{\theta }$ ($\mathrm{where} {\mathrm{\lambda }}_0 \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{constant}$). Then the moment of inertia of the ring about z-axis is:

• A

  ${\mathrm{MR}}^2$

• B

  $\frac{1}{2}{\mathrm{MR}}^2$

• C

  $\frac{1}{2}\frac{\mathrm{M}}{{\mathrm{\lambda }}_0}\mathrm{R}$

• D

  $\frac{1}{\mathrm{\pi }}\frac{\mathrm{M}}{{\mathrm{\lambda }}_0}\mathrm{R}$