Consider a sphere of mass \(M\) and radius \(R\) centered at origin. The density of material of the sphere is \(\rho=A r^\alpha\), where \(r\) is the radial distance, \(\alpha\) and \(A\) are constants. If the moment of inertia of the sphere about the axis passing through centre is \(\frac{6}{7} M R^2\), then the value of \(\alpha\) is

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.