A thin non-uniform rod of length \(L\) and total mass \(M\) lies on a frictionless horizontal surface and is pivoted at one end (\(x = 0\)) about a fixed vertical axle. The linear mass density of the rod is given by \(\lambda(x) = C x\), where \(C\) is a constant. A small block of mass \(m\) slides across the surface with speed \(v_0\) perpendicular to the rod, strikes the free end (\(x = L\)), and sticks to it. Which of the following expressions represents the magnitude of the horizontal linear impulse exerted on the rod-block system by the pivot during the collision?