A hexagonal pencil of mass M and sides length a has been placed on a rough incline having inclination thetaFriction is large enough to prevent sliding. If at all the pencil moves, during one full rotation each of its 6 edges, in turn, serve as instantaneous axis of rotation. (a) Show that for theta gt 30^(@)the pencil cannot remain at rest. (b) For inclination of incline theta lt 30^(@)the pencil will not roll on its own. A sharp impulse J is given to the pencil parallel to the incline at its upper edge (see figure). Friction does notallow the pencil to slide but it begins to rotate about the edge through A with initial angular speed omega_(0). Find omega_(0). Moment of inertia of the pencil about its edge is I. (c) Find minimum value of J so that the pencil will turn about A, and B will land on the incline. (d) If kinetic energy acquired by the pencil just after the impulse is K_(0), find its kinetic energy just before edge B lands on the incline