A hexagonal pencil of mass M and sides length `a` has been placed on a rough incline having inclination `theta` Friction 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 not allow 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