page 06 MARKS 12. A bead travels along a wire modelled by part of the curve with equation 3 2 2 4 33 x y x y     . The bead passes through two points with coordinates of the form (2, k). Determine the value of k for which the gradient is positive. 13. A planet has a radius of R metres. A satellite, at a height of 5R metres above the surface, moves in a circular orbit about the planet. The acceleration due to gravity at the surface of the planet is 3 m s−2. Show that the period of the orbit of the satellite is given by π 12 2R seconds. 14. Solve the differential equation 2 2 9 12 4 0 d y dy y dx dx    given that when 0 x , 6 y  and 3 dy dx . 15. A bullet of mass m kg is fired at a block of wood of mass M kg which hangs vertically and at rest at the end of a light inextensible string. The bullet enters the block horizontally while travelling at a speed of u m s−1, and becomes embedded in the block. The block then swings until it reaches a height h metres above its original position. Show that 2 1 2 mu h g M m         . 5 4 5 5