page 06 MARKS 12. A box of mass m kg is at rest on a rough slope inclined at an angle θ ° to the horizontal. It is held in place by an external force of magnitude F newtons, which is at the same angle θ ° to the slope, as shown in the diagram. F θ ° θ ° (a) The box is on the point of sliding down the slope when F is equal to 1 2 mg . Show that tan tan 2 ° 1 2 ° θ μ θ    . When ° 30° θ  , the external force is increased so that the box is on the point of moving up the slope. (b) Determine the magnitude of the new force. Express your answer in the form kmg, where k is a constant. 13. A function is defined as sec ( ) tan 1 x f x x   , where 0 2 x <   . (a) Show that tan ( ) ( ) tan 1 1 x f x f x x           , given that tan sec 2 2 1 x x   . (b) Hence find tan tan 1 1 x dx x    . 14. A particle of mass 5 kg is initially at rest. It is projected horizontally from an origin, O, along the positive direction of the x‑axis. The particle moves with variable acceleration given by   , 2 2 15 2 ms 0 a x x x      where x is measured in metres. (a) Calculate the displacement from O at which the particle reaches its maximum speed. (b) (i) Calculate the work done in reaching this maximum speed. (ii) Hence, or otherwise, calculate the maximum speed. 5 5 3 2 2 3 2