page 07 MARKS 13. A security spotlight is situated 10 metres from a straight fence. The spotlight rotates at a constant speed and makes one full revolution every 12 seconds. The situation at time t seconds is modelled in the diagram below, where: • L is the position of the spotlight • G is the point on the fence nearest to the spotlight • P is the position where the light hits the fence • θ is the angle between LG and LP • x is the distance in metres from G to P. fence x m G P 10 m beam of light θ spotlight (L) (a) Show that: (i) π 6 dθ dt  radians per second (ii) π sec2 5 3 dx θ dt  metres per second. (b) Prove that tan sec 2 2 1 θ θ   . (c) Hence, or otherwise, find the exact value of dx dt when P is 5 metres from G. [END OF QUESTION PAPER] 1 4 1 3 page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE