MARKS Page eight 7. (a) Find 3 2 1 cos x dx + ( ) ∫ . (b) Show that 3cos 2x + 1 = 4cos2x − 2sin2x. (c) Hence, or otherwise, find sin cos 2 2 2 x x dx − ( ) ∫ . 8. A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river. Crocodiles travel at different speeds on land and in water. The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, P, x metres upstream on the other side of the river as shown in the diagram. 20 metres x metres P The time taken, T, measured in tenths of a second, is given by ( ) ( ) 2 = 5 36 + + 4 20 – T x x x (a) (i) Calculate the time taken if the crocodile does not travel on land. (ii) Calculate the time taken if the crocodile swims the shortest distance possible. (b) Between these two extremes there is one value of x which minimises the time taken. Find this value of x and hence calculate the minimum possible time. 2 2 2 1 1 8