AJC_H2Maths_2012Prelim_P2_Question

Page 1 of 5 AJC / 2012 Preliminary Examination / 9740 / P2 Anderson Junior College Preliminary Examination 2012 H2 Mathematics Paper 2 (9740/02) Section A: Pure Mathematics (40 marks) 1 The Environment Authority wishes to build a fence surrounding a park shown by the figure below. BCEF is a square with sides measuring 2 kilometres. ECD and ABF are two identical triangles with CD = CE and angle DCE =  radians. The cost of building the fence is $28000. Given that  is sufficiently small for 2 and higher powers of  to be neglected , show that the unit cost of building the fence is approximately $ ()ab  per kilometre, where a and b are constants to be determined. [5] 2 It is given that   2 4 , 0 1 f ( ) 3 , 1 3 xx x xx     and f ( ) f ( )xx  . It is also known that f ( ) f ( 6)xx for all real values of x. (i) Show that f (4) = 1 . [2] (ii) Sketch the graph of y =f(x) for 69 x   . [3] (iii) Find the exact value of 7 5 f ( ) dxx   . [3] 3 Relative to an origin O, the points C and D have position vectors 23 3 and 2 cd                    respectively, where  and   . (i) The straight line l, passing through C and D, has c artesian equation s 322 3 yx , z    . Find the values of  and  . [3] (ii) Find the point of intersection of the line l and the y-z plane. [2] (iii) The reflection of C in the y-z plane is 'C . Find the position vector of 'C . [3] F A B C D E

Page 2 of 5 AJC / 2012 Preliminary Examination / 9740 / P2 4 (a) By using the substitution u x y , solve the differential equation 2 2 24( ) cos sindy x y x xdx   . [5] (b) A new drug for the treatment of diabetes is administered to a patient at a constant rate of R mg per day. The rate at which the drug is lost from the patient’s body is proportional to the square of the amount x (mg) of the drug present in his body at time t (days). (i) If the amount of drug in the patient remains consta nt at the instant when it is 2R (mg), show that 22d4 d4 x R x tR  . Given that, when x = 0 when t = 0, find x, in terms of R and t. [5] (ii) Explain the significance of this result in the long run. [1] 5 The complex number z satisfies the equation 3 221 21zi a  , where a is a real number. It is given that   3arg 4z   . (i) Show that 1 2a . [2] (ii) Hence solve the equation 3 221 21zi a  , expressing the solutions in polar form  cos sinri  , where 0r and      . Give r and  in exact form. [3] The complex number w satisfies the equation 3 2 1 i1 w i   . Describe how the point representing w can be obtained from the point representing z by a series of transformations. [3]

Page 3 of 5 AJC / 2012 Preliminary Examination / 9740 / P2 Section B: Statistics (60 marks) 6 Find the n