TJC_H2_MATHS_P2

Section A: Pure Mathematics [40 marks] 1 Given that    sin 1 sin 1 2cos sinnx nx n x x    , show that  11 22 1 1 2 sin sincos 2sin n r nx xrx x  . [4] Hence express  22 2 2 31 1cos cos cos ... cos22 2 x xxx               in the form sin sin bxad cx   , where ,,abc and d are real numbers. [3] 2(a) The diagram above shows two curves 1C and 2C which are reflections of each other about the line y = x. State with justification, whether the following statement is true: “If 1C is the graph of fy x , then 2C is the graph of   1fy x .” [1] (b) The functions f and g are defined as follows 2 1f: , 6x x x  x , x < 0, 1g: t a n 2 xx     , x . (i) Sketch the graph of  fy x . Determine whether 2f e x i s t s . [ 3 ] (ii) Find  1f x . [2] (iii) Given that gf(a) = ,4  find the exact value of a. [2] y = x C1 y (1, 2) (2, 1) C2 x 0

3 Given that ee s i ny x x  . Show that 22 22 2 dd2e 4e sin 0 dd yy yy xxx   . [2] (i) Find the values of y, d d y x and 2 2 d d y x when 0x . Hence, find in terms of e, the Maclaurin’s series for  ln e sinx x , up to and including the term in 2x . [4] (ii) By using appropriate standard series expansions from the List of Formulae (MF26), verify the correctness of the first three terms in the series expansion for  ln e sinx x found in part (i). [3] (iii) Use your answer to part (i) to give an approximation for  1e 20 2e 4 d,el ne s i n x xxx    giving your answer in terms of e. [3] 4 With reference to origin O, the points A, B, C and D are such that OA a  , OB b  , OC a   and 2OD b   . The lines AB and DC meet at E. Find OE  in terms of a and b. [4] Hence show that 3BE AB  . [1] It is given that A and E have coordinates (1, 4, 3) and (3, 15, 5) respectively. (i) Show that the lines AC and BD are perpendicular. [4] (ii) Find the equation of the plane p that contains E and is perpendicular to the line BD. [2] (iii) Find the distance between the line AC and p. [2] B C D E O A

Section B: Statistics [60 marks] 5 Four classes CG40, CG41, CG42 and CG43 are task ed to organise a College event. Each class sends 3 representatives for a meeting. (i) In how many different ways can the 12 re presentatives sit in a circle so that representatives from CG40 are not seated next to each other and representatives from other classes are seated with their respective classes? [3] The 12 representatives are to be split up in to 3 groups for bonding activities. Each group must consist of a representative from each class. (ii) In how many ways can the groups be formed? [2] 6 In a game at the carnival, a player rolls discs onto a board containing squares, each of which bears one of the numbers 1, 2, 5 or 10. I