AJC_H2_MATHS_P1_Question

Page 1 of 5 AJC / 2014 Preliminary Examination / 9740 / P1 Anderson Junior College Preliminary Examination 2014 H2 Mathematics Paper 1 (9740/01) 1 Solve the inequality 5 11 x x   . [3] Hence solve cos 4 1cos x x   for 0 x   . [2] 2 A circular cake of radius r is cut into 22 sectors. The areas of the sectors form an increasing arithmetic progression. The area of the eighth sector is twice the area of the smallest sector. Find, in terms of , the angle of the largest sector. [4] [Area of Sector = 21 2 r  where  is the angle of the sector] 3(a) Find 2 62 d 14 x x xx   . [5] 3(b) The diagram above shows the region R bounded by the curve C with equation ln ,1xyx x , the x-axis and the line L with equation 1 2(2 )yx ee  . Find the exact volume of the solid of revolution when R is rotated completely about the x-axis. [5] 4 It is given that 2 2f( ) 1 axx x   where a is a real constant and 1a . Find the range of values of a such that the curve of f( )yx has a maximum point. [4] Given that 1a . (i) Sketch f( )yx , showing clearly the coordinates of the turning point, any intersections with the axes and the equation(s) of any asymptote(s). [2] (ii) By drawing a sketch of another suit able curve in the same diagram, find the number of roots of the equation   22 1 1t a nax x x   . [2] L CR e x y

Page 2 of 5 AJC / 2014 Preliminary Examination / 9740 / P1 5 The function f is defined by 21 1:f xx  , .01  x (i) Define 1f . [3] (ii) Sketch the graphs of   1ffyx  and   1ffyx    on the same diagram. Hence, solve the equation     11ff f f x x  . [3] Figure 1 shows the graph 1 2 xye  , 0x , which undergoes a sequence of two geometrical transformations as shown below. 1P and 2P are points corresponding to the point P after each transformation. The resulting graph in Figure 3 shows the graph of the function h. (iii) Find h(x). [2] (iv) Explain clearly why the composite function f h does not exist. [2] (v) Find the maximal domain of h for f h to exist and hence find the range of f h. [4] 6 The points (, )x y on a curve satisfy the equations 21d 1 2a n d2d 2 1 yxtt x t   where t is a parameter, 2t  . It is given that 33l n 74y  when 4t . (i) Find y in terms of t . [4] (ii) Show that    2 22 d2 d 21 2 y x tt    . [2] (iii) Find the Maclaurin’s series for y in terms of x up to and including the term in 2x . [3] 1 2 3 2  111, 2P e   Figure 1 O

x y

 Figure 2 1 111, 2P e    x y  Figure 3 x y 2 110, 2P e   

Page 3 of 5 AJC / 2014 Preliminary Examination / 9740 / P1 7 The figure below shows a sketch of part of the graph of f( )yx for 02 x a . The vertical asymptote 2x a is also a line of symmetry of the gr