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 graph. Sketch, on separate diagrams, the graph of (i) f( )yx for 04xa ; [2] (ii) 1 f( )y x for 04xa ; [3] (iii) f 2 xya   for 26ax a  . [3] indicating clearly the asymptote( s) and the axial intercept(s). 8   The equations of planes P1, P2 are P1: 32 1 10 12 1 , , 33 1                       r P2: 1 01 1    r Find the coordinates of the foot of perpendicular from the point A 3,10, 3 to the plane P2 and show that the point B 2,10, 2 is the reflection of point A in P2. [5] The planes P1 and P2 meet in a line L. Find a vector equation of line L. [3] Plane P 3 is the reflection of P1 in P2. Using the results above, find a vector perpendicular to P3. Hence find, in scalar product form, the equation of P3. [3] 0 -2 a 2a y x

Page 4 of 5 AJC / 2014 Preliminary Examination / 9740 / P1 9 The graph of 40yx is shown in the diagram below. By considering the shaded rectangle, show that 1 40 40 d n n nx x     . [1] Deduce that 41 40 123 . . .8 0 4 0 d x x      . [2] Show also that 1 41 40 d n n nx x     . [2] Deduce that 41 40 1 2 3 ... 8 1 4 0 d x x      . [1] Hence, deduce the value a , where a , that satisfies the following inequality,  91 2 3 . . . 8 0 9 1aa  . [2] 10 Solve the equation 5 16 16 3 zi  , giving the roots in the form ire , where 0r  and   . Show the roots on an Argand diagram. [5] The points A and B represent the two roots with the two smallest positive arguments. Point P, the mid-point of AB, represents the complex number w. (i) Find, in exact form, the modulus and argument of w. [2] (ii) w i s a l s o a n n th root of k , where n is a positive integer and k is a real number. Find the least possible value of n and find the corresponding value of k, leaving your answer in trigonometric form. [3] 0 40yx -40 41 n n +1 x y

Page 5 of 5 AJC / 2014 Preliminary Examination / 9740 / P1 11(a) A sequence 1,u 2u , 3u , … is such that 1 10 eu  and 1 12 nn n euu   , for all 2n . Use the method of difference to show that 1 111 1010 2 n n eu     . [4] State, with a reason, whether nu is a convergent sequence. [1] 11(b) The sequence 123a , a , a ...... is defined by 1 123 ra ..... r   , where r is a positive integer. Another sequence 12 3b, b , b  is defined by  1 1 4 n nr r ba    , where n is a positive integer. (i) Find the terms 12b, b and 3b . Hence, make a conjecture for nb and express your answer in the form of f n n . [3] (ii) Prove your conjecture using mathemat ical induction. [5] END OF PAPER