2012 H2 Math Prelims Paper 1 Qn Paper (final)

© DHS 2012 This question paper consists of 6 printed pages (including this cover page). Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9740/01 Paper 1 11 September 2012 3 hours Additional Materials: Answer Paper List of Formulae (MF15) READ THESE INSTRUCTIONS FIRST Write your Name, Index Number and Class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, attach the question paper to the front of your answer script. The total number of marks for this paper is 100. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total Score Max Score 3 4 5 7 8 9 9 10 11 11 11 12 100

2 DHS 2012 Year 6 H2 Math Preliminary Examination 1 By using the substitution ,yx solve the inequality 2 3 1. 1 x x    [3] 2 Obtain a formula for  1 2 21 2 tan 2 d14 n x xx   in terms of n, where 0.n Hence evaluate  1 2 21 2 tan 2 d14 x xx   exactly. [4] 3 Express 2 9 x x   as a series in ascending powers of x , up to and including the term in 2.x [3] By using only the first two terms in the series expansion above and substituting 1 9x , find an approximation for 5 in the form p q expressed in its lowest terms, where and pq are integers to be determined. [2] 4 Given f( ) (1 ) ! rr r  , show that 2 31f( 1) f( ) . (1 ) ! rrrr r     [2] (i) Hence find 2 2 31 (1 ) ! n r rr r   . [3] (ii) State the value of 2 2 31 (1 ) !r rr r      , justifying your answer. [2] 5 A function f is defined as f ( ) ln(5 ), 5.xx x  (a) Find the volume of revolution when the region bounded by the curve of f( )yx and the x- and y-axes is rotated completely about the y-axis. Give your answer correct to 2 decimal places. [2] (b) (i) State the set of values of x for which    f| | fx x . [1] (ii) Evaluate the value of   9 02 90 2 ,fd f | | dx xx x   giving your answer in the form ln 2,ab where a and b are constants to be determined. [5] [Turn over

3 DHS 2012 Year 6 H2 Math Preliminary Examination 6 It is given that 1cose. xy   (i) Show that  2 2 2 dd1. dd yyx xyxx  [3] (ii) Find the Maclaurin’s series for y with exact coefficients, up to and including the term in 3.x [3] (iii) Hence find the expansion of 2 1cose 1 x x    up to and including the term in 2x and estimate the gradient of the tangent of 1cose xy   at 0.5.x [3] 7 Referred to the origin O, the points A and B are such that OA a  and .OB b  The point C lies on OB such that OC pOB     , where p is a constant. D is on AC such that AD : DC = 2 : 3 and E is on AB such that AE : EB = 1 : 3. (i) Find OD   and OE  in terms of a, b and p. [2] (ii) Given that O, D and E are collinear, find p. [3] (iii) If OB = 5, show that the s hortest distance from E to OB can be expressed as k ab , where k is a constant to be found. [3] (iv) Give a geometrical interpretation of ˆab . [1] 8 The curve C has equation   2 1 1 x xy x   , 1,x  where  is a constant. Find (i) the equations of the asymptotes of C, [2] (ii) the range of values for  such that C has 2 stationary points for 0.x [4] For the range of values for  obtained in (ii), sketch, on separate diagrams, the graphs of (iii) C, [2] (iv) f'y x , where    2 1f, 1 , 1 xxxx x   [2] indicating the coordinates of the points where the graphs cut the x-axis and the equations of asymptotes, if any. [Turn over

4 DHS 2012 Year 6 H2 Math Preliminary Examination 9 (a) Solve 4 44 3 i ,z   expressing your answers in the form ie,r  where 0r and .  (You do not need to list out the roots.) [3] Hence solve 4 13 i ,w   expressing your answers in a similar form. [3] (b) Given 22, arg( ) , 7 and arg( )33pp q q      , determine the modulus and argument of 7 3 p q . Hence express 7 3 p q in the form ix y , where ,.xy   [4] State the smallest positive integer n such that 7 3 n p q    is real and negative. [1] 10 The functions f and g are defined by f : x 2 24 , 1xx x   and g : x  1ln 2 1 , 2xx  . (i) Find an expression for   1f x and state the domain of 1f. [3] (ii) Sketch the graphs for  fyx and   1fyx  on the same diagram. What can you say about the solution of the equation 1f ( ) = f ( )?x x State your reason clearly. [3] (iii) Determine if the composite func tion gf exists. If so, find gf( x) and the exact range of gf. [5] [Turn over

5 DHS 2012 Year 6 H2 Math Preliminary Examination 11 (a) Given that 2 de d x yz x , express d d z x in terms of x, d d y x and 2 2 d d y x . Hence show that the differential equation 2 14 2 dd 2edd xyy xx  can be reduced to 12d ed xz x  . Solve this differential equation, expressing y in terms of x. [5] (b) An aviary keeper started an insect br eeding programme with an initial number of 5000 insects in a controlled environmen t to feed the birds in the aviary. The rate of birth of the insects is 1 25 of the number I (in thousands) of insects, at time t days after the start of the breeding programme. The insects are also being fed to the birds at a constant rate of 0.25 (in thousands) per day. Show that the population of insects can be modelled with the differential equation d 0.04( 6.25).d I It  Hence solve the differential equation to obtain the particular solution of I in terms of t. [4] By sketching the solution curve, show that it is possible for the insect breeding programme to be depleted of insects af ter some time. After how many complete days will this happen? [2] [Turn over

6 DHS 2012 Year 6 H2 Math Preliminary Examination 12 (a) A Dunman High alumnus plans to donate money to build two koi ponds in the middle of Zheng Xin Yuan. The ponds are in the shape of “ D” and “ H” as shown below (all dimensions are in metres and the diagrams are not drawn to scale). The shaded areas represent the pond surface. The “ D” pond is made up of a 3 h by h rectangle and a curved area formed from two concentric semi-circles of radii 2r and r respectively. The “H” pond is made up of seven identical squares of side h. (i) Show that the total surface area of the two ponds, A m 2, is given by 22 310 . 2A hr  [2] (ii) The landscape designer proposes that h and r be related by the equation 2rh k , where k is a constant. As r and h vary, find the exact value of r h at the stationary value of A. Determine the nature of this stationary point and use a calculator to evaluate the stationary value of A if 1.k  [6] (b) The equation of a graph C is given by 425 .yx x  By differentiation, find the exact set of values of y for which there are no points on C. [4] END OF PAPER 3h h 2r r r h h