DHS H2 MATHS P1 QP

This document consists of 7 printed pages (including this cover page). Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS 9740/01 (Higher 2) 17 September 2014 Paper 1 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 signi ficant 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 Total Score Max Score 4 6 7 8 8 8 9 12 11 13 14 100

2 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 1 (i) A fisherman has 800kg of fish, consis ting of mackerel, salmon and tuna. He may choose to sell all his fish at either Market A or Market B. The rates offered by the respective market s and his total returns are as follows: Market Price (in dollars) per kg Total Returns (in dollars) Mackerel Salmon Tuna A 7 21 39 20 300 B 5 23 49 23 900 Determine the weight of the salmon that the fisherman has. [3] (ii) Another fisherman has 600kg of the same t ypes of fish and he claims that he can obtain the exact same total returns as the fisherman in part (i) at the respective markets. Determine whether his claim is possible. [1] 2 (a) The diagram below shows the curve with equation f( )yx which passes through the origin O, has a vertical asymptote 2x and a stationary point at 15, .4  Sketch the graph of 2 f( ) ,y x indicating the coordinates of the points where the graph crosses the axes, statio nary point(s) and the equations of any asymptotes. [2] x = 2 O y x f( )yx

© DHS 20 3 4 014 (b) The ( 1 4 x of the Given that (i) Sho w (ii) By f u terms (iii) By s o in (ii) 4 dec Do not use Relative t o p ai j (i) The p in term (ii) The p AE   (iii) Given Year 6 diagram s h 21) (2 )y  e solid form e1 3y x   w that 2 d d y x   urther diffe r s in Maclaur olving1 3 x ) to find an imal places a calculato o the orig i 3 k and b point C is o ms of p. He point D is o .OC   Find t n that the an 6 H2 Mathemat hows the r 2 1 and th med when R 22,x 2 2 2 d 4ed y x rentiation o rin’s series 221 . 0x x approxima s. or in answe in O, two  i respectiv on AB suc h ence find the on OC prod the area of t ngle betwee y O C 3 tics Preliminary region R b he line 2x is rotated th e .y of the resul t for y. 0302, use a ate value for ering this q points A vely. h that :AC e exact area duced such trapezium O en a and b is y Examination bounded b y 2. Find the hrough 2π r t in part (i) suitable va r ln 1.030 2 question. and B ha v 2:1CB . a of triangle that OD OAED. s 135 , find x = 2 R y the curv e numerical v radians abo ), find the f lue of x and . Give you ve positio n Find the p e OAC. 2.CD The p d the value o [Tu e C with value of the out the x-axi first three n d the result ur answer c n vectors g position ve c point E is s of p. x urn over equation e volume is. [4] [2] non–zero [3] obtained correct to [2] given by ctor of C [3] such that [2] [3]

4 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 5 The complex number z is given by ie,zr  where 02 r  and ππ 62  . (i) Given that 23 i ,wz find w in terms of r and arg w in terms of . [2] (ii) Given that  has a fixed value, draw an Arga nd diagram to show the locus of z as r varies. On the same diagram, show the corresponding locus of w. You should identify the modulus and ar gument of the end-points of each locus. [4] (iii) Find the range of values that 2 *2 w z can take. [2] 6 Show that the differential equation  2d 30d  yxy x yx may be reduced by using the substitution ux y to 2d 30 .d  u ux Hence find the general solution of y in terms of x. [4] (i) Sketch the solution curve that passes through 11, ,3   indicating any stationary points and asymptotes clearly. [3] (ii) State the particular solution for which y has no turning point. [1]

5 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 7 The curve C has equation e xy   , where  is a constant, 1  . (i) Sketch C. [2] Hence, on separate diagrams, sketch the graphs of (ii) e xy  , [2] (iii) 1 exy   , lnx  , [3] including the coordinates of the points where the graphs crosses the axes and the equations of any asymptotes. For 1x  , use graphs in parts (ii) and (iii) to deduce the number of real solutions for the equation   ee e e1xx  . [2] 8 (a) Find 2sec ( ) d , x xa x where a is a constant. [3] (b) Find 2 1 d22   x xx x . [2] Hence find (i) the exact value of 2 21 4 d,22   x xx x [4] (ii) 22 1 d22 p p x xxx   where p is a constant, 1.p Leave your answer in terms of p. [3]

6 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 9 (a) A sequence of positive numbers 123, , ,..., nx xx x is a geometric progression. Show that the sequence of numbers 123, , ,..., ny yy y given by the relation, log , where is a positive constant,nk nyx k k is an arithmetic progression. [4] (b) Thomas takes an education loan of $20 000 from a bank for his undergraduate studies. Starting from the month he obtai ned the loan, Thomas makes a monthly payment of $ x to the bank in the middle of every month. At the end of each month, the bank charges him an interest of 1% for the remaining amount owed. The repayment plan continues until he has fully paid for his loan. (i) During his studies, Thomas is only ab le to make a monthly payment such that the amount owed remains at $2 0 000 at the end of every month after the interest is char ged. Find the value of x for this payment plan. [2] After completing his studies, Thomas st ill owes the bank an outstanding amount of $20 000. He adjusts the repayment amount in the month after completing his studies so that he can repay his loan in full after a certain time. (ii) Show that starting from the month the repayment amount is adjusted, the monthly payment of $x such that the loan is fully paid after the n th payment is given by 1200(1.01 ) .1.01 1 n nx    [ 3 ] (iii) Hence determine the monthly amount that Thomas should make to fully pay up the loan in exactly 3 years’ time. [2]

7 © DHS 2014 Year 6 H2 Mathematics Preliminary Examination [Turn over 10 The function f is