DHS H2 MATH P1 with ans

© DHS 2015 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 17 September 2015 3 hours Additional Materials: Answer Paper Graph 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 an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total Score Max Score 5 6 6 6 6 8 8 9 9 11 13 13 100

2 DHS 2015 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 1 A graphic calculator is not to be used in answering this question. (i) Find the value of 2(1 4i) , showing clearly how you obtain your answer. [1] (ii) Given that 12 i  is a root of the equation 2 (i ) 0 ,zza b   find the values of the real numbers a and b. [2] (iii) For these values of a and b, solve the equation in part (ii). [2] 2 Using partial fractions, find 2 2 62 d.(1 2 )( 1) xx xxx   [6] 3 A curve C has parametric equations cos , sin , for 0 π.xy       (i) Sketch the curve .C [2] (ii) The point P on the curve C has parameter p and the point Q has coordinates ( π,0). The origin is denoted by O. Given that p is increasing at a constant ra te of 0.1 units per second, find the rate of decr ease of the area of triangle OPQ when 3 4 π.p [4] 4 The complex number z is given by 1 6 πi e.)3   (i) Given that (1 i) ,wz find w and arg w in exact form. [2] (ii) Without using a calculator, find the smallest positive integer n such that nw is purely imaginary. State the modulus of nw when n takes this value. [4]

3 DHS 2015 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 5 The diagram shows the graph of f( ).y x The graph has a minimum point at (1 ,1 ) and a maximum point at (4 ,7 ) .  It intersects the axes at 2, 1x x and 2 3 .y  The equations of the asymptotes are 2y x and 3.x (i) Sketch the graph of 1 ,f( )y x giving the coordinates of any stationary points, points of intersection with the axes and the equations of any asymptotes. [3] (ii) Solve the inequality 1f0 .x  [3] 6 (i) By using the Maclaurin series for ex and cos ,x find the Maclaurin series for g( ),x where g( ) e cos2 ,xx x up to and including the term in 2.x [3] (ii) Use your answer in part (i) to give an approximation for 0 g( ) d a x x in terms of a, and evaluate this approximation in the case where 1 3 e,a giving your answer correct to 5 significant figures. [3] (iii) Use your calculator to find an accurate value for 1 3 0 e g( ) d ,x x up to 5 significant figures. Why is the approximation in part (ii) not very good? [2] x y 1 O

4 DHS 2015 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 7 Relative to the origin O, two points A and B have position vectors a and b respectively. It is given that 2, 1 and 3 2 (37).   ab a b (i) By considering the scalar product (3 2 ) (3 2 ), abab show that 1 4ab and give the geometrical meaning of | | .ab [4] (ii) Give the geometrical meaning of ()  ab b and find its exact value. [3] (iii) Write down, in terms of and ,ab a vector equation of th e line that passes through O and bisects the angle AOB. [1] 8 A curve C has equation 223 2 80.xx y y  (i) Show that d3 .d yx y x yx   [2] (ii) Show that the curve C has no stationary points. [3] (iii) The normal to the curve at the point (6, 2)P  meets the curve again at the point Q. Find the coordinates of Q. [4] 9 The diagram shows the curve with equation   1 1 2sinyx  for 02 . x  (i) Find the area of the region R1 bounded by the curve, the lines 1 1 12 6π , πyy and the y-axis. [2] (ii) Find the volume of revolution when the region R1 is rotated through 2π radians about the x-axis. [3] (iii) Without using a calculator, find the exact area of the region R2 bounded by the curve, the lines 1, 3xx  and the x-axis. [4] R2 x y 1 O R1

5 DHS 2015 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 10 (a) Show that the substitution 2wx y reduces the differential equation 24 2d24 1d yxy x y yx   to the form 2d ,d w aw bx   where a and b are to be determined. Hence obtain the general solution in the form 2 f( ) . yx [5] (b) A certain species of bird with a population of size n thousand at time t months satisfies the differential equation 12 4 2 d e.d tn t   Find the general solution of this differential equation. [2] Sketch three members of the family of solution curves, given that 30n when 0.t  [4] 11 Functions f and g are defined by 1 3f : ln(3 1) for , ,xxx x   4g : for , 1 4.(1 ) ( 5)xx x xx   (i) Describe fully a sequence of transfor mations which would transform the curve lnyx onto the curve of f( ) .yx [3] (ii) Sketch the graph of g( ).yx   [2] (iii) Find the exact range of fg. [2] (iv) If the domain of g is further restricted to 4,kx  state with a reason the least value of k for which the function 1g exists. [2] In the rest of the question, the domain of g is defined as ,4 ,xk x  where k is the value found in part (iv). (v) Find 1g( ) .x   [2] (vi) If h is a function such that gh is well-defined and the point  16 15, lies on the graph of gh( ),y x find the value of h( ). [2]

6 DHS 2015 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 12 The diagram shows a cuboi d with rectangular base OABC and top EFGH, where 4u n i t s ,OA 3 unitsOC  and 2 units.OE  The point O is taken as the origin and unit vectors i, j and k, are taken along OA, OC and OE respectively. (i) Find the cartesian equation of the plane p which contains the points A, C and E. [3] (ii) Find the acute angle between p and the base OABC. [2] The line l, passing through O, is perpendicular to p and intersects the plane containing B, C, G and H at the point T. (iii) Find the position vector of the point T and deduce the perpendicular distance from T to p. [5] (iv) A point Q lies on the line passing through C and T such that its distance from p is twice that of the distance from T to p. Find the possible position vectors of the point Q. [3] O A BC E F GH i j k

7 DHS 2015 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over Marking