2019 J2 SAJC H2 Math Prelim (with ans)

[Turn Over] ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS HIGHER 2 9758/01 Wednesday 28 August 2019 3 hrs Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) NAME:_________________________________________( _____ ) C.G.: __ ________ TUTOR’S NAME: _________________________________________ SCIENTIFIC / GRAPHIC CALCULATOR MODEL: _______________________ READ THESE INSTRUCTIONS FIRST Write your name, civics group, index number and calculator models on the cover page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Total marks : 100 Write your answers in the spaces provided in the question paper. Give non-exact numerical answers correct to 3 significant figur es, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphi ng calculator are allowed unle ss 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. The number of marks is given in brackets [ ] at the end of each question or part question. Question 1 2 3 4 5 6 7 8 9 10 TOTAL Marks 6 7 1 0 9 1 01 01 21 21 21 2 1 0 0 This document consists of 27 printed pages and 1 blank page including this page. www.KiasuExamPaper.com 681

2 [Turn Over 1 The diagram below shows the graph of 2f (3 )yx . The graph passes through the origin O, and two other points A 93, 4   and B(3,0). The equations of the vertical and horizontal asymptotes are 1x  and 2y  respectively. (a) State the range of values of k such that the equation f( 3 )xk has exactly two negative roots. [1] (b) By stating a sequence of two transformations which transforms the graph of 2f (3 )yx onto f( 3 )yx , find the coordinates of the minimum point on the graph of f( 3 )yx . Also, write down the equations of the vertical asymptote(s) and horizontal asymptote(s) of f( 3 )yx . [5] 2( i) On the same axes, sketch the curves with equations 226 4yx x  and 34 ,yx indicating any intercepts with the axes and points of intersecti on. Hence solve the inequality 234 2 6 4xx x  . [4] (ii) Find the exact area bounded by the graphs of 34yx , 226 4yx x  , x = 3 and x = 1 . [3] y xO A B(3,0) www.KiasuExamPaper.com 682

3 [Turn Over 3 The functions f and g are defined as follows: 4f: 1 xx x  6 , ,1xx 2g: 2 2xx x 6 , ,1xx  (i) Show that f has an inverse. [1] (ii) Show that 1ff  and hence evaluate 101f (101) . [5] (iii) Prove that the composite function fg exists and find its range. [4] 4 It is given that cose. xy  (i) Show that d2s in 0d y yxx  . Hence find the Maclaurin’s expansion of y up to and including the term in 2.x [4] Deduce the series expansion for 2sin 2e x   up to and including the term in 2x . [3] (ii) Using the series expansion from (ii), estimate the value of 2sin2 2 0 ed x x     correct to 3 decimal places. [2] 5 A curve C is determined by the parametric equations 2, 2 , where 0.x at y at a  (i) Sketch C. [1] (ii) Find the equation of the normal at a point P, with non-zero parameter p. [2] Show that the normal at the point P meets C again at another point Q, with parameter q, where 2qp p  . Hence show that 2 22 3 4 16|| ( 1 )aPQ p p . [4] (iii) Another point R on C with parameter r, is the point of intersection of C and the circle with diameter PQ. By considering the gradients of PR and QR, show that 22 22 rpr p     . [3] www.KiasuExamPaper.com 683

4 [Turn Over 6 (a) (i) Express 13 ip   in exponential form. [1] (ii) Without the use of a calculator, find the two smallest positive whole number values of n for which (* ) i np p is a purely imaginary number. [4] (b) Without the use of a calculator, solve the simultaneous equations 67 i 0zw   and 2i * 1 9 3 i 0wz   , giving z and w in the form ixy where x and y are real. [5] 7 The position vectors, relative to an origin O, at time t in seconds, of the particles P and Q are (cos )t i + (sin )t j + 0k and 3 cos24 t     i + 3sin 4t     j + 33 cos24 t      k respectively, where 02 t  . (i) Find |OP JJ JG | and ||OQ JJJG . [2] (ii) Find the cartesian equation of the path traced by the point P relative to the origin O and hence give a geometrical description of the motion of P. [2] (iii) Let  be the angle POQ at time t. By using scalar product, show that 32 1cos cos 2 .84 4 t      [3] (iv) Given that the length of projection of OQ JJJG onto OP JJ JG is 5 units, find the acute angle  and the corresponding values of time t . [5] www.KiasuExamPaper.com 684

5 [Turn Over 8 (a) Meredith owns a set of screwdrivers numbered 1 to 17 in decreasing lengths. The lengths of the screwdrivers form a geometric progression. It is given that the total length of the longest 3 screwdriv ers is equal to three times th e total length of the 5 shortest screwdrivers. It is als o given that the total length o f all the odd-numbered screwdrivers is 120 cm. Find the total length of all the screwd rivers, giving your answer correct to 2 decimal places. [4] (b) Meredith is building a DIY workbench, and she needs to secure several screws by twisting them with a screwdriver drill. Each time Meredith presses the button on the drill, the screw is rotated clockwise by nu radians, where n is the number of times the button is pressed. Each press rotates the screw more than the previous twist, and on the first press, the screw is rotated by 2 3  radians. It is given that 1 13cos cos sin22 nn nuu u  and 1 13sin sin cos22 nn nuu u  for all 1n  . (i) By considering  1cos nnuu  or otherwise, and assuming that the increase in rotation in successive twists is less than  radians, prove that nu is an arithmetic progression with common difference 3  radians. [3] (ii) Each screw requires at least 25 complete revolutions to ensure that it does not fall out. Find the minimum numbe r of times Meredith has to pres s the drill button to ensure the screw is fixed in place. [3] (iii) The distance the screw is driven into the workbench on the nth press of the drill, nd , is proportional to the angle of rotation nu . If the total distance the screw is driven into the workbench after 21 presses is 144mm, find the distance the screw is driven into the workbench on the first press. [2] www.KiasuExamPaper.com 685

6 [Turn Over y x 9( i) By using the substitution 15sin 15x  , find the 15 22 0 15 ( 15) dxx leaving your answer in terms of  . [5] (ii) A sculptor decides to make a stool by carving from a cylindrical block of base radius 30 cm and height 35 cm using a 3D carving machine. The design of the stool based on the piecewise function g x where  22230 15 ( 15) for 0 15g 3 30 for 15 35. xxx x         The figure below shows the 3D image of the stool after the desi gn ran through a 3D machine simulator. Figure 1: 3D Image of the stool (a) Find the exact area bounded by the curve y = g, x x = 15 and the x-axis and y-axis. [3] (b) The curve defined by the function g( )yx when rotated 2 radians about the x –axis gives the shape of the stool that the sculptor desires, as shown in Figure 1. Find the exact vo