ACJC 2019 H2 Math Prelim

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/01 Paper 1 29 August 2019 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on all the work you hand in. 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. Write your answers in the spaces provided in the question paper. 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. The use of an approved graphing calculator is expected, where appropriate. Unsupported answers from a graphing 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. The total number of marks for this paper is 100. __________________________________________________________________________________ This document consists of __ printed pages. [Turn Over www.KiasuExamPaper.com 3

2 ANGLO-CHINESE JUNIOR COLLEGE 2019 H2 MATHEMATICS 9758/01 1 The points A(2,3) and B(3,1) are on a curve with equation f( )yx . The corresponding points on the curve y f a(x b)  are A (7,3) and B (1,1). Find the values of a and b. [3] 2 Use differentiation to find the area of the largest rectangle w ith sides parallel to the coordinate axes, lying above the x-axis and below the curve with equation 244 4yx x . [5] 3 Solve the equation 2 3 231 xx xx   exactly. [4] Hence, by sketching appropriate graphs, solve the inequality 2 3 231 xx xx   exactly. [2] 4 A kite 50 m above ground is being blown away from the person ho lding its string in a direction parallel to the ground at a rate 5 m per second. Assuming that the string is taut, at the instan t when the leng th of th e string already let out is 100 m, find, leaving your answers in exact form, (i) the rate of change of the angle between the string and the ground, [3] (ii) the rate at which the string of the kite should be let out, [4] 5 Given that   3tan 1 e xy , show that  32d e1d xy kyx  , where k is a constant to be determined. By further differentiation of this result, or otherwise, find the first three non- zero terms in the Maclaurin series for   3tan 1 e x . [5] The first two terms in the Maclaurin series for   3tan 1 e x are equal to the first two non- zero terms in the series expansion of x ab x . Find the constants a and b. [3] www.KiasuExamPaper.com 4

3 ANGLO-CHINESE JUNIOR COLLEGE 2019 H2 MATHEMATICS 9758/01 [Turn over 6 The diagram below shows the graph of 21xy for 01 .x Rectangles, each of width 1 n , are also drawn on the graph as shown. Show that the total area of all n rectangles, ,nS is given by 1 1 2 1 (2 1) n n n S n   . [3] Find the exact value of lim nn S  . [2] 7 (a) F i n d sin cos dpx qx x where p and q are positive integers such that p ≠ q. [2] (b) S h o w t h a t 2 cos sinsin d xn x n xxn x x cnn   where n is a positive integer and c is an arbitrary constant. [1] Hence find (i) π 0 sin dxn x x , giving your answers in the form πk n where the possible values of k are to be determined, [2] (ii) π 2 0 sin 3 dxx x in terms of . [3] y ……… x 1 n 2 n 3 n 3n n  2n n  1n n  21xy O 1 www.KiasuExamPaper.com 5

4 ANGLO-CHINESE JUNIOR COLLEGE 2019 H2 MATHEMATICS 9758/01 8 Do not use a calculator in answering this question. (a) The complex numbers z and w satisfy the following equations 29wz , *31 7 3 0 iww z  . Find w and z in the form iab , where a and b are real and Re 0z  . [4] (b) (i) G i v e n t h a t i is a root of the equation  32 82 2 i 8 i 0zk z z    , where k is a constant to be determined, find the other roots, leaving your answers in exact cartesian form ixy , showing your working. [3] (ii) Hence solve the equation  32i2 2 8 i 8 i 0zk z z    , leaving your answers in exact cartesian form. [2] (iii) Let 0z be the root in (i) s u c h t h a t  0arg 0z  . Find the smallest positive integer value of n such that  0i n z is a purely imaginary number. [2] 9 (a) The diagram below shows the graph of y 1 f(x) with asymptotes 0,x 2,x and 1y  , and turning point (1,2). (i) Given that f(0) f(2) 0 , s k e t ch th e g r a p h of f( )yx , stating clearly the coordinates of any turning points and points of intersection with the axes, and the equations of any asymptotes. [3] (ii) The function f is now defined for xk such that 1f exists. State the smallest value of k. On the same diagram, sketch the graphs of fyx and  1fyx  , showing clearly the geometrical relationship between the two graphs. [3] O x y www.KiasuExamPaper.com 6

5 ANGLO-CHINESE JUNIOR COLLEGE 2019 H2 MATHEMATICS 9758/01 [Turn over (b) The function g is defined for x0 as 1 11g: 2 1, 22 n nnxx x  a , where n . (i) Fill in the blanks. 11 , 42g( ) 1 , 12 x x x     Hence sketch the graph yg(x) for 1 4  x 1. [3] (ii) S h o w t h a t g(x) g x 2     . [2] (iii) Find the number of solutions of g(x) x for 0.001 x1. [2] 10 David is preparing for an upcoming examination with 9 practice papers to complete in 90 days. The examination is on the 91st day. He is planning to spread out the practice papers according to the following criteria, and illustrated in the diagram below.  He only completes 1 practice paper a day.  He attempts the first practice paper on the first day.  The duration between the first and the second practice paper is a days.  The duration between each subsequent paper decreases by d days.  He completes the last practice paper as close to the examination date as possible. (i) By first writing down two inequalities in terms of a and d, determine the values of a and d. [4] The mark for his n-th practice paper, un , can be modelled by the formula un  92 65(b)n where 0 b1. (ii) What is the significance of the number 92 in the formula? [1] (iii) F i n d m, his average mark, for the nine practice papers he completed, leaving your answer in terms of b. [3] (iv) Given that he scored higher than m from his fourth practice paper onwards, find the range of values of b. [2] days 1st paper 2nd paper 3rd paper a days days 4th paper www.KiasuExamPaper.com 7

6 ANGLO-CHINESE JUNIOR COLLEGE 2019 H2 MATHEMATICS 9758/01 11 A toy paratrooper is dropped from a building and the attached p arachute opens the moment it is released. The toy drops vertically and the distance it drops after t seconds is x metres. The motion of the toy can be modelled by the differential equation 22 2 dd 10dd xx ktt  , where k is a constant. By substituting velocity, d d xv t , write down a differential equation in v and t. [1] Given that d 6d v t  when 10v , and that the initial velocity of the toy is zero, show that 4 4 5(1 e ) 1e t tv     , and deduce the velocity of the toy in the long run. [6] The toy is released from a height of 10 metres. Find the time it takes for the toy to reach the ground. [5] 12 In air traffic