ACJC 2020 H2 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/01 Paper 1 25 August 2020 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 calculat or 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. 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 Question Marks 1 /6 2 /6 3 /7 4 /8 5 /8 6 /9 7 /10 8 /10 9 /12 10 /12 11 /12 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/01 1 (i) Use the substitution u = 2x – 1 to find 2 d 1 (2 1) x x x−−∫ . [4] (ii) Hence find 1sin (2 1) dxx− −∫ . [2] 2 The diagram above shows the graph of f( )yx= . The curve has turning points ( )1, 2−− and ( )2, 0.5− . It cuts the x-axis at 1x = and 2.5x = . The curve has asymptotes with equations 0x = , 3y = and 3yx= + . Sketch, on separate diagrams, the graphs of (i) ( )f'yx= , [3] (ii) ( ) 1 fy x= , [3] including the coordinates of the points where the graphs cross the axes, the turning points , and the equations of any asymptotes if possible. x y O

3 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/01 3 It is given that ( )fyx= , where ( ) 3f e sec 2 xxx = . (i) Show that ( ) ( )f ' 3 2 tan 2xy x= + . [1] (ii) Find the Maclaurin series for ( )f x , up to and including the term in 2x . [3] (iii) Hence find the first two non- zero terms in the Maclaurin series for ( )( )3e sec 2 3 2 tan 2x xx + . [1] (iv) Using the standard series from the List of Formulae (MF26), verify the result found in part (ii). [2] 4 (i) Given that 0<a and 1<−b , find the roots of the equation b xa xa b−= − , where a and b are constants. [2] (ii) If 0ab<< , (a) write down a condition for b xa xa b−= − to have a negative real root. [1] (b) using the same condition from (a), on the same axes, sketch the graphs of y x ab= − and y bx a= − . [2] Hence solve the inequality b xa xa b− ≥− . [1] Deduce the set of values of x for which eexxb a ab− +≤ + . [2] 5 (i) Let 3f( ) n n n , where n is a positive integer. Show that 3 (2 1)f( 1 ) f( ) ( 1) n nnn nn    . [1] (ii) Hence find 1 31 112 N n n nn=   −  +  ∑ in terms of N. [4] (iii) Deduce 1 3 (2 1) ,( 1)( 2) nN n n nn= + ++∑ leaving your answer in the form 23 2 N ab N + − + , where a and b are constants to be determined. [3]

4 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/01 6 The function f is defined by ( ) 2 1 3 1 for 0 2, : 2 4 for 2 4. f xx x xx − − ≤ ≤ − − <≤  It is given that f( 4 ) f( )xx+= for all real values of x. (i) Sketch the graph of f( )yx= for 19 x−≤ ≤ , stating the coordinates of the end points. Find the range of f. [4] The function g is defined by ( ) 2 1g: , , 2 2 . 11 x xx x ∈ −≤≤ +−  The graph of h( )yx= is obtained by applying the following transformations in succession to the graph of g( )yx= : (A) Reflection in the y-axis (B) Stretch with scale factor 2 parallel to the x-axis (C) Stretch with scale factor 3 parallel to the y-axis (ii) Find h(x) and state the domain of h. [3] (iii) Prove that hf(x) exists and find its range. [2] 7 A hot air balloon company gives a sticker to each pass enger who is going for a ride. The sticker is in the shape of a curve C given by the parametric equations 2cos sin 2 , 2 2sin , for ππxy θθ θ θ= + =+ −<≤ . The curve has a y-intercept at (0, 4). (i) Use a non-calculator method to find the exact area of the sticker in terms of π. [5] (ii) Prove that cosxy θ= . Deduce that the Cartesian equation of the curve C is 2 3444x yy= − . [3] (iii) A model of a hot air balloon is made by rotating the curve C about the y-axis through π radians. Find the volume of the model. [2] (0,4) O x y Curve C

5 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/01 8 A hollow circular cylinder of diameter 42 mm is wrapped with n layers of thin film. The first layer of film is in contact with the outer surface of the cylinder and it has length 42 π mm. The second layer is in contact with the first layer and has length (42 + 2x)π mm, where x mm is the thickness of the thin film. There is no gap between any two layers. The nth layer has length 124π mm. (i) Show that the thickness of the film satisfies the equation x(n – 1) = 41. [2] (ii) Hence find x, given that the total sum of the lengths of the n layers of film is 16766π mm. [3] Another roll of film which has length 52580 mm is to be cut into n pieces of increasing lengths, where the lengths form a geometric progression with common ratio r. It is given that the sum of the lengths of the first ten pieces of film is k times the sum of the length s of the first five pieces of film, where k is an unknown constant. (iii) Show that 5 1kr= + . [2] (iv) Given that k = 33 and that the first piece of film is of length 50 mm, find the largest possible value of n. [3] 9 (i) Given that πsinsin cos 3 Rxxx  +=+3√  , find R , where 0R > . [1] (ii) The function f is defined by : sin cos , , 0 π.f x x xx x +3 ∈ ≤≤√ Using a graphical method, explain why 1f − does not exist. [2] (iii) The function g is defined by g : f ( ), , 0 πx xa x x + ∈ ≤≤ , where 0a > . State in exact form the minimum value of a such that 1g− exists. [1] Using this value of a, sketch on the same diagram the graph s of g( )=yx and 1g ()yx −= , indicating the equation of the line of symmetry and the coordinates of the endpoints. [3] Determine also the solution to ( ) ( ) 11gg g gxx−− = . [1] (iv) The function h is defined by 2 h: e , . xxx x ∈ Show that h is an increasing function. Hence , using the value of a found in (iii), find the range of values of x for which ( ) 1h g 0.5x− ≤ . [4]

6 ANGLO-CHINESE JUNIOR COLLEGE 2020 H2 MATHEMATICS 9758/01 10 In the study of light, a straight line is used to model a ray of light. A scientist wishes to conduct an experiment using light rays. Two light rays 1L and 2L have equations as follows. 1 2 : 2 ( 2 2 ), 9: 5 (2 2 3 ),2 ri j ij k r j k i jk Ls Lt =+ + − ++ =− + −+ where s and t are parameters. (i) Show that line