ACJC 9758 2023 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/01 Paper 1 21 August 2023 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 appr oved graphing calculator is expected, where appropriate. 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. 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 27 printed pages and 1 blank page. Question Marks 1 /3 2 /4 3 /5 4 /6 5 /7 6 /8 7 /9 8 /10 9 /11 10 /12 11 /12 12 /13 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 [Turn over

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 [Turn Over 1 A curve has equation ( )fyx= , where ( ) 32f x x ax bx c=+ ++ . The curve ( )fyx= has a maximum point at ( )1, 32− , while the curve ( ) 1 fy x= has a vertical asymptote with equation 5x= . Find the equation of the curve ( )fyx= . [3] 2 Do not use a calculator in answering this question. The complex numbers z and w satisfy the following equations. * 2 4i 2 3i wz zw + = −+ += Find z and w, giving your answers in the form iab+ , where a and b are real numbers. [4] 3 Let ( )f 1 xx x = + . (i) Show that ( ) ( ) 1 0 2f d 22 3xx = −∫ . [2] (ii) The diagram below shows a sketch of the curve with equation ( )fyx= . Rectangles width 1 n are drawn as each of shown. By considering the total area of these n rectangles, deduce that [3] y x 0 ……… 1

4 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 4 The points P, Q and R have position vectors p, q and r respectively, with respect to the origin O. The vector p is a unit vector and the vectors q and r are non-zero vectors. (i) Write down the geometrical meaning of ⋅pq . [1] It is given that 2⋅=pq and that the point M is the foot of the perpendicular from Q to the line OP. (ii) Show that 2OM = p   . [1] It is further given that ( )k= −r pq for an arbitrary constant k. (iii) Show that the area of triangle RQM is 2 unitsa ×pq , where a is a constant to be determined in terms of k. Hence describe the geometrical relationship of points R , Q and M when 2k =− . [4] 5 (i) Find 2d ed x x − . Hence find 23 edxxx − ∫ . [3] (ii) By using the substitution 2 e xzy −= , find the general solution of 3d 2d y xy xx−= , expressing y in terms of x. [4] 6 Two of the roots of the equation 4 33 4 0z kz k z k−+ −= , where k is a positive real number, are 1zk= and 2zk=− . The other two roots are denoted 3z and 4z , where ( )3Im 0z > and ( )4Im 0z < . (i) Show that 3 3 i22 kkz = + . [3] (ii) Find 3z and 3arg z , giving your answer in terms of k if necessary. [2] (iii) Find the two smallest positive integer values of n for which 3 1i n z −+ is purely imaginary. [3]

5 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 [Turn Over 7 The graph below shows the graph of ( )fyx= . (i) Sketch on the same diagram the graph of ( ) 1fyx −= , clearly stating the equations of any asymptotes and the coordinates of the points where the curve crosses the axes. State also the geometrical relationship between these two graphs. [3] The function g is defined by 2g : 6 5, xx x x ++ ∈ . (ii) Find the range of the composite function fg . [2] (iii) It is given that ( )fg 0a = . Find the value of ( )g a , where a is a negative constant. [1] The function h is defined by 2h : 6 5, , 4xx x x x+ + ∈ <− . (iv) Find 1h− and state the domain of 1h− . [3] 8 The curve C is defined by the parametric equations 2sin , sin , 0 πx t ty t t=− = ≤≤ . The point A on C has parameter π 2 and is a maximum point. The line l passes through the origin and the point A. The region R is bounded by C and l as shown in the diagram below. y = 4 y (0,3) x (-4,0) y x A O R

6 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 (i) Find the coordinates of A. [1] (ii) Using an algebraic method, find the exact area of R. [5] (iii) Show that the Cartesian equation of C is given by 1sinx yy −= − . [1] (iv) Find the volume of the solid formed when R is rotated by 2π radians about the y- axis. [3] 9 (a) The following diagram shows the graphs of two differentiable functions f( )yx= and g( )yx= . At xc= , the distance between the two functions is a maximum. Show that f () g ()cc′′ = where f( )x′ and g( )x′ are derivatives of f( )x and g( )x respectively. Using the second derivative, explain clearly why the distance is a maximum. [3] (b) Two runners Andy and Ben, with different strategies and abilities, participate in a 10000 m race. The distance in metres ran in x minutes by Andy and Ben are given by the functions A(x) and B(x) respectively as defined below. 320.16 12 , 0 50A( ) . 10000, 50 xx xx x − + ≤≤= > ( )3B( ) 5000log 9 10000, 0 72xx x= +− ≤≤ . (i) Sketch the graphs of A( )yx= and B( )yx= on the same diagram for 0 72x≤≤ . [2] (ii) Andy and Ben are furthest apart at xc= . By using (a) or otherwise, show that xc= satisfies the equation 2 9 RPx Qx x+= + , where P, Q and R are constants to be determined, and find the value(s) of c. [4] (iii) Hence or otherwise, find the furthest distance between Andy and Ben and state who was leading at that moment. [2] y x c

7 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 MATHEMATICS 9758/01 [Turn Over 10 It is given that the curve C has equation 24 6 11 , ,122 xxy xxx − −−= ∈ ≠−+  . (i) Without using a calculator, find the set of values that y cannot take. [3] (ii) Sketch C, clearly stating the equations of any asymptotes, the coordinates of the stationary points and the point(s) where the curve crosses the axes. [3] (iii) Deduce the range of values of a such that the equation ( ) ( ) ( ) ( ) 222 222 21 22 4 8 1 3 22x x a x x ax+ + − − −− = + has no root, where a is a positive constant. [3] (iv) Describe fully a sequence of transformations which transform the curve 92 2yx x= + onto the curve 24 6 11 22 xxy x − −−= + . [3] 11 A famous artist wants to create an artwork that uses squares only. On the first day, he draws the perimeter of a square with length x m as shown in Figure 1. On the second day, he divides the square into 9 equal squares and draws the perimeter of the middl e square as shown in Figure 2. On the third day, he divides the remaining 8 empty squares from Day 2 into 9 equal squares, and draws the perimeters of the middle squares as shown in Figure 3. This procedure of “drawing the perimeters of the middle squares” is then repeated for subsequent days (see Figure 4 for his finished effort on Day 4).