TMJC 9758 2023 Prelim P1

Tampines Meridian Junior College 2023 JC2 Preliminary Examination H2 Mathematics CANDIDATE NAME: ___________________________________________________ CIVICS GROUP: _______________________________________________________ __________________________________________________________________________________ H2 MATHEMATICS 9758/01 Paper 1 13 SEPTEMBER 2023 3 hours Candidates answer on the question paper. Additional material: List of Formulae (MF26) ________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and Civics Group 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. 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. 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 6 printed pages and 0 blank pages. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION For Examiners’ Use 1 2 3 4 5 6 7 8 9 10 Total [Turn Over

2 1 The hyperbola 1C has equation ( ) 2 2 22 2 1.32 x y− −= (a) Sketch 1C , labelling the equations of the asymptotes and coordinates of the axial intercepts. [3] The curve 2C has equation ( ) 2 222x yk−+= , where 0k > . (b) State the value of k such that 1C and 2C intersect exactly twice. [1] 2 (a) Differentiate 2 2exx+ with respect to x and hence find the exact value of ( ) 21 2 0 1e dxxxx ++∫ . [3] (b) Using the substitution sinxt= , where 0 2t π<< , find ( ) 3 2 2 1 d 1 x x− ⌠ ⌡ in terms of x. [4] 3 In an art lesson, wires are cut and bent to create a series of shapes. (a) Student A cuts a piece of wire of length 100 cm to create a series of squares. The perimeter of the smallest square is 5 cm and the perimeter of each succeeding square is increased by 3 cm. Find the maximum number of squares Student A can form. [3] (b) Student B cuts another piece of wire to create a series of 12 circles, such that the radius of the first circle is cmx . The circumference of each succeeding circle is 2 3 that of the preceding circle. Given that the t otal length of wire used by Student B is exactly 100 cm, find the value of x . [4]

3 [Turn Over 4 The curve C has equation 32 ay xa= + − , where a is a positive constant. (a) Describe fully a sequence of transformation s which would transform the curve 1y x= onto the curve C. [3] (b) Sketch the graph of 32 ay xa= + − , labelling the equations of the asymptotes and coordinates of the axial intercepts. [3] (c) Use an algebraic method to solve 323 1x+= − and state the solution for 32 3. 1x+< − [4] 5 (a) The polynomial ( )P z is given by 3 8z za−+ , where a is a real constant . Given that 3i+ is a root of the equation ( )P0z = , solve exactly the equation ( )P 0.z = [5] (b) It is given that 3iw= −+ . Write down the exact value of arg( )w and hence find the three smallest positive integer values of n such that nw is purely imaginary. [3]

4 6 A curve C has equation 2y ax bx c= ++ , where , and ab c are constants. It is given that C passes through the points ( )2,17− , 13,24   and ( )5, 3 . (a) Find the equation of C. [3] The function f is defined by 2f: , 0x ax bx c x++ ≤ . (b) Using your answer from part (a), find ( )1f. x− [3] The function g is defined by ( ) 2 3 1 for 0 2,g: 5 4 for 2 6. xxx xx  − ≤< −− ≤<  (c) It is given that g( ) g( 6)xx= + for all real values of .x Sketch the graph of ( )gyx= for .4 9x− ≤< [4] (d) For the rest of the question, let the domain of g be [ )0, 6 , as originally defined. The function h is defined by 33h : cos , 0 π.22x xx + ≤≤ Prove that the composite function gh exists and find the range of gh . [3]

5 [Turn Over 7 (a) An ellipse has equation 22 22 1,xy ab+= where a and b are positive real constants. Show that 2 2 d d y bx x ay=− if 0.y≠ [1] (b) The point ( )cos , sin ,Pa b θθ where 0 2 πθ<< , lies on the ellipse. S how that the equation of the tangent to the ellipse at P is cos sin 1.xy ab θθ+= [4] (c) The tangent found in part ( b) meets the x-axis and y -axis at points R and S respectively. Find the area of triangle ORS in terms of , and ,abθ where O is the origin. [3] (d) Explain why the minimum area of triangle ORS is ab and state the value of θ which gives the minimum area of triangle .ORS [2]

6 8 (a) Find the exact value of 6 3 1 d.xxπ∫ [1] (b) The diagram shows a sketch of the curve 3 2 ,yx= where 0.x≥ The region under the curve between 1x= and 6,x= shown shaded in the diagram, is .R It is required to estimate the volume of the solid formed when R is rotated through 2π radians about the x-axis. The region R is split into n vertical strips of equal width as shown in the diagram below. Each vertical strip can be approximated by a rectangle. Each rectangle, when rotated through 2π radians about the x -axis, will result in a circular disc. The sum of the volumes of the n circular discs with equal thickness is denoted by V . Suppose V gives an overestimation of the volume of revolution of the solid formed when R is rotated through 2π radians about the x-axis. Show that 3 1 5π5 1. n r Vr nn =  = +  ∑ [3] (c) Find an expression for ,V leaving your answers in the form 2 5π 259 ,4 ab nn  ++ where a and b are constants to be determined. You may use the results ( )( ) 2 1 1 12 1 6 n r r nn n = =++∑ and ( ) 232 1 1 1. 4 n r r nn = = +∑ [5] (d) Using your answer in part (c), state the limit of V as .n→∞ Explain how this could be verified by the answer in part (a). [2] y 1 O x ...

7 [Turn Over 9 A home interior designer is looking to design a pull-down wardrobe lift . To test his concept, the designer built a small-scale model of it, in which points ( ),,xyz are defined relative to a corner of the model at ( )0, 0, 0O . The rods used in the model can be modelled by equations of straight lines. The main rod can be modelled by the line 1l with equation 1 , 5 0 5 8 1λλ   ∈  = −+ r  . Another rod, line 2l , passes through the points A( )5, 5, 8− and B( )8, 4, 3− . (a) A supporting wire connects B to a point F on 1l such that BF   is perpendicular to 1l . Find the position vector of F. [3] (b) A retractable rod, line Rl , is placed such that it can be modelled by reflecting 2l about 1l . Find a vector equation of Rl . [3] The panels of the small -scale model can be modelled by equation s of planes. The main panel, plane 1p , contains the main rod 1l and an attachment point C ( )0, 0, 6 . A side panel, plane 2p , can be represented by the equation 2 : 17 37 4 24p x yz− − += . (c) Show that 1p can be represented by the cartesian equation 5 30xy z−++ = . [3] (d) A structural rod, line Sl , needs to be placed at the intersection between 1p and 2.p Find a vector equation of Sl . [2] (e) A decorati