EJC 9758 2023 Prelim P1

2023 JC2 H2 Mathematics Preliminary Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 1 [100 marks] 9758/01 12 September 2023 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on 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. DO NOT WRITE ON ANY BARCODES. Answer all 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. This document consists of 23 printed pages and 1 blank page. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total

2 2023 JC2 H2 Mathematics Preliminary Examination Paper 1 1 The general solution of a differential equation is given by 22ee exx xyA B C −= ++ , where A, B and C are arbitrary constants. Given that a particular solution satisfies 2y= , d 3d y x =− and 2 2 d 11d y x = when 0x= , find the values of A, B and C. [4] 2 A sequence of numbers 1 23, , , ...uuu has sum nS where 1 n nr r Su = =∑ . It is given that 1 2 218 3 n n nS + −= − . (a) By finding an expression for nu , or otherwise, show that the sequence is a geometric progression. [3] An arithmetic progression has first term 4− . The sum of the first 9 terms of the arithmetic progression is equal to 1 r r u ∞ = ∑ . (b) Find the common difference of the arithmetic progression. [3] 3 With reference to the origin O, the points A , B and C are such that 2 1 1 OA  =−    , 0 1 2 OB  =−   and 1 3 0 OC  =   . Point P lies on AB, between A and B, such that :1:AP PB λλ −= where 0 1λ< < . Point Q lies on BC , between B and C, such that 1::BQ QC λλ= − . (a) Express PQ   in terms of λ. [3] The point X has coordinates 22 5, 1, −− . Points P, Q and X are collinear. (b) Find the value of λ. [3] 4 (a) Using the method of differences, find 2 2 2 1 n r r= −∑ . [4] (b) Hence, find ( ) 31 11 2 2 n r rr − = +∑ . [3] 5 (a) Find sin 3 cos dx xx∫ . [2] (b) Find 2e cos3 d .x xx∫ [5]

3 2023 JC2 H2 Mathematics Preliminary Examination Paper 1 6 A curve R has equation f( )yx= . The point P with coordinates (7, 9)− lies on R. The tangent to R at point P has gradient 10. (a) The curve C is obtained from R by applying the following two transformations: • A scaling parallel to the y-axis with scale factor 3, and • A translation in the negative x-direction by 4 units. Find the coordinates of the point on C corresponding to P, and state the gradient of C at this point. [2] (b) The curve D has equation 1 f( )y x= . Find the coordinates of the point on D corresponding to P, and find the gradient of D at this point. [3] 7 The parametric equations of a curve 1C are 3232xt t= + and 3 221yt t= −+ + for 11 t− ≤≤ . (a) Sketch 1C , stating the exact coordinates of the endpoints and any axial intercepts. [3] Another curve 2C has equation ( ) 22 1xy k+−= where k +∈  . (b) Describe 2C geometrically. [1] (c) For 16k = , find the coordinates of the point of intersection between 1C and 2C . [4] (d) State all the possible number of point(s) of intersection between 1C and 2C as k varies. [1] 8 The function f is defined by ( ) 2 1f: 4 11 x x + −+  , 0x≥ . (a) Sketch the graph of f( )yx= , indicating clearly the coordinates of any stationary points, the value of the y-intercept, and the equation of any asymptotes. [3] The function g is defined by g : sinxx , [ ]2π0,x∈ . (b) State the exact coordinates of two points on the graph of g( )yx= , which demonstrate that g does not have an inverse. [1] (c) Explain clearly why the composite function gf exists. [2] (d) Using a suitably-labelled sketch of the graph of g( )yx= , find the range of gf. [3]

4 2023 JC2 H2 Mathematics Preliminary Examination Paper 1 9 (a) Using standard series, find the Maclaurin series expansion of 22 1 1 ax− up to and including the term in 4x , where a is a positive constant . Find also the range of values of x , in terms of a, for which the expansion is valid. [5] (b) Hence, by using a suitable value for a, show that 1 3513cos 2 6 40 πx xx x− ≈−− − . [3] (c) Using the result in part (b), estimate 1 12 0 cos dxx− ∫ to 5 significant figures. [2] (d) Comparing your estimate in (c) with the actual value of 1 12 0 cos dxx− ∫ , comment on the accuracy of your estimate and suggest how it can be further improved. [2] 10 Do not use a calculator in answering this question. The complex numbers z and w have the same modulus r, and have arguments α and β respectively. (a) Show that 2 cos cos isin22 2zw r αβ αβ αβ−+ ++= + and hence write down expressions for zw+ and for arg( )zw+ . [4] It is now given that πi122ez= and 5πi 122ew= . (b) Find the values of zw+ and arg( )zw+ . [2] Let v be the complex number such that 2zv w= . (c) Find v and arg( )v . [3] (d) Let O be the origin and P and Q be points representing zw+ and v respectively on an Argand diagram. Find the area of triangle OPQ . [2]

5 2023 JC2 H2 Mathematics Preliminary Examination Paper 1 11 Nozzles are often used at the end of hoses to direct and speed up the flow of water. The diagram below represents the side view of a nozzle. The nozzle itself consists of solid material between an outer surface and an inner surface (shaded in the diagram). Water flows through the hollow region ABCD enclosed by the inner surface. The outer surface of the nozzle is obtained by rotating the curve with equation ( )( ) 2 2 12 kyk xx= + ++ , for 0 xk≤≤ , where ,1kk∈≥ , completely about the x-axis. Similarly, the inner surface is obtained by rotating the curve with equation 2 3 xyk= − , for 0 xk≤≤ , completely about the x-axis. The units of x and y are centimetres. (a) Show that the volume of material used to make the nozzle is 222π 2 ln 26 kkk k  + +  + cm3. [6] (b) Find the area of ABCD in terms of k. [3] Using the Principle of Conservation of Mass, the following relationship can be found. in in out outAv A v= inA and outA are the cross-sectional areas of the circular openings at the inlet and outlet of the nozzle respectively, and inv and outv are the speeds of the water at the inlet and outlet of the nozzle respectively. (c) Calculate the ratio of relative speeds i out n v v for the given nozzle. [3]

6 2023 JC2 H2 Mathematics Preliminary Examination Paper 1 12 Trinity de Rale is a budding installation artist. Her latest mobile artwork comprises of two fixed rails placed at an angle of 60° to each other at O (refer to diagram). A rod PQ of fixed length L cm is placed such that its ends P and Q glide freely along each rail. It is given that the length OP is x cm, and that the angle OPQ is θ . (a) Show that 1co ins s 3 xL θθ + =   . [2] A motor is placed at P to move that end of the rod along the rail, allowing θ to vary between 30° and 90°. The motor is controlled such that t