2024 JC2 H2 Math prelim - EJC

Error! Reference source not found. EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2024 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 1 9758/01 09 September 2024 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 22 printed pages and 2 blank page(s). Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total

1 The sum of the digits of a certain three-digit number is 15. The sum of the digit in the tens placing and twice the digit in the hundreds placing is equal to the digit in the units placing. If the digits are reversed, the new number is 21 more than 5 times the old number. What is the number? [4] 2 Do not use a calculator in answering this question. Find the complex numbers z and w which satisfy the following simultaneous equations 2 5i,zw−= 3i * 5.zw− =− Give your answers in the form iab+ , where a and b are real numbers. [5] 3 The curve with equation 1yp xq=− + , where p and q are constants, is transformed by a reflection in the x-axis, followed by a translation of 3 units in the negative y-direction, followed by a stretch of scale factor 4 parallel to the x-axis. A point on the curve 1yp xq=− + has coordinates 3 ,34 −− . (a) Find the resulting coordinates of this point after it undergoes the same sequence of transformations. [1] The resulting curve has asymptotes 1x=− and 2y= . (b) Find the values of p and q. [5] 4 A curve C has equation 2 25 114 1 xxy x ++= + . (a) Sketch C, indicating on the diagram the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts. [4] A curve D has parametric equations 23sin 2 1x =− , 23 23cos2y =+ for 22  −   . (b) Find the cartesian equation of curve D. [2] (c) Hence, on the same diagram as in part (a), sketch the curve D and determine the number of intersection points between curve C and curve D. [3]

Error! Reference source not found. 5 (a) (i) It is given that 2 4 2d2 2 . d yxy x x y x− = + Using the substitution 2wy x= , show that the differential equation can be transformed to 2d 1 2 .d w wx =+ [3] (ii) Hence, solve the differential equation 2 4 2d2 2 . d yxy x x y x− = + [3] (b) It is given that u satisfies 2 2 2 d ed xu x = and has a stationary value of 2e 4 when 1x= . Find u in terms of x. [3] 6 (a) Show that 2 31 ( 2)! rr r ++ + can be expressed as 11 ! ( 2)!rr− + . [2] (b) Find an expression in terms of n for 2 1 31 ( 2)! n r rr r= ++ + . You should not simplify your answer. [3] (c) Explain why 2 1 31 ( 2)!r rr r  = ++ + is convergent, and state the limit. [2] (d) Using your answer to part (b), find the exact value of 2 6 1 !r rr r  = −−  . [3] 7 For a right -angled triangle ABC with BC a= and AC b= , CAB x= and 2 πABC= as shown below, a sector with radius a is extended from the vertex C. This sector meets the side AC at the point Y. (a) Show that ( ) 2 2 1 1 sin AC AY x  = − . [2] (b) Using appropriate expansions from the List of Formulae (MF26), find the Maclaurin series for ( ) ( ) 2 f 1 sinxx − =− , up to and including the term in 3x . [3] (c) Deduce the series expansion for ( ) 3 2cos 1 sin x x− , up to and including the term in 2x . [2] (d) Using your answer in part (b) or otherwise, show that 21AC xxAY  + + , when x is sufficiently small. [3] a A b B C Y x

8 Do not use a calculator in answering this question. (a) The equation ( ) 2 2 i 2 1 i 0z a z b+ + + + = , where a and b are real, has a root 11 i22− . (i) Find the values of a and b. [2] (ii) Find the second root of this equation. [3] (b) The complex numbers 1w and 2w are given by 1 3i−+ and 2 2i− respectively. (i) Find the modulus and argument of 1 2 w w in exact form. [3] (ii) Hence show that 11 3 1sin π12 22 −= . [3] 9 The function f is given by ( ) 2 1f , , 2, 2. 4x x x xx=  − − (a) Sketch the graph of f ( ).yx= Give the equations of any asymptotes and the coordinates of any turning points. [2] (b) If the domain of f is further restricted to xk , state the least value of k for which the function 1f − exists. For this value of k, find ( ) 1f x− and state the domain of 1f.− [4] For the rest of the question, the domain of f is , 2, 2x x x −  as originally defined. The function g is given by ( ) 1 1 3g , , , 1, . 1 2 2x x x x xx=    − (c) Find ( )fg .x [2] (d) Find the range of fg. [3] 10 A mining company has identified a mineral layer below ground. Points ( , , )x y z are defined relative to the mining office located at (0, 0, 0) , where units are metres. The ground is modelled as a horizontal plane with equation 0z= . The top surface of the mineral layer is modelled as part of the plane containing the points (8, 4, 50)A − , (12, 14, 42)B − and ( 6, 20, 60)C −− . (a) Show that a cartesian equation of the top surface of the mineral layer is 19 6 17 1026x y z+ − = . [2] (b) Find the acute angle between the ground and the top surface of the mineral layer. [2] The mineral layer is found to be of thickness 14 14 metres, with the bottom surface modelled as part of a plane parallel to the top surface. (c) Find a cartesian equation of the bottom surface of the mineral layer. [3] The mining company plans to drill vertically downwards from a point on the ground to reach the mineral layer. After the drill touches the top surface, it continues to penetrate through the mineral layer until it touches the bottom surface.

Error! Reference source not found. (d) Find the length of the drill that is found inside the mineral layer. [2] It is found that in fact, cost effectiveness and safety could be increased by performing directional drilling. As such, the mining company proposes a new plan to drill at a certain angle from the point ( 14, 3, 0)D − towards the mineral layer. (e) If the shortest path is taken, find the position vector of the point at which the drill touches the top surface of the mineral layer. [3] 11 On his 45th birthday on 1 July 2024, Mr Eu purchases an annuity plan called EuRetire for a principal sum of $x. The plan works like this: Phase 1: The plan pays compound interest at a rate of 3% per annum on the last day of June each year, until Mr Eu turns 65. The total sum at the end of Phase 1 is called the “accumulated sum”. Phase 2: Upon turning 65, Mr Eu receives a monthly payout, $ y, on the 1 st of each month, from the accumulated sum. The amount remaining in the plan continues to draw interest at 0.2% per month on the last day of each month. (a) Write down an expression for the accumulated sum at the end of Phase 1, giving your answer to 5 significant figures. [1] (b) Show that the amount remaining in the plan at the end of the nth month of Phase 2, after