SJI 2024 Year 6 HL MAA Prelim Examination Paper 3

STUDENT NAME: _____________________________ TEACHER NAME: _____________________________ ST. JOSEPH’S INSTITUTION YEAR 6 PRELIMINARY EXAMINATION 2024 MATHEMATICS: ANALYSIS AND APPROACHES HIGHER LEVEL PAPER 3 Wednesday 21 August 2024 1 hr 1300 – 1400 hrs/ 1630 – 1730 hrs INSTRUCTIONS TO CANDIDATES • Do not open this examination paper until instructed to do so. • Answer all the questions using the foolscap paper provided. • The use of a scientific or examination graphical calculator is permitted in this paper. • TI-Nspire calculators must be in Press-to-Test mode and cleared of all previous data. • TI-84+ graphical calculators must only have permitted apps and be ram cleared. • Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. • A clean copy of the mathematics: analysis and approaches formula booklet is required for this paper. • The maximum mark for this examination paper is [55 marks]. • Number of printed pages = 5. __________________________________________________________________________ FOR MARKER USE ONLY: Q1 Q2 TOTAL /55

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P3 2 Answer all questions in the answer booklet provided. Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Solutions found from a graphic display calculator should be supported by suitable working. For example, if graphs are used to find a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. ………………………………………………………………………………………………… 1. [Maximum mark: 24] In this question, you will be exploring the Steiner inellipse, which is a unique ellipse that is tangential to the midpoints of a triangle. Consider the cubic polynomial 32( ) 2p z z z= + − . It is given that 1 2 3, andz z z are the roots of the equation ( ) 0,pz = such that 1 2 3arg arg argz z z−     . In an Argand diagram, points P, Q and R represent the complex numbers 1 2 3, andz z z respectively. (a) (i) Find 1 2 3, andz z z . (ii) Indicate P, Q and R on the Argand diagram. [5] P, Q and R are joined together to form the triangle PQR. It is given that A is the midpoint of PR, B is the midpoint of QR, and C is the midpoint of PQ. (b) Indicate A, B and C clearly on the same Argand diagram. [1] Let 45andzz be the roots of '( ) 0pz = . Points H and K represent 4z and 5z respectively such that AH AK . (c) (i) Solve '( ) 0pz = . (ii) Indicate H and K on the same Argand diagram. [4]

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P3 3 Now, treat the Argand diagram as the real Cartesian x-y plane. Marden’s theorem states that the coordinates of H and K form the foci of the unique ellipse which is tangential to the triangle PQR at its midpoints A, B and C. This ellipse, known as the Steiner inellipse of triangle PQR, is illustrated in the following diagram. (d) Find the coordinates of the midpoints A, B and C in your diagram. [2] When H and K have coordinates ( ),hc and ( ),kc , the geometrical centre (or centroid) of the ellipse is ( , )mc , where 2 hkm += . The Steiner inellipse is then given by the following equation: ( ) ( ) 22 22 1x m y c ab −− += , where 2 2 2a b m−= . (e) (i) Show that 2 4 9a = . (ii) Find the equation of this ellipse in the form above. [6] (f) By considering a transformation of the equation in part (e)(ii) or otherwise, show that the equation of a Steiner inellipse of a triangle with vertices ( )X 1, 2−− , ( )Y 1,0 and ( )Z 1, 2− is given by: 2 2143 33xy + + = [2] (g) Find the gradient of the ellipse in part (f) at the points ( )0,1 and ( )0, 1− . Deduce XYZ. [4] [Turn Over]

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P3 4 2. [Maximum mark: 31] In this question, you will be investigating Euler’s and modified Euler’s method. The function ( ), where 0, 0,y y x x y=   satisfies the differential equation d 2 (*) d y yx =− and 0 0 0 ( ) , .y x y x x= (a) Find the exact solution ()yx in terms of 00, and .x x y [4] (b) Using Euler’s method with step size 0 ,xx n − where ,n + show that the estimate of ()yx after n steps is given by 0 01 2 . n n xxyy n  − =+   [2] Let 1. n wv n =+ (c) By considering ln ln 1 , wvn n =+  use l’Hôpital’s rule to find lim ln . n v → [4] (d) Deduce that 1 e as . n ww nn + → → [1] (e) By considering the result obtained in part (d), show that the Euler’s estimate converges to the exact solution ()yx in part (a). [2] Modified Euler’s method is given by  111 f ( , ) f ( , )2 nnn n n n hy y x y x y− −−= + + , where ny is the value of y when nxx= , df ( , ) , d yxy x= and h is the step size. (f) Using modified Euler’s method with step size 0 ,xxh n −= where ,n + show that the estimate of ()yx after n steps is given by 0 0 0 1 . 1 n n xx nyy xx n − + = − − [3]

Year 6 Mathematics: Analysis and Approaches HL Preliminary Examination 2024/P3 5 Consider the differential equation d 2 (*) d y yx =− for which 1y= when 0x= . (g) Find the exact value of of y when 0.5x= . [2] (h) Without computing the approximate value of y, explain why the Euler’s method in part (b) gives an under-estimation of the true value of y when 0.5x= . [1] (i) Let 0.1.h= Find an approximate value of y when 0.5x= using (i) the result obtained in part (b); (ii) the result obtained in part (f). [Leave your answers to 3 decimal places.] State which of the methods: the Euler’s method or the modified Euler’s method, gives a better estimate of e . [4] (j) Determine the minimum value of n such that 3e 10ny −− when 0.5x= for Euler’s method and modified Euler’s method respectively. [8] End of Paper