SJI 2023 Year 6 HL MAA Prelim Examination Paper 3

155 TAI n paper is [55 marks]. FOR MARKER USE ONLY: 01 02 TO required for this paper. The maximum mark for this examinatio Number of printed pages = 4. alysis and approaches formula booklet is have permitted apps and be ram cleared. on, all numerical answers should be given ures scap paper provided. raphical calculator is permitted in this paper. Lo-Test mode and cleared of all previous data. til instructed to do so. 1130-1230 hrs CHES 24 August 2023 1 hr A clean copy of the mathematics: ana TI-84+ graphical calculators must only Unless otherwise stated in the questi exactly or correct to three sianificant fic Answer all the questions using the fools The use of a scientific or examination g TI-Nspire calculators must be in Press-t Do not open this examination paper un Thursday IMSTRUCTIOIS TОСNDIDATES MATHEMATICS: ANALYSIS AND APPROA HIGHER LEVEL PAPER 3 TUTION RY EXAMINATION 2023 EXAMINATION CODE 8 8 2 3 7 1 0 3 CANDIDATE SESSION NUMBER 0 2 5 이 1 2 YORAEtLABORA. ST. JOSEPH'S INSTI YEAR 6 PRELIMINAR TEACHER NAME: STUDENT NAME:

agel re equatlon reuuces to 1 2 [1] sof c. oocuntion rodugog to [2] (b)(ii) in the form, 1. M(This auestion continues ou the following r 2 2 leaving h and k in term: Iot c sчеuu thet th- (c) (i) Express your equation in 2 2 X y 2 1-2 n(b)(i) can be further simplified to 22y² =c² (4-c²) 2 [2] ±c can be expressed as [3] 2 (x+1)² +y² ression for z+1. [1] 2 2 + , [2] -iy, where x, y∈R, (ii) Show that the equation in 4(4-c²)x²-4c (b) (1) Show that equation z-1 4x+c² =±2c√ (ii) write down a similar exp (i) show that |z−1|=√(x-1 (a) By expressing z in the form x+ +1[=±c oh of hyperbolas and investigate how complex ncorrect, some marks may be given for a correct . You are therefore advised to show all working. ed. Please start each question on a new page. Full wer with no working. Answers must be supported d from a graphic display calculator should be phs are used to find a solution, you should sketch Approaches HL Preliminary Examination 2023/P3 where zEC and 0 <c<2. Let z be a complex number such that z-1-2- 1. [Maximum mark: 26[ In this question you will explore the graр numbers are linked to hyperbolas. these as part of your answer. Where an answer is method, provided this is shown by written working i Answer all questions in the answer booklet provide marks are not necessarily awarded for a correct ans by working and/or explanations. Solutions foun supported by suitable working. For example, if gra Year 6 Mathematics: Analysis and

[5] 1 can be rotated about the origin to coincide 2 २ X (f) Find the possible values of p. 22 The hyperbola with equation x²- y² = with the curve defined by v = 2 DEH neaning of |z-1|. [1] lues of |z-1[-2+1. [1] plex number z on the Argand diagram. [1] 11 ne nyperdola, x -y 2 2 2 1of x²-y² = 2 ts (1,0) and (-1,0) respectively, and 22 1 (iii) Deduce the geometrical (iv) Hence write down the va The point P represents the com (i) find the length of FP ii find EP. -EP. )x,) represent any point on If P is the positive x-intercept (e) Let F and F₂ represent the poin che equation of each oblique asymptote. [4] V 2x 1 12 , stating clearly the coordinates of 21 Dola, x -y 2 an be expressed as y = ±x1 1 [2] k 2 12 2 2 1, where h,k eR, is called2 12 Approaches HL Preliminary Examination 2023/P3 any axial intercepts and t (ii) Sketch the graph of x²- NOW Consider tne equation or the nyper (d) (i) Show that the equation c a hyperbola. (Question 1 continued( A curve with the equation in the form Year 6 Mathematics: Analysis and

Paper how that 2<e<3 where e is [31 n 2+x < [5] 2-x + 2 3 X + + 2 2 2 + n n-1 2" [2] End of (f) Using results from (a) and (e), s Fuler's number (e) For 0<x < 2, show that X 1+ и n X (d) Hence show that <1+x n [2] [2] where 2≤r≤n. [1] 2 a =1 n n [4] a, x" n an + 2≤r≤n. r! n! n n it can be shown that X 1+ n (+1)10 X 1 1 (ii) r!21 (c) Show that for 2≤r≤n, (i) 0<a, <1, (ii) Find an expression for a 1 (b) (i) Show that a₂ =1-- and n 1+x+a2x² 3 а3 X + 2! 3! By considering the binomial expansion can be written in the form een y in (a)(11) and fn(x). [2] n dx (0) and hence deduce, for large values L and when n is large, (f,(x))z f,(x). [3]dx =0solve the equationd = 11 [51 Ix€[0,0]. ow binomial expansion and limits are used upper bound. Approaches HL Preliminary Examination 2023/P3 of n, a relationship betwe (iii) Determine the value of (a) (i) Explain that for a given ii Given that y=1 when r= Let f()=[1+ where n EZ+ and n 2. Maximum mark: 29]| In this question you will investigate h to determine Euler's number and its Year 6 Mathematics: Analysis and