(ML 926457 z Please check the examination details below befor ing you Of A FRANCUSCO M&M Centre Number Candidate Number CLIT) Pearson Edexcel International Advanced Level Friday 12 January 2024 Morning (Time: 1 hour 30 minutes) WEMO1 /01 (eK Mathematics International Advanced Subsidiary/ Advanced Level Further Pure Mathematics F1 ‘You must have: (Total Marks Mathematical Formulae and Statistical Tables (Yellow), calculator Candidates may use any calculator allowed by Pearson regulations. Calculators just not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Instructions © Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Fill in the boxes at the top of this page with your name, centre number and candidate number, © Answer all questions and ensure that your answers to parts of questions are clearly labelled. © Answer the questions in the spaces provided ° = there may be more space than you need. You should show sufficient working to make your methods clear. ‘Answers without working may not gain full credit. © Inexact answers should be given to three significant figures unless otherwise stated. Information © A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. © There are 10 questions in this question paper. The total mark for this paper is 75. @ The marks for each question are shown in rackets = use this a5 a guide as to how much time to spend on each question. Advice © Read each question carefully before you start to answer it. © Try to answer every question. © Check your answers if you have time at the end. © Ifyou change your mind about an answer, cross it out and put your new answer and any working underneath, Turn over > evoon (MINIMUM, Escaneado con CamScannerLD | eae — 1 (23 k where & is a constant k+7 k+4 (a) Show that M is non-singular for all real values of k. (b) Determine M" in terms of k. = at ke aie 6) |M[ = f- [3 kal = \Vauy sini nl 221M Lon Oo [kt? key wPeeR K-24 | Cyeallkre) 2 2 = 3 kty = kerakeg Jo VER a ke -2209-6 gs VBE - - _ 2 4 e _ — [axa keo\ wt A —olat) te eee t W3UY SIHLNI SLIM LON 00 V3MY SIHLNI LIM LON Od AUN a = el scaneado con CamScannerwhere p and q are integers. The complex number 5 — 4i is a root of the equation f(z) = 0 (a) Write down another complex root of this equation. f(@)=22) + pz + qz—4l (a (b) Solve the equation f(z) = 0 completely. a 4) 2 (©) Determine the value of p and the value of g. & @ 2 When plotted on an Argand diagram, the points representing the roots of the equation £(2) = 0 form the vertices of a triangle. (@) Determine the area of this triangle. Q) Se $= 44 = St ye 5) {zsh wf - ) =e] === *WaMY SIHLNI@LINWLON'OG = 26 244 ana ean {= (22+ W)(2% a2 1) = (22-1) (2502 144 oO = 2271 (k20)2" + kz 41k WAUV SIHLNIaiNIMLON 00” - 4 CMM Cuma . Escaneado con CamScannerTHIS AREA DO.NOTWRITEINT ~ DO NOT WRITEIN THIS AREA DO NOT WRITE INTHIS AREA. © com ANN ONE 0 runever > Escaneado con CamScanner3. The hyperbola HT has equation xy = c* where c is a positive constant. ‘The point P(e £), where ¢ > 0, lies on H. The tangent to H at P meets the x-axis at the point A and meets the y-axis at the point B. (a) Determine, in terms of ¢ and f, (i) the coordinates of A, ii) the coordinates of B. (4) Given that the area of triangle AOB, where O is the origin, is 90 square units, (b) determine the value of c, giving your answer as a simplified surd. e ) VaUY SIHLNISLIIMLON OG ng mma © 4aav SIMLNI ALIN LON Od VaUy-SIHLNI3LRM1ONOG. Escaneado con CamScannerDO NOT WRITE INTHIS AREA, DO NOTWRITEIN THIS AREA DO NOT WRITE INTHIS AREA ac ) Question 3 continued = ait ()—__(@2E)) _ g _ A. 108s _=> 90 : c= YS => (Total for Question 3 is 6 marks) a IANO 00 0 Thenover > Csma ora ane an aa? Escaneado con CamScanner10 A= 03 (a) Describe the single geometrical transformation represented by the matrix A. Q) ‘The matrix B represents a rotation of 210° anticlockwise about centre (0, 0). (b) Write down the matrix B, giving each element in exact form. ay ‘The transformation represented by matrix A followed by the transformation represented by matrix B is represented by the matrix C. VaUY SIHLNISLIUM LON OG (©) Find C. @) ‘The hexagon is transformed onto the hexagon H' by the matrix C. (a) Given that the area of hexagon H is 5 square units, determine the area of hexagon H" @ leas 8 5 Cine Gy. a EES echcad shilling wll SER fy) oa [Ces 40 = Sin 240" \ = 9530" Sth 30 - in 24O— - |= Sino? | “WaUY SIHLNI3LINM LON OG | 3LuM.LON OG 10 Cuma Escaneado con CamScannerDO NOT WRITE INTHIS AREA DO NOT WRITEINTHIS AREA. DO NOT WRITE INTHIS AREA @ i as Question 4 continued : _ bt C= leJe Aven +" = 2x aad = %Sut2 1Sur | _ (Total for Question 4 is 7 marks) 7 cama I 0 Turnover > Escaneado con CamScanner5. The quadratic equation 2 -3x+7=0 has roots a and Without solving the equation, (a) write down the value of (a+ f) and the value of af © (b) determine the value of a? + Q) VaUY SIHLNTSLIEMLON Od SS (c) find a quadratic equation which has roots (gle) giving your answer in the form px* + gx + r= 0 where p, q and r are integers to be determined. se ed (oy (Hip) = as aapap 2% ep ah wip (3). = ag - 4 7< [a 4 =e shape ——— 7 a _———_— © WMV SIHLNISLRIM LONG my — 4 _ 304 4g 7 2” 7 “OR Bs a SIHLNI@LIuM1oNGa » 2 IA 0 0 0 cmma "-__ Escaneado con CamScannerQuestion 5 continued iene are = a-f+A “fe = wip (San Fe 2 4 a xt 5 al SAP eg St a Be ge - DO NOT WRITE INTHIS AREA | —DONOTWRITEINTHISAREA << — RLV DO NOT WRITEINTHIS AREA Turnover > Escaneado con CamScannerf(x) =x-4~cos(SVx) x >0 (a) Show that the equation f(x) = 0 has a root a in the interval (2.5, 3.5] @) (b) Use linear interpolation once on.the interval [2.5, 3.5] to find an approximation to a, giving your answer to 2 decimal places. Q) aul SIAL NISL LON Oa (a) Determine g'(x): @ The equation g(x) = 0 has a root f in the interval [6, 7] (b) Using x, = 6 asa first approximation to #, apply the Newton-Raphson procedure once to g(x) to find a second approximation to f, giving your answer to 3 decimal places. “Wau SIMLNIALINMIONOG ~ Since, od ee couhjunous in Lb thee Ts ‘ach = ce. "peal Re oh uggs, certo “S$ . “Eis bounds He Howe - fone rank 4 Os wilhin the tales ey (2. 3 35) _ Lix) = Vauv SIHLNI LIM LON 0G Escaneado con CamScannerDO NOT WRITE INTHIS AREA ~ DO NOT WRITE IN THIS AREA x DO NOT WRITE INTHIS AREA Question 6 continued 0. 498 AUS 3.$-% k= 2,5 = OMG hoe = 2.5" OMIG = LAS ©3S. 148K xX = 1,45» 3,5 + 2,5x 0,998 EE AE J [x = 3, 244 363 3.24 CAMO 00 0000000 Eee a Escaneado con CamScanner® Questién 6 continued a ¢ 8 8 g a eS _ = Tyg = Xe 4s) = 3 = jj —_—__——|8 a (6) = ee g oo y g - = om $2, 204 62 —— dtr $4 3g tt a ; _ _ . —— We 6 Bs ly, 6.644 an 2.204624 Pe: | |s — § 5 A Zz z B g & : g 3 a 2 q = 5 — 5 S IAEA UA A Cuma Escaneado con CamScanner— 7. The parabola C has equation y? = te ‘The point (fe, 2), where 1# 0, lies on C. (a) Use calculus to show that the normal to C at P has equation Bat 3y=r +2 <\walw SIHDNTanaMoNod SS. @ ‘The normal to C at the point where 1 = 9 meets C again at the point O. (b) Determine the exact coordinates of Q. ® 2 yee - __ | % 2 /) Mai SIHLNIZiRIM LON 0a (0) €29_ 2: Pb, 209) => Rife, Met 3x6 2-9 ——het_) = = 94 249 ng y= 6 = 4 (X22) UNAM SIHL NISL LON Oa” (iq = -2- 7 ——— = 2x > 814"- Lupys Sol = Fx 0 =o ;caneado con CamScannerQuestion 7 continued / DO NOT WRITEIN-THIS AREA. Wf | © DONOT WRITE THIS ARE EINTHIS AREA $ 2) a Law ° § (Total for Question 7 is 7 marks) 2 cama ANU 0000 0 0 Tum over Escaneado con CamScanner a x &§ [es Use the standard results for summations to show that, for all positive integers 1, Velo’ -3r=1) = bale) (2) § a 4 2 (b) Hence show that, for all positive integers », 3 Ser(2r? — 37-1) = Sul = 1)(an + (on +) 3 2 ; where a, b, c and d are integers to be determined. a 4 q _ ae / ©) Erbe: 2 IB age Br = pe = 3xEn(neDlanes) ~ 4nlnea) = |g La oe - 8 - 4? ~4 _ A 2 g ton (net) = 4. Soot 2 =_ 4, niowe)[nn lanes) = aE Fine) [nn ana) i : =[f re 3 _ zines), (n= 1-2) = F nlnrd) (n= -a)(n4) = z = = |4 wnat (mca) VAUV SIHLNI3LIUM10N 0G Escaneado con CamScannerQuestion 8 continued en a : 3 ah (ant 4) Oo) Brak 3r-s)= z r(ar® ar) — Briar a) a (an = a- 4 ie ne (n-3) = cinta ” DONOT Wi = eS - “= Bae [a ( anv) = = run-3)] = Lf) 16nd ~n*+3n |= tat 2) (15 n+ ices) Snes 19m +4) = = Salad) (ise : Diet) ay DONOTWRITEINTHIS AREA” DONOT WRITE INTHIS AREA 23 Turnover > Escaneado con CamScanner& 9. Given that where 2 is a real constant, (a) determine z, giving your answer in the form x + yi, where x and y are real and in terms of A. 4 Given also that argz = = 4 (b) find the possible values of 2. o 6) 2A AeGe ATH 5 32-1 - HE 20» We 2 Ree Ne RN ZH =UO FANT ; 7 ze 1+ 4 - = a nis 16 ge DO BN 2t Mt ae AE OM _ 3 N16 A246 [22 Ay 1 en + ee eT a 3-17 comned bo a proper. colubon “hecause it will ba {te couplx number have both eel) and jnragincn gina a iM MM OE Escaneado con CamScanner10. (i) Prove by induction that for n€Z* 5 HY get Qn+3 0 -n ) 4 1° 4n 3-2n (ii) Prove by induction that for eZ" 6) f=" 4! is divisible by 7 -& — ate aks -K 20K mek te ale] 7 v x vt es = 3k-3_\ (nakoaa ek 43 28 Omsa Escaneado con CamScanner~ | Question 10 continued eet ka (et) La (ket) +3. —_ aa — 3 y ; ~ healed) = 2(ked)+3 _ As te apron s_vabol fer nek ifs abo vabiol foc ne k+l Since it tas util fo no tndiely, He a proven. By indchia fo: all 1 @a foyer sce _ _ — I< + 6 = Sif — 74v2 Dinsile by 7 DO NOT WAITEINTHIS AREA” 2k Assume F< tg a lade by F— so fikay= 9" eget ghia Pts | °SIDONOT WRITE'IN THIS AREAS" 25g" 2 —fltea) = = 64 (fe et fA “fa 64 (XK) - gage Both terms tin ee Expressions are cee Ee Hina, (ea was antl ? ond dley THis piven by nebo Hed ideo Sh, Al “DO NOT WRITE IN THIS AREA °° men COO MAN zs | 7 4 ;caneado con CamScanner