ACSI 2022 Promo Paper 1 ans

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 1 / Solutions 1 FINAL EXAMINATION 2022 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS : ANALYSIS AND APPROACHES HIGHER LEVEL Paper 1 Qn Solution Marker’s Comments 1. 9 12 3 27 0xx−  + = 23 12 3 27 0xx−  + = Let 3xy= 2 12 27 0yy− + = ( 9)( 3) 0yy− − = 93y or= 21x or= General well done with some careless mistakes.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 1 / Solutions 2 2. 33 2 9 9log 1 log lg lg 2lg 9 125 5xy= + = − + 2 9 9 2 125 81lg lg 2lg lg9 125 5 9 9 25 − + =  lg10 1== Therefore 3log 1 3xx=  = 331 log 1 log 0yy+ =  = Hence 1y= Many students made mistake when trying to simplify the log expression on the RHS. 3. (a) 32ln ln 3ln 2lnx x x x− = − = ln x 2ln ln 2ln lnx x x x− = − = ln x Since the common difference is the same, they form 3 consecutive terms in AP Students are required to show. But many simply find the difference and did not explain why are they finding the difference. A handful find the general terms when the question clearly state that there are only three terms.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 1 / Solutions 3 (b) 10 10 1 10ln (ln ln )2 k k x x x = =+ 5(11ln )x= 55ln 55 ln 1 x x = = xe= Some students used GP formula instead of AP. Did not realise that (a) and (b) are related. Some treat x as the dummy variable.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 1 / Solutions 4 4. (a) General term 26 6 12 3 6 ( ) ( ) 6 () rr rr px xr pxr −− −− = = Constant term: Power of x is 0 12 3 0r − = 4r= Term = 226 154 pp = Generally well done. Most errors are due to wrong computation of 6C4 or 6C2 (b) 215 60p = 22p or=− Generally well done. Most errors are due to carelessness. 5. (a) Let ( ) 32 99f x x x x= − − + ( )10f = Therefore, by factor theorem, 1x− is a factor of ( )fx . Let ( )( ) 3 2 2 9 9 1x x x x ax bx c− − + = − + + By inspection, 1a= and 9c=− Comparing coefficient of 2x : 1 ba− = − 1 1 1 0ba=− + =− + = ( )( ) ( )( )( ) 3 2 2 9 9 1 9 1 3 3x x x x x x x x− − + = − − = − − + Generally well done. (b) 32 99 02 x x x x − − + − ( )( )( )1 3 3 02 x x x x − − + − using answer from (a) 31 x−   and 23 x Many students could not work with equations with inequalities. This is an area for further work. –3 1 2 3

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2022 Final Exam / Paper 1 / Solutions 5 6.