ACJC_9758_2022_Promo_Solutions

1 2022 ACJC H2 Math Promo Marking Scheme Qn Solution Remarks 1(i)     3 3 1 2 eln cos ln e ln cos 3ln cos d 1 sin 3d 2 cos 1 3tan 2 x y x xx xx yx xxx x x        Setter: YXF M1 apply laws of logarithm A1 (ii)     1 ln 2 2 1 ln ln ln ln ln 1 d 1 ln 2 lnd d ln (ln 2)d xxyx y x xx y x x y x x xy x x y y x xx       Alternative: 1 ln 2 1 ln ln ln 1 1 d 1 1 ln 2 lnd 1 d 1 ln 2lnd d1 2ln lnd xxyx y x xx y yxx y x x x y yxy x x y y x yxx       B1 apply ln on both sides and apply laws of logarithm M1 apply implicit differentiation A1 OR B1 apply ln on both sides and apply laws of logarithm M1 apply implicit differentiation A1

2 2(i)        2 2 2 8851 10 , 22 51 2 88 10 2 02 51 14 10 20 02 10 31 14 02 10 31 14 02 2 7 5 2 02 272 or 52 xxx x x x x x x x x xx x xx x xx x xx                         Setter: NSH M1 move to LHS and combine M1 factorising the numerator A1 (ii) 88 10 51 or 012 88 10 51 or 01 2 1 2 1 72 or or 052 22No solution. 77 22 77 x xxx xx x xxx x x                   M1 replace x with 1 x A1 3(i) 2 31 ax bx cy x   Substitute ( 1, 4) and ( 3, 2) into equation, 8 ----- (1) 9 3 16 ----- (2) a b c a b c              2 2 2 3 1 3d d 31 ax b x ax bx cy x x        Substitute d3 and 0d yx x  ,     0 6 8 3 9 3 21 3 0 ----- (3) a b a b c a b c           From GC, 1, 0 and 7a b c   . Setter: NSH M1 substituting in ( 1, 4) or ( 3, 2) . M1 d 0d y x  A1

3 (ii)   2 7 31 1 64 3 9 9 3 1 xy x x x       2 2 1 39 3 1 0 7 3 7 3 1 39 1 7 9 x x x x xx x x        2 7 31 xy x   B1 shape B1 vertical asymptote and y-intercept [ECF allowed based on value of c found in part (i).] B1 equation of oblique asymptote 4(i) Setter: YKX B1 equations of asymptotes B1 shape of graph B1 coordinates of axes intercepts (Do not penalise if not in coordinate form.) –6 –4 –2 2 4 6 5 10 x y –6 –4 –2 2 4 6 5 10 x y (0, 7) y x O C x y

4 (ii) B1 equations of asymptotes B1 shape of graph B1 coordinates of axes intercept and turning point (Do not penalise if not in coordinate form.) 5(i) 62 1 3 6 1 32 6 2 10 1 3 2 2 6 1 0 1 6 4 6 0 2 2 2 1 ON OA AN AN ON                                                         Setter: LCH B1: AN M1: 2 10 1 AN   A1 (ii) '2 2 3 1 2 2 6 2 1 1 3 OA ON OA                         