RI 9758/02 2022

1 | P a g e Section A: Pure Mathematics Question 1 No. Suggested Solution Remarks for Student    2 2 2 3 3 2 2 2 d2 1d 2 d 2 d2 2 2 d 12 23 2 2 4 23 u x u x uu x x u x u u ux u u u u C x x C                       Question 2 No. Suggested Solution Remarks for Student (a) Volume 72 0.9 1000k h   900 12.572 hk h   (b) 2 2 2 2 2 d d 1 25 2 4 0 when 0 0 25 4 25900 900 4 144 Since 0, 12 V ktt V kt C ht C V t C V ht V h h ht t t t s                  Raffles Institution H2 Mathematics (9758) Solution for 2022 A-Level Paper 2

2 | P a g e (c) 2 2 d 25 12.5 25d 6.25 25 0 when 0 0 6.25 25 10, 900 900 625 250 250 10m275 11 V kt htt V ht t C V t C V ht t t V h h h h                       Question 3 No. Suggested Solution Remarks for Student (a)   i61 2i 52 2 7i i5 6 303 1 2 3 3 3 i 3 12e 1e2 12e 3e2 73, arg 30 z z z z z z z                     Need to specify 3 3 and argz z explicitly. Answer left as 7i303e  is not acceptable. (b) Let A, B and C represent complex numbers z1, z2 and z3 respectively

3 | P a g e (c)     7 7i i30 303 1515 7 3 15 3 3e 3 e 7 3 5 7 9, , , ,30 2 2 2 2 2 7 15, 45,75,105,... smallest positive integer 15 3 3 3 2187 3 105 7arg 30 2 24 4 n nnnz n n n z z                                 

4 | P a g e Question 4 No. Suggested Solution Remarks for Student (a)     2 1 1 1 1 9 3 2 3 2 3 1 3 3 1 3 3 2r r r r r r        (b)    3 3 2 1 1 1 1 9 3 2 3 3 1 3 2 1 1 1 3 3 1 3 2 1 1 3 2 3 5 1 1 3 5 3 8 . . . 1 1 9 4 9 1 1 1 9 1 9 2 1 1 1 3 3 1 9 2 2 1 3 1 9 2 m m r m r mr r r r m m m m m m m m m m m m m m m                                               You are required to combine and simplify the answer as a single fraction. (c)    21 2 1 1 1 1 1 1 1 9 3 2 3 2 3 2 6 3 3 2 1 1 1 1lim9 3 2 6 3 3 2 6 n r nr r r n n r r n                                      (d)   2 2 1 1 1 1 0.0049 3 2 9 3 2 1 0.004 27.1113 3 2 Least 28 n r r r r r r nn n                         Note the correct use of inequalities leading to the answer 28 which has to be an integer.

5 | P a g e Alternatively, n 2 2 1 1 1 1 9 3 2 9 3 2 n r r r r r r                   27 0.004016 > 0.004 28 0.003876 < 0.004 Least n is 28.

6 | P a g e Question 5 No. Suggested Solution Remarks for Student (a) Volume of spherical cap is      2 2 2 2 3 33 3 2 3 3 2 3 2 2 3 2 3 2 d d 1 3 1 1 3 3 2 1 1 3 3 3 1 3 1 33 r r h r r h r r h x y r y y r y y r r r r h r h r r r h r r h rh h rh h h r h                                       