2023 VJC Promo (Soln)

2023 VJC H2 Math Promo Solutions 1 (a)     1 12 2 2 2 2 2 2 2 d1 1tan 1 1 2d2 11 1 11 1 1 1 x xxx x x x x xx          (b)  cos cos ln cos ln 1d 1 cos ln sind dc o s sin lnd x x yx yx x y x xxyx x yx x xxxx      

2 (1)   replaced by ff yyyx y x     (2)   1 replaced by 2 1ff 2 yy yx y x    (b) f1yx . (1)    replaced by 1ff 1 xxyx yx     (2)   replaced by f1 f 1 xxyx y x     

3             2 2 2 2 11 21 01 2 01 2 01 17 24 01 ax a xa x ax a x a x xa x ax a x a x ax xa x xx xa x x xa x                 Since 2 17 7 024 4x    for all x  ,   10xa x  . 1 or xx a For   2 11 ax a xa x   , replace x with x in the result above to get the following. 1x  or xa 11 x  or xa or xa  : or 1 1 or xx a x x a    

4 tan 2 tand es e c ed x xxx     4t a n 22 t a n 2t a n t a n 2t a n t a n 2 2 tan tan tan 2 tan 2 tan tan tan 2 sec e d sec sec e d sec e e 2sec sec tan d sec e 2 e sec tan d sec e 2 e tan e sec d es e c 2 et a n2 e e sec 2 tan 2 x x xx xx xx x xx x x xx xxx xx x x x xx x x x xx x xx C xx C                     

5  Replace with 22 2 2 2 2Replace with 222 2 44 4 42 4 yy xx xy x y xy x y              (1) Scale the graph of C by a factor of 2, parallel to the y-axis, followed by (2) translating the resultant graph by 2 units in the positive x-direction.

6  22 22 21 dd 54 54 d 54 92 2 3 n 24 2 1 25 4 s i 42 1 3 xx xx xx x xx x xx xC xx x                    43 4 22 3 3422 23 d 63 33 d 3 d 3322 99 9132 81 2 922 2 xxx x x x xx xx                              3 22 22 2 1 d 1 x x x                                      dsin cos d xx uu u  π π π ππ π 23 2 4 π 23 4 π 23 4 ππ 33 44 π 3 π 4 1s i n 1s i n 1s i n d 1s i n d 1 d d cos d coscos ππcos cos 0cos 4 3 1c o s 2 3c o s 2 22 3s i n 2 24 3 ππ 12 ππsin sin23 4 4 3 2 π 31 884 u u uu u uu u u uu u uu uu uu uu                                

7 Since all the coefficients are real, 12 i is also a root.     2 2 Quadratic factor = 1 2 i 1 2 i 12 1 23 xx x xx       Let  32 233 2 3x px qx x x ax b     .  By inspection or by comparing coefficient of 3x and constant term, 3a and 1b .     32 233 2 3 3 1xp xq x x x x     By comparing coefficients, 16 5 29 7 p q        32 233 2 3 3 1 0xp xq x x x x      Other than 12 i , the other roots are 12 i and 1 3 . Alternative          32 3 1 2i 1 2i 1 2i 3 0 3 132 i 6 22 i 1 22 i 2 1 2 i 3 0 12 3 2 2i 0 pq pq pq pq                Comparing real and imaginary parts, 12 0 12 5, 732 0 2 3 pq pq pqpq pq            . Since all the coefficients are real, 12 i is also a root.     2 2 Quadratic factor = 1 2 i 1 2 i 12 1 23 xx x xx         32 2 357 3 0 23 310 112 i o r 3 xxx xx x xx          Other than 12 i , the other roots are 12 i and 1 3 .

*2i *2iwz z w       2 2i 3i 2i 3i 0 ww ww      2 2i 2i 4 3i 2 2i 44 i1 1 24 i 2 2i 9 2 2i3 i 2 w            12 iw or 1iw When 12 iw : *2i 1 2 i 1 i 1i z z     (reject since it’s given that wz ) When 1iw : *2i 1 i 1 2 i 12 i z z    Hence, 1iw and 12 iz .

8 ln 3 2.25 1.5 ln 5.5 42 l n 9 ab c ab c ab c       By GC, 0.227a , 1.78b and 0.455c (to 3 s.f.).     2322 2 2 2 4 6 11 8fl n 1 2 2 2 2 2 2 23 3xx x x xx x x      The expansion is valid when 212 1 x  .    22 2 2 1 2 0 for all 21 0 21 210 xx x x xx       11: 22 xx     22 4 6 8ln 1 2 2 2 3x xx x   Substitute 2 1 6x  : 23 11 1 8 1ln 1 2 2 266 6 3 6 44 7ln 31 6 2                47m       2 222 2 234 24 2 3 4 12ln ln 1 2 ln 1 ln 1 2 2ln 1 12 2322 2 2 3 234 3 2 x xx x x xx xxxxx x x x x x                      Alternative           2 22 2 2322 4 24 2 3423 4 2 4 2 4 234 12ln ln 1 2 ln 1 2 12 22 222 2 . . . 23 4 83 444 2 3222 4 2 3 23 3 2 x xx x xx xx xx xxx x x xxxx xxxx x x x x x x                       

9 5πi63i2 e    and πi3ππcos i sin e33   555 2 32** 1        5 5ππ 27π 9π5arg arg * 5 63 6 2 πarg *2               5 ππ32 cos isin*2 2     Alternative 5πi63i2 e    and πi3ππcos i sin e33   55πi6 5 πi3 25πi6 πi3 πi 25π 25π6 ii 4 πi66 πi3 πi2 2e * e 32e e 32e ee e 32e ππ32 cos isin22                