VJC_H2_MATH_P2_ANS

1 2015 VJC JC2 Prelim Paper 2 Solutions Q1(i) Given  1e sin 2 (1)2 y x   Differentiating with respect to x,  de 2cos 2 (2)d y y xx   Differentiating again with respect to x,   22 2 dde e 4sin 2dd yy yy xxx   22 2 d d 1e e 4 ed d 2 y y y yy xx             22 2 22 2 dde e 4e 2dd dd 4 2e (3)dd y y y y yy xx yy xx         Q1(ii) Differentiating result in (i) with respect to x, 32 32 d d d d2 2e (4)d d d d yy y y y x x x x    At 0x , (1): 11e ln 22 y y     (2):  1 d d 2cos 0 42 d d yy xx   (3):   22 2 22 d 2 d 4 4 16 1dd 2 yy xx     (4):    33 33 dd 2 4 16 2(2)(4) 112dd yy xx     23 23 1 16 112ln 4 ...2 2! 3! 1 56ln 4 8 ...23 y x x x x x x                      Q1(iii)  1ln sin 22yx       3 3 21ln 2 ...2 3! 14ln 2 ...23 xx xx           318ln ln 1 4 ...23 xx      

2 233 3 31 8 1 8 1 8ln 4 4 4 ...2 3 2 3 3 3 x x xx x x                           3 32 23 1 8 1 1ln 4 16 ... 4 ... ...2 3 2 3 1 56ln 4 8 ...23 xx x x x x x               Alternative  1ln sin 22yx                     23 233 33 3 23 23 1ln ln 1 2sin 22 1 1 1ln 2sin 2 2sin 2 2sin 2 ...2 2 3 2214ln 2 2 ... 2 ...2 3! 2 3! 28 2 ... ...3 3! 1 8 8ln 4 2 4 ... 8 ... ...2 3 3 1 56ln 4 8 ...23 x x x x xxxx xx xx x x x x x                                                 Q2) ,,OA OB OC   a a c c Q2(i) 2 OB OXON      1 432 1 522 ON      a c c a ca Q2(ii) Area of triangle OAB = 1 2 OA OB     12 2 1 2 1 02            a a c a a a c a c a a 4  ac   4 4 43 OX OA AX OB AC       a c a ca

3 Area of triangle AXB = 1 2 AB AX     1 42 2 2 24 8           c c a c c c a ac Q2(iii) BXAX BX  is the length of perpendicular from A to BX . Area of triangle AXB 8 1 82 BXAX BX BX  16 16 16 43 16 34 BXAX BX BX OX OB          c a a c ca 16, 3, 4k m n    Q3(i) Q3(ii)  : 1 11aa    Re Im (5, –7) 6 (–1, –7) (11, –7) 2y O (5, –7) 6 Re Im O

4 Q3(iii) 2 4izk   is a circle with radius k and centre  2, 4 Distance between the two centres = 227 3 58 Possible exact values of k = 58 6 and 58 6 For 58 6k  58 6 42 77 58 58 58 6 18 33 58 58 a a b b         42 182 7 4 3 i 58 58 42 185 7 i 58 58 z                        42 18Similarly for 58 6, 5 7 i 58 58 k