SJI 2025 AMATH PRELIM P2 MS

ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2025 (YEAR 4) ADDITIONAL MATHEMATICS 4049/02 Qn Solutions 1(a) 2 12 5(2 ) 3x x   Let 2xu 22 5 3 0u u   1(b) (2 1)( 3) 0u u   1 2u or 3u  2 0x  therefore 12 2 x   2 3x  lg 2 lg 3x  lg3 lg 2x = 1.6 (2 sf) 2(a)                3 3 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1x x x x x x x x                  =   2 2 22 4 4 1 4 1 4 4 1x x x x x         =   22 12 1x  2(b)    22 2 1 12 1 0x x   212 1 0x   or 2 1 0x     2Discriminant 0 4 12 1 = 48 0     No real roots 1 2x is the only real root.

ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2025 (YEAR 4) ADDITIONAL MATHEMATICS 4049/02 3 (a) ( 3) 1y x x       1 2 1(1) 1 ( 3) 1 1 2 dy x x xdx           31 2 1 xx x     2( 1) 3 2 1 x x x     2 2 3 2 1 x x x     3 1 2 1 x x   3(b) ∫ 3𝑥 − 1 2√𝑥 + 1 3 0 𝑑𝑥 = [(𝑥 − 3)√𝑥 + 1]0 3 3 33 0 00 3 1 1 16 d = 6 ( 3) 1 d 2 1 2 1 2 1 x x x x x x x x                       3 33 0 00 9 1 d = 6 ( 3) 1 d 1 2 1 x x x x x x x                       3 1/2 0 1 12 = 6 (3 3) 3 1 (0 3) 0 1 1 2 x                           1/2 1/2 = 6 (3) 3 1 0 1        = 6 (3) 2 1    = 24

ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2025 (YEAR 4) ADDITIONAL MATHEMATICS 4049/02 4(a) Let E be a point below C (and to the left of D) In , cos 80 CECDE   80cosCE  Let G be a point below D (and to the right of A) In , sin 185 DGADG   185sinDG  h CE EF  185sin +80cos (shown)h   4(b) 185 sin +80 cosh   185sin +80cos sin( )R     sin 80 cos 185 R R     2 280 185 40625 25 65 R    1 80tan 185 23.385          25 65 sin( 23.4 )h     4(c) To clear the kitchen door, max h is 180cm, When h = 180cm 25 65 sin( 23.385 ) 180    180sin( 23.385 ) 25 65     36sin( 23.385 ) 5 65     Basic angle =63.259 23.385 63.259      39.9  

ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2025 (YEAR 4) ADDITIONAL MATHEMATICS 4049/02 5(a) 22 5y x  2 22 5 3x x  2 5 0x   ( 5)( 5) 0x x   5 5 x   5(b) 22 5y x  ---- (1) 3y kx  3y kx  ---- (2) 22 5 3x kx   22 2 0x kx   2 4 0b ac  2( ) 4(2)(2) 0k   2 16k  4k   5(c) 22 4 2 0x x   2 2 1