2007_AJC_Prelims_Paper_2_solutions

2007 H2 Mathematics Prelim Paper 2 solutions No. Solutions 1a () 32 2 13 3 1( l n ) 19 ( l n ) dy dx x x xx ⎛⎞== ⎜⎟+ +⎝⎠ 1b 2 22 33 2 2, 2 1 (2 1)(2 1) 22 22 dx dyttdt t dt dy t t ttdx t t −=+ = − ⎛⎞ −=− = ⎜⎟ −−⎝⎠ At intersection, 3 2 22 0 1 2 131 3 tt x p −=⇒= =+= = 2 23 2 (1 ) (1rr r r+= ++− − ) ))() ( )( ([]∑ ∑ == − + + += + n r r n r r r r rr 11 11 2 2 2 32 () ()[]∑ = + − − + + = n r rrr r r r 1 1 2 1 2 2 1 () ()[]∑ = −+ − − + + = n r rrr r r r 1 11 2 1 2 2 21 where []∑ = − − + + = n r r f r f r f 1 ) 1 ( 2 ) ( ) 1( rr r f2 ) (= (2) (1) 2 (0) (3) (2) 2 (1) (4) (3) 2 (2) .............................. f ff f ff f ff =+− ++− ++− + ................................................. (2 ) (3 ) 2 (4 (1 ) (2 ) 2 (3 ) () ( 1 ) 2 ( 2 ) (1 ) ( ) 2 (1 ) fn fn fn fn fn fn fn fn fn fn fn fn + +− +− − − +− +− − − ++− − − ++ + − − ) ) (1 ) 2 ( ) ( 1 ) 2 ( 0f nf n f f=+ + −− () 112 22 2nnnn +=+ + − ()22 22 2n nn=+ + − ( )24 2 2n n=+ − 2 () 121 2 nn +=+ − Page 1 This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:19:05 GMT -06:00 https://www.coursehero.com/file/5568602/2007-AJC-Prelims-Paper-2-solutions/

Page 2 No. Solutions 1 2 2 ) 3 2 (− = ∑ + r n n r r = ()∑ = + n n r rr 2 2 3 22 1 () () ⎥⎦ ⎤ ⎢⎣ ⎡ + − += ∑∑ = − = n r n r rr rr 2 1 1 1 2 3 2 2 3 22 1 () ( ) ()[] 2 2 1 1 2 2 2 1 42 1 1 2 + + − − − + =+ nn nn ()() () 21 11 41 2 21 2 24 1 22 1 nn nn nn nn − −+ =+ −− ⎡ ⎤=+ − +⎣ ⎦ 3a ∠ AOB = ∠ AOD = 90° O B = 2 O A , O D = O A 3a Minimum |z - w| = EC = 3 2 a 3b 1arg arg zz ⎛⎞ =−⎜⎟⎝⎠ 1sin 22 a a B E D O AC α y 2zi aa−= ww i a= + x 6 παα== ⇒ = arg arg arg26 26az a π ππ⎛⎞ ⎛+ − ≤≤+ +⎜⎟ ⎜⎝⎠ ⎝ π ⎞ ⎟⎠ ⇒ 2arg arg arg33az aπ π+ ≤≤+ ⇒ 21arg arg arg33aa z π π⎛⎞− −≤ ≤ − −⎜⎟⎝⎠ 4a 22 45xy y−−− = () () ()() 22 2 22 22 450 24 5 0 21 0 21 xyy xy xy ⎡⎤−+ + =⎣⎦ ⎡⎤−+− + =⎣ −+ − = −− − = 0 ⎦ 2xy ⇒ A translation of – 2 units in the y – direction. 4bi) This study source was downloaded by 100000851604598 from CourseHero.com on 11-29-2022 21:19:05 GMT -06:00 https://www.coursehero.com/file/5568602/2007-AJC-Prelims-Paper-2-solutions/

Page 3 No. Solutions 4bii) 5i 1Π : 5 41 3 ⎛⎞ ⎜⎟•− =⎜⎟⎜⎟⎝⎠ r y ( )2y fx= − 5 Distance of A from 1Π 65 24 63 15 50 50 ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟•−⎜⎟ ⎜⎟⎜⎟ ⎜⎟−⎝⎠ ⎝⎠=− 41 5 1 1 5050 50 =−= 15 4 50 50 > A and O are on the same side of ⇒ 1Π 5ii Vector parallel to =2Π 56 22 0 86 1 2 −⎛⎞ ⎛⎞⎛⎞ ⎜⎟