ASRJC FM 2023 Prelim P1 (Solutions)

1 2023 ASRJC H2 FM Prelim Paper 1 Solutions 1 031cos 60 223     031cos 54.7363 3     0 0 060 54.736 5.264      0.09193 3 cos0.0919 sin 0.0919 2.99 0.275ia e i i      2 Using the reflective property of ellipse,             2 2 2 2 ' 2 ' cos 2    c FP F P FP F P          2 2 2 2 ' 2 ' cos 2   c FP F P FP F P         22224 ' 2 ' 1 2sin    c FP F P FP F P           22224 ' 2 ' 4 ' sin    c FP F P FP F P FP F P      2224 ' 4 ' sin   c FP F P FP F P      2224 2 4 ' sin c a FP F P    2 2 2' sin  FP F P a c    2 2' sin  bFP F P , shown

2 3(i) 1 nuy      ddd 1 d 1d d d d 1 nn yyuu n y yx x x x n          23 1 d 1 d1 1 nn yuyy xn x n x  1 23 d 1 1 d nun yx xx  23 d 1 1 d un ux xx    , shown (ii) 23 d 2 1 d u ux xx Integrating factor = 2 2 2 d ee x xx   2 2 2 32 1 1 2e e d e d 2 x x xu x x xxx      22 2 11e e d2 2 xx xx x   22 11e + e24 xx Cx  2 11+e24 xuC x    2 2 11+e24 xyC x      1 2 2 11+ e , 024 xy C y x      4(i) 1sin cos 22r    21sin 1 2sin2   2 31 sin42     maximum value of r is 3 4 . Alternatively,  d cos sin 2 cos 2sin cos cos 1 2sind r               d 0 cos 1 2sin 0d r      cos 0 2    or 1sin 2  5,66 

3 2   , 1 2r  5,66   , 3 4r  maximum value of r is 3 4 . (ii) (iii) 1sin 26    Area of triangle 1 3 3 1 3 3 3tan 32 4 4 6 2 4 4 32                   Area of sector 26 0 1 d2 r    2 6 0 11 sin cos 2 d22      226 0 11 sin cos 2 sin cos 2 d24              6 0 1 1 1 11 cos 2 1 cos 4 sin 3 sin d2 2 8 2            6 0 1 5 4cos 2 cos 4 4sin 3 4sin d16           6 0 1 1 45 2sin 2 sin 4 cos3 4cos16 4 3           1 5 1 2 4 42sin sin cos 4cos 416 6 3 4 3 3 2 6 3                      5 9 1 396 128 6    Required area 3 5 9 13332 96 128 6     3 5 13128 96 6    where 3 5 1,,128 96 6a b c   1 2

4 5 Since   11 2 2 0 1d xx is the area of the quarter circle 21yx with centre at origin and radius 1, therefore     11 22 2 0 111 d 1 . 44I x x     Using Simpson’s Rule,   0 1 2 3 4 222 2 1 4 ( 4 2 4 )3 1 1 1 31 4 1 2 1 4 1 1 112 4 2 4 1 15 3 71 4 2 4 012 16 4 16 0.7708988 0.7709 (4 ) I y y y y y dp                                       From symmetry , area under the curve from 1 2 to 1 is 1 1 1 2 4 2 8 4       1 1 2 22 0 1 1 12 1 d 2 2 8 4 4 2 2x x I           Using Simpson’s rule,   2 2 2 211 2 22 0 1 1 2 3 1421 d 1 4 1 2 1 4 1 13 4 2 4 2 4 2 2 0.642648365 xx                                         12 0.642648365 0.78529673 0.7853 (4 )2I dp    The second method gives a better approximation to the first method because the width of the interval is smaller and so the quadratic approximation for Simpson’s Rule in the second method give a better approximation to the area under the curve.

5 6 (a)   1 (1 birth rate death rate) cull = 1 , nn n uu b d u c         (b)       1 2 2 2 2 3 32 3 1 2 3 1 1 1 1 , ..... ..... 1 1 nn n n n n n n n n n u pu q p pu q q p u pq q p pu q pq q p u p q pq q p u p q p q pq q qp pu p                                    (c) If x is the equilibrium population, then 1nnuu  , 1.08 160 2000 xx x   (d) (i)       11 1 1 1 11 1 1 1.08 2000 1 1.08 1.08 2000 2000 If 2000, then 2000 0 and 1.08 2000 as , ie nn n n n n uu u uu un u                 The mathematical model which we have developed predicted unlimited growth in population. (ii) The most likely scenario is a reducing rate of growth as resources are depleted. 7(a) When 22, 2 ' 2u v a a         When 22, 2 ' 2u v a a         224 dvv u a u du   

6 2 3 4 2 3 2 2 11 24u u au b du u u a a b du          2 2 2 2 2 2 3 2 2 2 2 2 2 2 1 1 1 .2 4 4 1 1 1 4 2 2 4 a a a a u u a a b dv u va v a b dv                             Hence   221( ) 4 16f v v a v a b    (b) 1C :   223 , 2 3 where 0.x t y t t t      2 30 2 3 0 0 2y t t t             223 2 0 3 222 0 3 22 0 dd ddd 6 6 ( 1) 6 1 1 xyrt tt t t t dt t t dt                       dLet w 1 1 d When 0 1 31When 22 ut t tw tw             1 22 1 6 1 1 dr w w w      Or By translating the curve by 1 unit in the negative direction of the t-axis,   3 22 0 6 1 1t t dt   1 22 1 6 1 1 dttt       1 22 1 6 1 1 dw w w     2C : 333 3 , 3x t t y t   22dd 1,dd xy tttt  

7   223 2 0 3 3 2242 0 3 3 4 22 0 3 3 4 22 0 dd2d dd 21 3 2 2 2 13 2 2 1 32 xys y t tt t t t dt t t t dt t t t dt                                    Using part (a),     1 22 1 1 22 1 2 2 1 1 1 2 d3 16 2 1 1 dw24 2 .24 6 2 144 s w w w ww r r                   8(i) sinxL , cosyL dd cosdd x Ltt , d dsindd y Ltt 222 22 d d d cos sin ddd    x LL ttt , 22 2 22 d ddsin cos ddd    y LL ttt , shown (ii) 22 2 ddsin cos sin dd          T m L L tt ----- (1) Applying Newton’s 2nd law, 22 2 ddcos sin cos dd          T mg m L L tt 22 2 ddcos sin cos ddT mg mL tt           ----- (2)     22 2 22 2 ddsin cos2 cos dd 1 sin ddcos sindd         g L L tt LL tt

8 2222 22 22 d d d dcos cos sin sin sin cos sind d d d                          L L g L L t t t t   2 22 2 dcos sin sin d     Lg t   2 22 2 d sin cos sin 1d       g tL If  is small, sin 2 2 d d    g tL , shown (iii) Characteristic equation: 2 0   ggm m iLL sin cos   ggA t B t LL where A,B are arbitrary constants 0 12   12  B d cos sind   g g g gA t B tt L L L L d d 12  g tL ,  Lt g 12   gg ALL 12 A sin cos12      ggttLL 2 sin12 4    g tL Maximum 2 12   since 1 sin 1 4     g tL (iv)  d 1d  t  d sin 2d   g tL 00 1 0 0 , d dh t            00 1 0 0 0 , d 0 1 sin sind ggh t L L          