Anglo-Chnese Junor College H Mathematcs 97 8 JC PRELIM PAPER Solutons ( ( + + + + + 8 range of valdty: > or < but snce number, reject < wll result n the sq rt of a negatve <. range of valdty: >. no. of terms n the sum: n + n sum + ( m d ( m n d + + + + n + + ( m+ n d Method ( cos e cos d e + ( d e cos e d dv I e cos d u e, cos d sn du sn e, sn e d e v d sn dv e ( e cos + e cos d u e, sn d du e + e I e v d 5 sn I e + e cos sn I e + e cos 5 5 sn cos, cos Thus, e ( cos d e sn e cos + e + C 5 5 sn cos e + e e + C 5
Method ( e cos d e cos d e d dv e cos d e, u cos d du v e, sn d dv e cos + e sn d u e, sn d du e, v cos d cos cos dv e e + e cos d u e, cos d du e, v sn d cos cos e e + ( e sn e sn d cos cos sn e e + e e sn d Let I e sn d Then, e cos + I e cos e cos + e sn I 5 sn I e e cos sn I e e cos 5 5 cos sn e d e e cos e + C 5 5 5 5 5 :,,,, 8 5 5 5 y : 5,,,, Thus, ( : GP wth a and y: GP wth a 5 and r r : S 8, y 5 : S + + the ant wll eventually end up at ( 8,.
5 For, ( ( ( ( + (N.A. or + Hence, +, ( + Let u +, uu B u + +, u + B ALTERNATIVE: For < <, ( + ( No real soluton ecept OTHERWISE sketch of y ( + hence,, ln(cos θ, y ln(sn θ d snθ dy cosθ dθ cosθ dθ snθ dy cosθ cosθ d snθ snθ cos θ sn θ dy when θ, d y ln ln The equaton of tangent s y ln. If ths tangent meets the curve agan, ln(sn θ ln(cos θ ln ln(sn θ sn θ θ θ < θ < The tangent wll not meet the curve agan as there s only one soluton n range < θ <.e. θ.
7 By Newton s Law of Coolng, dθ k ( θ dt dθ kdt θ ln θ kt + C Method Gven: when t t, θ A When t t +, θ 5 B From : ln kt + C, ln 5 kt + k + C ln 5 k k ln 5 8 Also gven: t t, θ? t t +, θ 5 C From & : ln θ 5 ( ln 8 t+ C, ln 5 ( ln 5 5 t+ ( ln 8 8 + C ln 5 ln 5 [Modulus can be removed as ntal θ 8 temperature s hgher than 5 C, hence θ >.] θ 5 ( + 8 C 5 8 Method Gven: t, θ A t, θ 5 B From : ln C, ln 5 k+ C ln 5 k k ln 5 8 Also gven: t, θ? t, θ 5 C From & : ln θ 5 ( ln 8 t+ C, ln 5 5 ( ln 8 + C ln 5 ln 5 [Modulus can be removed as ntal θ 8 temperature s hgher than 5 C, hence θ >.] 5 5 ( θ 8 θ 5 ( + 8 C 5 8
8 a, b Asymptotes y, Aal ntersecton : (,, (/, Asymptotes y ±, Aal ntersecton : (,, (,- y + I : A translaton of unt n drecton of the postve as II : A scalng parallel to the as wth scale factor unts III: A translaton of unt n drecton of the postve y as OR I : A translaton of unt n drecton of the postve y as II : A translaton of unt n drecton of the postve as III : A scalng parallel to the as wth scale factor unts OR I : A translaton of unt n drecton of the postve as II : A translaton of unt n drecton of the postve y as III: A scalng parallel to the as wth scale factor unts OR I : A scalng parallel to the as wth scale factor unts II : A translaton of unt n drecton of the postve as III: A translaton of unt n drecton of the postve y as OR I : A scalng parallel to the as wth scale factor unts II : A translaton of unt n drecton of the postve y as III: A translaton of unt n drecton of the postve as OR I : A translaton of unt n drecton of the postve y as II: A scalng parallel to the as wth scale factor unts III :A translaton of unt n drecton of the postve as
9 y sn ( dff. w.r.t, dy. d dy d d y dy + d d ( ( ( d y dy d d dff. w.r.t, d y d y d y dy + d d + d d d y d y dy ( QED d d d dff. w.r.t, ( ( d y d y d y 5 d d d dff. w.r.t, 5 d y d y d y 7 9 5 d d d f(, f '(, f ''(, f '''(, f ''''(, f '''''( 9 5 y + + +... d sn d 5 d... d + + + + + +... 8 Usng, + + +... 8 + + +... 8 5 7
(a (b Area d Let sn θ d cos θ dθ When, θ ;, θ. Substtutng, Area sn sn θ θ cos θdθ 8 sn θcos θdθ sn θ dθ cos θ dθ θ sn θ 8 y e y ln y e From G.C., appromate coordnates of the ponts of ntersecton of the.875,.87. two curves are (.759,.55 and (.875 ( y ln Volume ln e d.7 ( s.f..759 arg( z + arg[ z ( ] arg+ arg( z arg( z N P(, X A Q(, NPQ ˆ + 5 PQ + sn 5 NQ NQ. ( sg fg m., m
( ( A symptotes y + a, dy d a ( dy For statonary ponts, d a ( ± a w hen + a, y + 5a when ay, a ( k w w e, k,, ( z+ + ( z ( z+ ( z ( ( z z+ z+ z+ k e z+ ( + e z + e ( + cos + sn + cos + sn + sn + sn cos + cos + sn cos ( ( ( ( k ( k ( + sn cos + sn cos cos sn tan
ALTERNATIVE: k e ( + z + e ( + e e. + e e ( e + e e + e (sn cos tan ( ( ( z z z( z ( z+ + ( z z + z z + z z z + z, ± Snce z tan s negatve tan. ( ( BOC Area of. cos 7.5. ( d.p 9 OBC OB OC sn 7.5 8sn ( 7.5. unts Δ ( ( t t CP + t 7+ t t t Snce, CP. CB CB, CB 9 5 t 7+ t. 9 t 5
t+ 8 or t+ 8 t or (.7 (v (v PA t AB Snce PA t AB, hence, P, A and B are collnear for all values of t. ( [OR PB ( t ( AB, OR PB + ( PA ] t Δ OBC and Δ OPC share the same base of OC. Snce OC // BP, Δ OBC and ΔOPC also has the same perpendcular heght. Area of ΔOPC Area of Δ OBC. unts.