2a a a 2a 4a. 3a/2 f(x) dx a/2 = 6i) Equation of plane OAB is r = λa + µb. Since C lies on the plane OAB, c can be expressed as c = λa +

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Francis Atkinson
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1 / GCE A Level H Mths Nov Pper i) z + z z 9 From GC, poit of itersectio ( 8, 9 6, 5 ). z + z From GC, there is o solutio. So p, q, r hve o commo poits of itersectio. Sice oe of the ples re prllel to ech other, the ples form trigulr prism. ) ( ) + + For this qudrtic equtio to hve solutio, ( ) ( + ) or or + i) (, ) (, ) / Let + + solutio is < or >. i) w ( + i) + (i) + (i) + (i) + 6i + ( ) 8i i ( i) + 5( + i ) + 7( + i) + b + + b + i( + 5) Comprig comple prts: Comprig rel prts: + + b b 95 (iii) Sice + i is root, so i is lso root. [ z ( + i) ] [ z ( i) ] z z + 5 B comprig coefficiets, 7z + 5z + 7z + 95 (z z + 5)(7z + 59) The roots re ± i d i) / f() d / π/ si θ cos θ dθ π/6 π/ cos θ dθ π/6 π/ + cos θ dθ π/6 si θ [ θ + ] π/ π/6 [ π + π 6 ] π 6i) Equtio of ple OAB is r λ + µb. Sice C lies o the ple OAB, c c be epressed s c λ + µb. ON + c 7 (iii) Are of ONC + c c 7 7 c 7 (λ + µb) µ 7 b 6 Are of OMC b c b c b (λ + µb) λ b µ 7 b λ b λ 8 7 µ 7i) p 8 l p l 8 + ( )l 7 l + ( ) l ( ) l ( + 6) l + ( + ) l A, B 6, C, D. S 8 8 totl legth cot be >8 cm. (iii) S 8 8 [ ] > 8 < 96 log 96 > log.6 pieces must be cut off. Method :

2 From GC, pieces must be cut off. 8i) w i z r rg w rg ( i ) + rg z r π + θ locus of z k+ (k + )[(k+) + k + ] k+ (k + )(k + k + ). LHS k r(r +) + (k+)[(k+) +] r k (k + )(k + k + ) + (k+)[(k+) + ] k+ [k +k +k+ (k +k+)+] k+ (k + 5k + 9k + 6) k+ (k + )(k + k + ) RHS Sice P() is true, d P(k) is true P(k+) is true, b Mth Iductio, P() is true for ll π/6 Z +. π/ r f(r) f(r ) r + r + r + [(r ) + (iii) locus of w (r ) + r + ] r + r + r + [(r rg z rg w π θ θ π r +r ) + (r r+) + r + ] r + r + r + [r r + r + ] 8θ π 6r θ π 6 r [ f(r) f(r )] r r f() f() 9) Let P() be the + f() f() sttemet r(r +) : : r + f() f( ) (+)( + +). Whe : LHS ( + ) RHS r ()() LHS r P() is true. Assume tht P(k) is true for f() f() ( + + ) 6 ( + + ) some k Z +, i.e. ( + )( + ) k r(r +) k 6 (k+)(k + k +). r (iii) f(r) To prove tht P(k+) is lso true, r k+ i.e. r(r + ) ( ) r r r(r + ) + r + r r r (+)( + +) + (+)( + ) + i) z dz d l ( z) + C l ( z) C z e C z e C z e e C + Ae where A e C (iii) d d + Ae A e + D d d Ae ( d d ) d d +, b (iv) Let A. members of the fmil re d +. Let A, D : + e. + e i) d d 6t, dt dt 6t d d 6t 6t t Equtio of tget is t t( t ) t t + t t t q q : p p q q Equte p p with

3 (p q) p q (p q)(p + pq + q ) p q p + pq + q p(p + pq + q ) p p(p + pq) pq(p + q) + p q (p + q) + (p + q) + p + pq + q + p + pq + q pq p + q + pq R lies o the curve +. (iii) Substitute t, t ito + : t (t ) + t 6 + t 6 t + (t + )(t t + ) (t + )(t ) t t sice >,. M (, ). (iv) Are / d / d / t ( )/ 6t dt [ / ] / / t dt [ ] [ t5 5 ] / 5 5 GCE A Level H Mths Nov Pper i) R g R / R\{} D f fg does ot eist. gf() g ( + ) + Let (gf) (5) gf() (i) Legth of side of bse t / Volume of prism, V ( ) si 6 ( ) dv d ( ) + ( )( ) ( )( ) ( )( 6 ) or 6 d V d ( )( 6 ) ( ) ( ) 6( ) Whe, d V >. d Whe 6, d V <. d mimum V 6 ( ) ( ) 5 cos (i) f '() + si f "() (+ si )( si ) ( cos ) ( + si ) si si cos ( + si ) si ( + si ) f '"() (+ si ) ( cos ) + ( si + )(+ si ) cos ( + si ) f() f '() f "() f '"() + 6 f() e si ( )( ) , Third o zero term 6 + ( )( 6 )

69 (5i) Prit the mes of ll emploees o slips of pper d put i bo. Rdoml pick 9 mes from the bo. There m be too m represettives from some coutries, d oe from some coutries. Use strtified smplig.

Use rdom smplig to pick required umber of emploees from ech coutr. (6) P(Y < ).95 P(Z < µ σ ).95 µ σ.685 µ.685σ () P(Y < ).5 P(Z < µ σ ).5 µ σ.679 µ.679σ () () ().9σ µ +.679.9.9 F ~ B(, ) P(F ).

4 6 (i) cos θ θ. + z z From GC, equtio of lie l is r /6 / 7/6 + λ 5/, λ R (iii) Distce from A to p c 9 + c Distce from A to p 6 + c 9 c 7 + c c 7 9( + c) 9(c ) 9( + c + c ) 9(c 56c + 96) c + 6c 75 c 9,.69 (5i) Prit the mes of ll emploees o slips of pper d put i bo. Rdoml pick 9 mes from the bo. There m be too m represettives from some coutries, d oe from some coutries. Use strtified smplig. The umber of emploees from ech coutr to be ivited to the prt is proportiol to the umber of emploees i tht coutr. Eg, if Sigpore hs emploees, the ivite 9 emploees from Sigpore. Use rdom smplig to pick required umber of emploees from ech coutr. (6) P(Y < ).95 P(Z < µ σ ).95 µ σ.685 µ.685σ () P(Y < ).5 P(Z < µ σ ).5 µ σ.679 µ.679σ () () ().9σ µ F ~ B(, ) P(F ).77 (iii) F ~ B(6, ) Sice 6 > 5 d p < 5, F ~ Po() pproimtel. P(F 5) P(F ).85 P(B A') (8i) P(B A') P(A') P(B A').8.. P(A') P(A' B) + P(A' B') P(A' B')...6 P(A B') (iii) P(A B') P(B') P(A B').88 P(A B') + P(A' B').88 P(A B') P(A B'). P(A B').58 P(A B'). P(A) P(A B) + P(A B').7 P(A B) +. P(A B).6 (9i) (7i) The probbilit of pickig pcket cotiig free gift is costt. Whether pcket cotis free gift or ot is idepedet of other pckets. From GC, ubised estimte of µ.8 d ubised estimte of σ We ssume tht. is good estimte of the ukow the popultio vrice. H : µ.8 H : µ <.8

Sice p vlue.5 >.5, we do ot reject H. There is isufficiet evidece t 5% sigificce level to s tht the distce trvelled per litre of fuel is too high.

of ws 6 P 9 seem to lie close to curve with egtive grdiet d cocve Cse : sme letters d dowwrds, model A is the most sme digits pproprite. No. of ws 6 9 Probbilit + + 6 6 6 9 9.

99 (iv) From GC, regressio lie is 89.78.698 Distce trvelled 89.78.698 ( ).8 km (i) 6 5 9 8 6 6 6 9 9.789 P(sme digits) 9 9 (i) The me o. of bsetees per uit time is costt.

5 Sice p vlue.5 >.5, we do ot reject H. There is isufficiet evidece t 5% sigificce level to s tht the distce trvelled per litre of fuel is too high. (i) A: + b B: c + d l 8 9 (iii) Cse : ectl sme letters d differet digits No. of ws 6 5!! 9 P 7 Cse : ll letters differet d 88 8 sme digits (iii) Sice the dt poits No. of ws 6 P 9 seem to lie close to curve with egtive grdiet d cocve Cse : sme letters d dowwrds, model A is the most sme digits pproprite. No. of ws 6 9 Probbilit (iv) Cse : differet cosots, vowel d eve digit No. of ws C 5 C! C 5 C! 5 Cse : sme cosots, vowel d eve digit No. of ws C 5 C!! C 5 C! 6 Probbilit C: e + f From GC, r.99 (iv) From GC, regressio lie is Distce trvelled ( ).8 km (i) P(sme digits) 9 9 (i) The me o. of bsetees per uit time is costt. The probbilit of emploee beig bset is idepedet of other emploees. The me o. of bsetees m be higher durig epidemic. If emploee is ill, his illess m be cotgious d will icrese the probbilit of other emploees becomig ill. Let A o. of bsetees from Admi Dept o ds. A ~ Po(.) P(A ) e. <.. < l. >.8 Smllest o. of ds 5

6 (iii) Let T o. of ds of bsece from the depts i 5 ds T ~ Po( (. +.7)5 ) Po(9.5) P(T > ) P(T ).97 (iv) Let S o. of ds of bsece from the depts i 6 ds. S ~ Po( (. +.7)6 ) Po() Sice λ >, S ~ N(, ) pproimtel. P( S 5) P(99.5 S 5.5).88 6

Important Facts You Need To Know/Review:

Important Facts You Need To Know/Review: Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t