VICTORIA JUNIOR COLLEGE PROMOTIONAL EXAMINATION MATHEMATICS 970 (HIGHER ) Frida 6 Sept 0 8am -am hours Additioal materials: Aswer Paper List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write our ame ad CT group o all the work ou had i. Write i dark blue or black pe o both sides of the paper. You ma use a soft pecil for a diagrams or graphs. Do ot use staples, paper clips, highlighters, glue or correctio fluid. Aswer all the questios. Give o-eact umerical aswers correct to sigificat figures, or decimal place i the case of agles i degrees, uless a differet level of accurac is specified i the questio. You are epected to use a approved graphig calculator. Usupported aswers from a graphig calculator are allowed uless a questio specificall states otherwise. Where usupported aswers from a graphic calculator are ot allowed i a questio, ou are required to preset the mathematical steps usig mathematical otatios ad ot calculator commads. You are remided of the eed for clear presetatio i our aswers. At the ed of the eamiatio, faste all our work securel together. The umber of marks is give i brackets [ ] at the ed of each questio or part questio. This documet cosists of prited pages VJC 0 VICTORIA JUNIOR COLLEGE [Tur over
9 A curve C has equatio. (i) Fid the equatios of the asmptotes of C. [] Prove, usig a algebraic method, that C caot lie betwee t t < <, where t is a costat to be determied. [] (iii) Sketch C, idicatig its mai features. [] A sequece of real umbers,,, satisfies the recurrece relatio ( + ) + + for. It is give that, ad that λ as. (i) Determie the eact value of λ. [] If >, show that > +. [] (iii) Determie the smallest value of such that + < 0.0. [] I his later ears, the Frech mathematicia Abraham de Moivre oticed that he was gettig more lethargic ad recorded how ma hours he slept dail. He recorded a sleep duratio of 6 hours o the first da. O the eighth da, he recorded a sleep duratio of 7 hours 5 miutes. He suspected that his dail sleep duratios followed a arithmetic progressio. O the ith da, he recorded that he had slept for 8 hours. (i) Does his latest record support his suspicio? Eplai our aswer clearl. [] State, with a reaso, whether it is possible to coclude that his dail sleep duratios followed a arithmetic progressio. [] Assume, as Abraham de Moivre did, that his dail sleep duratios followed a arithmetic progressio. (iii) Fid the total umber of wakig hours he had, from the da he started makig his records util the 8th da. [] (a) Fid ( ) ta d. [] (b) Fid + d. [] + 5 (c) Differetiate si ( ) with respect to. [] Hece fid the eact value of ( ) si d, simplifig our aswer. []
5 The Maclauri s series for sec is give b 5 sec + + +... + (i) Use the above result to fid the series epasio for sec, where,, up to ad icludig the term i. Epress our aswer i the form a + b + ( c + d ) +..., where abcdare,,, costats to be determied. [] 0 Hece, obtai a estimate for the value of sec 0. to decimal places. Suggest oe possible wa to obtai a better estimate. [] r 6 (i) Evaluate ( ) si cos. [] r For 0< < π, π, fid the values of such that r si cos + si r. [] 7 A curve C has parametric equatios t, e t. Fid d i terms of t. Sketch C, showig clearl the feature of the curve at the poit d t 0. [] Fid the equatio of the ormal to the curve at the poit ( 8 p,e p ) aswer i the form, epressig our e p M + N, where M ad N are costats i terms of p. [] This ormal meets the - ad -aes at poits A ad B respectivel. B usig the gradiet of the ormal or otherwise, fid the values of p whe triagle OAB is a isosceles triagle, where O is the origi. [] 8 The fuctio f is defied b f:,,. 5 + 5 (i) Eplai wh both the fuctio f ad composite fuctio f eist. [] Fid f ( ) ad state the rage of f. [] (iii) Determie the solutio of the equatio f( ) f ( ) The fuctio g is defied b g:,,,,. 5 + 5 5 5. [] (iv) (v) Verif that g ( ) f( ). [] 9 Give that k is a iteger, fid g ( k ), givig our aswer i terms of k. [] [Tur Over
9 A curve C has equatio (+ ) 6. Sketch C, idicatig clearl, the coordiates of the aial itercepts, equatios of asmptotes ad a other relevat features. [] Fid the rage of values of m such that there is o itersectio betwee the lie m ad C. [] Aother curve C is defied b the parametric equatios asiθ +, acosθ, where a > 0. Fid the cartesia equatio of C. Hece, fid the rage of values of a such that C itersects C at four distict poits. [] 0 (i) B epressig r i partial fractios, show that. [] r r + Use the method of mathematical iductio to prove the result i part (i). [] (iii) Eplai wh is a coverget series ad state its value. [] r r (iv) Use our aswer to part (iii) to deduce that <. [] r r (a) The diagram shows the graph of. (,8) f( ) O 5 Sketch, o separate diagrams, the graphs of (i) f( ), [] f( ) [] (iii) f '( ). [] (b) Describe precisel a sequece of trasformatios which trasforms the graph of + to the graph of +. []
0 VJC Promotioal Eamiatio Solutios (i) 9 9 ( )( + ) Asmptotes are, ad 0 9 9 + ( ) + 9 0 Discrimiat < 0 ( )(9 ) < 0 ( ) + 6+ 9 6+ 8 < 0 9 0+ 9 < 0 9( ) < 0 + < < t + (iii) 9 0 9 0, (,) 5, (,0) (i) As, λ ad + λ ( + ) λ λ λ + λ + λ λ+ λ λ λ 0 + ± 7 ()() λ ± or From GC, λ
0 VJC Promotioal Eamiatio Solutios ( ) Cosider + + + Sketch the graph of ( ) + + ( ) + + From the graph, for >, ( + ) > 0 --() + for >, + > 0 ( ) + > 0 --() + > + Alterative Cosider ( ) + + + + + + ( )( ) + for >, > 0, > 0 ad + > 0 ( )( ) > 0 + + > 0 > +
0 VJC Promotioal Eamiatio Solutios (iii) From the GC,.786 5.767 6.75 7.79 + 0.099 + 0.07 + 0.007 For < 0.0, + Least 6 (i) (iii) (a) (b) u 6 hrs 60 mi u 8 7 hrs 5 mis 65 mi If his sleep duratio follows a AP, The u8 60 + 7d 65 d 5 u 9 60 + 8(5) 80 8 hrs Hece his latest record supports his suspicio. It is ot possible to coclude AP as there is isufficiet evidece from fiite records. Total umber of sleepig hours i the first 8 das S8 8 [ (6) + (8 )(0.5) ] 6.5 Total umber of wakig hours i the first 8 das (8) 6.5 85.75 ( ) ( ) ta d sec d ta ( ) + C + d + 5 ( ) + 5 d + 5 d+ 5 d + 5 ( ) + l + 5 + 5 ta + C
0 VJC Promotioal Eamiatio Solutios 5 l ( + 5) + ta + C ( ( ) + > 0 for all real values of ) (c) ( si ( )) d d d d si ( ) si ( ) l si l si π π l l 6 l si ( ) l 5(i) sec ( ) 5 ( ) + + +... 0 + + +... 0 0 + + +... + + +... +... ( ) 0 + + ( )... + + 0 + + + ( ) +... + + ( 5 + ( ) ) +... + + ( + ) +... a, b, c, d Let 0, 0. ( ) 0 (0) 0. sec 0. + (0)(0.) + (0 + ) +.... (dp) Use the series epasio for sec [or cos ] up to higher powers of (e.g. 6 )
6 6 8 0 0 VJC Promotioal Eamiatio Solutios 6 (i) r si r cos ( ) cos + si cos + si cos +... + si cos cos si si cos Sum to ifiit, S si cos + si si cos si cos cos cos cos 0 ( ) cos 0 or cos π π, or 0, π (reject sice 0< < π ) π π Sice si <,,. π But whe, si, ad cos 0 r si cos 0 0 0 r + + + si 0 LHS RHS 7 d d d d dt dt t e t d At t 0, d π t, e t (0,) 0
0 VJC Promotioal Eamiatio Solutios At p d e t p, d 8p 8p Gradiet of ormal e p Equatio of ormal at t pis 8 p e 8p e ( ) p p e 8 p + 6 p + e p p B A O A For OAB to be a isosceles triagle, Gradiet of ormal or 8 p e p 8 e 0.0 0.79.08 p 0.0, 0.79,.08 (s.f.) 8 (i) 5 (,0) (0, ) f( ) 5 + k 5 Sice ever horizotal lie k ( k R) f cuts the graph of f( ) eactl oce, f is - f eists. R f (, ) (, ) or \ or 5 5 5 : 5 D f,, 5 5 Sice R f D f f eists. : < or > or 5 5
0 VJC Promotioal Eamiatio Solutios (iii) (iv) (v) f ( ) f 5 + 5 + 5 + 5 + + 0 5 0 + 5 + 9 9 R,, f 5 5 f( ) f ( ) f ( ), 5 g() g 5 + 5 + 5 + 5 + 0 8 0 0 + 0+ 6 6 0 + 6 f( ) 5 + g ( ) f( ) g ( ) f ( ) times 9 k g ( k) g[g g...g ( k)] g( k) 5k + 9 ( ) + ( ) + ( + ) Asmptotes: ( ) +
0 VJC Promotioal Eamiatio Solutios + ± or Cetre: (0, ) Vertices: (, ),(, ) Whe 0, 6() 60 ± 0 ( 0,0) ( ) + (, ) ( 0, ) (, ) ( 0,0 ) Ever lie m passes through (0, -) ad must have a steeper gradiet compared to the asmptotes of C. m or m asi θ () + acos θ () + () () : ( ) ( ) ( ) ( ) a si θ + a cos θ + + + + a For C to itersect C at four distict poits, a > 8 0(i) ( )( ) r r r+ r r+
0 VJC Promotioal Eamiatio Solutios r r r+ r r + 5 +... + + + + Let P be the statemet, r + r Whe, LHS of P () RHS of P () + P is true Assume P k is true for some k, k i.e. r k+ r We wat to show P k+ is true i.e. LHS of P k + k + r r k + r + + ( k + ) + k k + 8 k+ k+ + ( k+ )( k+ ) k+ k 6 + ( k+ )( k+ ) k + k+ k+ ( )( ) r k+ +
0 VJC Promotioal Eamiatio Solutios RHS of P k + P k true P k+ true k + Sice () P is true () P k true P k+ true B mathematical iductio, P is true for all + (iii) As, 0 + Series coverges. r r r r (iv) For all real values of r, r > r < r r < r r r r r < r r r () (show) (a) (i) f( ) (, 8 ) (0,) (,0) (5,0) (, 8 ) (0, )
0 VJC Promotioal Eamiatio Solutios f( ) 0, (, ) (,0 ) 8 5 (iii) f '( ) (,0) 0 (b) + () () + or or () or + The series of trasformatio is. Reflectio i the -ais. Stretch parallel to the -ais, factor.. Traslatio uits i the positive -directio