MATHEMATICS 9740/01 Paper 1 14 Sep hours

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1 Cadidate Name: Class: JC PRELIMINARY EXAM Higher MATHEMATICS 9740/0 Paper 4 Sep 06 3 hurs Additial Materials: Cver page Aswer papers List f Frmulae (MF5) READ THESE INSTRUCTIONS FIRST Write yur full ame ad class all the wrk yu had i Write i dark blue r black pe bth sides f the paper Yu may use a sft pecil fr ay diagrams r graphs D t use staples, paper clips, highlighters, glue r crrecti fluid Aswer all the questis Give -exact umerical aswers crrect t 3 sigificat figures, r decimal place i the case f agles i degrees, uless a differet level f accuracy is specified i the questi Yu are expected t use a apprved graphig calculatr Usupprted aswers frm a graphig calculatr are allwed uless a questi specifically states therwise Where usupprted aswers frm a graphig calculatr are t allwed i a questi, yu are required t preset the mathematical steps usig mathematical tatis ad t calculatr cmmads Yu are remided f the eed fr clear presetati i yur aswers At the ed f the examiati, faste all yur wrk securely tgether The umber f marks is give i brackets [ ] at the ed f each questi r part questi This dcumet csists f 6 prited pages

2 The plice wish t crack a 3-digit passcde The sum f the digits is 4 Whe the digits i the umber are reversed, the ew umber becmes 495 mre tha the rigial umber The digit i the tes psiti is 3 mre tha the digit i the hudreds psiti What is the passcde? [4] A C B C Fig Fig A Fig shws a circular card with cetre C A sectr CAB is remved frm the card, ad the remaiig card is flded such that AC ad BC meet withut verlappig t frm a ce, as shw i Fig (A will meet B) Use differetiati t fid the agle ACB exactly such that the vlume f the ce is as large as pssible [6] [It is give that a ce with radius r ad height h has vlume area rl where l is the slat height] 3 rh ad curve surface 4 3 (i) Shw that 4r + r+ 5 ca be expressed as A B +, where A ad B are r+ r+ 5 cstats t be determied [] (ii) Hece, fid a expressi fr (iii) Hece, fid the smallest value f fr which i terms f [3] r= 4r + r+ 5 is at least 99% r= 4r + r+ 5 f its sum t ifiity 06

3 3 4 A curve C has parametric equatis where (i) x a y a 3 3 = cs θ, = si θ, θ ad a is a psitive cstat A pit P lies C Fid, i terms f a, the exact crdiates f P, whse taget is parallel t the lie y = x [4] 3 3 (ii) The taget t C at the pit ( cs, si ) Q a t a t, where 0 < t <, meets the x- ad y-axes at R ad S respectively Fid a cartesia equati f the lcus f the mid-pit f RS as t varies [4] 5 The sum, S, f the first terms f a sequece is give by S 3 = ! 3! 4! ( + )! (i) Fid the values f S, S, S 3 ad S 4 [] (ii) By expressig S i the frm [ f( ) ] fr =,, 3, 4, fid a cjecture fr S i terms f [] (iii) Hece prve by mathematical iducti the result f S fr all psitive itegers [4] 6 Referred t the rigi O, pits A ad B have psiti vectrs a ad b respectively Pit C lies OA prduced such that OA: OC = :5 Pit D is AB, betwee A ad B such that AD : DB = 4: (i) Fid the psiti vectrs OC uuur ad OD uuur, givig yur aswers i terms f a ad b [] (ii) Fid a vectr equati f lie CD [] uuur (iii) Pit E lies CD prduced, ad it is als OB, betwee O ad B Fid OE ad the rati OE: EB 06

4 4 7 Newt s law f clig states that the rate f clig i t miutes is prprtial t the differece betwee the bdy temperature temperaturet T Cad its immediate surrudig kt C Shw that T = T + Ae, where A ad k are psitive cstats [3] Nurul is the chef f a dessert shp ad she leaves her wrk place at 9pm daily Befre she leaves, she is required t ck a big pt f dessert ad leave it t cl, befre placig it i the refrigeratr fr the ext busiess day She takes 30 miutes t ck the pt f dessert t00 C, ad the leaves it t cl After 5 miutes, the pt f dessert cls t 70 C The rm temperature f the kitche is accmmdate items with temperature f at mst 30 C, ad the refrigeratr ca ly 35 C By what time, crrect t the earest miute, must Nurul start t ck the pt f dessert s that she will be able t leave her wrk place time? [5] 8 A li eyes its prey which is k m away ad starts its chase with a leap f 5 m Each subsequet leap f the li is shrter tha its precedig leap by 005 m Its prey tices the li s chase ad rus away with a first leap f 5 m, with each subsequet leap 5% less tha the previus leap Yu may assume that the li ad the prey start ruig at the same mmet ad they cmplete the same umber f leaps after the first leap (i) Fid the ttal distace cvered by the li after leaps [] (ii) Fid the ttal distace cvered by the prey after leaps Deduce that the distace cvered by the prey ca ever be greater tha 30 m [3] (iii) Give k = 5, fid the least umber f leaps the li eeds t take t catch its prey [3] (iv) Assumig that the li ca cver a maximum f 30 leaps, fid the least iteger k, s that the prey will survive the hut 06

5 5 9 (a) (i) θ t If t = ta, shw that siθ = + t [] (ii) Use the substituti θ t = ta t fid the exact value f θ ta + dθ 0 siθ + [5] (b) Fid v e cs3vdv [4] 0 The pit A has crdiates ( 8,,0 ) The plae p has the equati x+ 3y+ z= a, where a is a cstat It is give that p ctais the lie l with equati x z = y = 5 (i) Shw that a = [] (ii) Fid the crdiates f the ft f perpedicular frm the pit A t p [3] (iii) B is give t be a geeral pit l Fid a expressi fr the distace betwee the pit A ad B Hece fid the psiti vectr f B that is earest t A [4] The plaes p ad p 3 have the equatis x+ z = ad x+ by+ z = 4 respectively, where b is a cstat (iv) Give that p ad p 3 itersect at l, shw that l is parallel t the vectr b By fidig a pit that lies bth plaes, fid a vectr equati f l b 06

6 6 i (a) The cmplex umber w is such that w= re θ, where r > 0 ad 0 < θ The cmplex cjugate f w is deted by w * Give that values f w = 3, fid the exact w * r ad θ Hece fid the three smallest psitive iteger fr which w is a real umber [5] 5 (b) The cmplex umber z is such that z i= 0 (i) Fid the mdulus ad argumet f each f the pssible values f z [5] (ii) Tw f these values are z ad z, where < arg z < ad < arg z < Fid the exact value f arg ( z z) i terms f ad illustrate the lcus arg ( z z ) arg ( z z ) = a Argad diagram 06