Mathematics Extension 1

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1 016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved calculators may be used A referece sheet has bee provided Show all ecessary workig i Questios Sectio I Pages marks Attempt Questios 1 10 Allow about 15 miutes for this sectio Sectio II Pages marks Attempt Questios Allow about 1 hour 45 miutes for this sectio

2 Total marks 10 Attempt Questios 1 10 All questios are of equal value Shade your aswers i the appropriate bo i the Multiple Choice aswer sheet provided 1 Which of the followig equatios best represets the graph below? y 1 (A) y si si 1 (B) y si si (C) y l e (D) y l e Let P ad Q be polyomials with real coefficiets Which of the followig statemets is ALWAYS true? (A) If P ad (B) If P0 Q0, the Q have the same set of roots, the P Q P ad Q have the same costat term (C) If P Q has a costat remaider, the P ad Q have the same degree (D) If P is a eve fuctio ad P Q, the Q is a odd fuctio 1

3 3 Let, ad be the real roots of the polyomial P, satisfyig Which of the followig statemets is NT always true? (A) The sum of the roots of P has magitude at least 3 (B) The sum of the squares of the roots of P is at least 3 (C) e of the roots of P is 1 (D) e of the roots of P is the reciprocal of the other root 4 Cosider the biomial epasio of The coefficiet of a b, where is a positive iteger ad ab, 0 is the largest i the biomial epasio Which of the followig statemets is true? (A) (B) (C) (D) a b a b b a b a

4 5 Let p be the probability of success i a trial If the probability of havig k successes i trials is higher tha the probability of havig the same umber of successes i 1 trials, which of the followig is true? (A) (B) (C) (D) k p 1 k p 1 k p k p 6 Let P be a poit that divides the iterval AB eterally i the ratio m:, where the legth of AP is larger tha the legth of BP What ratio does the poit B divide the iterval PA iterally? (A) : m (B) m : (C) m : m (D) m : m 3

5 7 Which of the followig best represets the graph of 1 y ta? 1 1 (A) y (C) y (B) y (D) y 4

6 8 Which of the followig is the correct value of 4 si d? 0 (A) (B) (C) (D) A particle moves i simple harmoic motio with amplitude 05 uits, period ad cetre of motio at 1 Which of the followig is a possible displacemet-time equatio of the particle? (A) t cos 1 (B) (C) (D) si t 1 t t si cos 1 t si 1 5

7 10 Cosider the followig velocity-time graph of a particle v t Which of the followig is a possible velocity-displacemet graph of the same particle? (A) v (C) v (B) v (D) v 6

8 Total marks 90 Attempt Questios All questios are of equal value Aswer each questio i a SEPARATE writig booklet Etra writig booklets are available Questio 11 (15 marks) Use a SEPARATE writig booklet (a) A glucose solutio eters the bloodstream at a costat rate r The huma body metabolises the glucose ad removes it from the bloodstream at a rate proportioal to the cocetratio at the time The cocetratio satisfies the differetial equatio dc r kc dt, where k is positive Let the iitial cocetratio be C 0 (i) Show that the cocetratio at time t is r r kc C e k k 0 kt (ii) Suppose C0 r Describe how Ct chages, as t gets large 1 k (b) Use the substitutio u si cos to fid ta cot d (c) Fid the set of solutios to the iequality Questio 11 cotiues o page 8 7

9 Questio 11 (cotiued) (d) (i) Write dow the geeral solutio for if ta 1 (ii) Hece, or otherwise, show that if ad y are positive where y 1, the y ta ta y ta 1 y (e) Use mathematical iductio to prove that 3 for positive iteger values of 1 1 k k 1 Ed of Questio 11 8

10 Questio 1 (15 marks) Use a SEPARATE writig booklet (a) A iverted coe with semi-vertical agle holds some amout of water 3 r h The volume of the water decreases at a rate proportioal to the eposed area At time t, the radius of the eposed area is r ad the depth of the water from the verte is h Show that the depth is decreasig at a costat rate (b) Suppose 3 Show that ta ta ta ta ta ta 1 Questio 1 cotiues o page 10 9

11 Questio 1 (cotiued) (c) The diagram below shows ABC where P, Q ad R are the midpoits of 3 AB, BC ad AC respectively, show i the diagram below A P R B S Q C Let S be the foot of the perpedicular from A oto BC Prove that PRQS is a cyclic quadrilateral Questio 1 cotiues o page 11 10

12 Questio 1 (cotiued) (d) A particle X with displacemet fuctio t is said to be periodic with period T if t T t The particle X moves i simple harmoic motio about the origi with period ad amplitude A (i) Aother particle Y has the displacemet equatio 3 where b is a costat y t t b, Fid the acceleratio equatio of particle Y i terms of the displacemet yt ad the costats, A ad b (ii) Deduce that if b 0, the particle Y also moves i simple harmoic 1 motio (iii) Show that if b 0, the particle Y is periodic, but does ot move i simple harmoic motio Ed of Questio 1 11

13 Questio 13 (15 marks) Use a SEPARATE writig booklet (a) A particle is projected from a cliff of height h with a fied iitial acute 3 agle from the horizotal ais, with variable iitial speed V y h V R The particle lads o the groud at a acute agle at most 45 horizotal ais from the Show that V gh You may state, without proof, ay relevat equatios of motio Questio 13 cotiues o page 13 1

14 Questio 13 (cotiued) (b) Let P ap, ap ad, Q aq aq be two poits o the parabola 4ay The chord PQ subteds a right agle at the focus S, ad the tagets at P ad Q itersect at T y Q P S M N T Let the feet of the perpediculars from T to the lies PS ad QS be M ad N respectively (i) Fid the coordiates of T (ii) Fid the equatio of the chords PS ad QS (iii) Hece, or otherwise, prove that SNTM is a square 3 Questio 13 cotiues o page 14 13

15 Questio 13 (cotiued) (c) Cosider a sequece of idetical dollar symbols ad m 1 idetical dots 1 arraged i a row The diagram below shows the case for whe 1 ad m 5 $$ $$$ $$$$ $$ $ Eplai why there are m1 ways of arragig the dollar symbols ad dots (d) Travis has idetical cois He distributes the cois amogst his m frieds (i) Use part (c) to eplai why the umber of ways that Travis may distribute the cois so that all his frieds have at least oe coi is 1 m 1 (ii) Hece, or otherwise, simplify the sum m 1 m 1 m 1 m m m1 Ed of Questio 13 14

16 Questio 14 (15 marks) Use a SEPARATE writig booklet (a) Cosider the fuctio f 1 1 It is possible to epress f as the sum f k 1 A k, k where A k is some real umber (i) Show that 1 A k k 1 1 k k 1!! (ii) Hece, or otherwise, simplify Questio 14 cotiues o page 16 15

17 Questio 14 (cotiued) (b) Let g a g be a smooth cotiuous fuctio i the domai : a b, where g b (i) With the aid of a diagram, briefly eplai why there eists 1 0 i such that g 0 0 (ii) Let f be a smooth cotiuous fuctio defied i 1 Defie b a f b f a f a b a g f f a a f a a Use (i) to show that there eists 1 i such that g 1 0 (iii) Hece, show that there eists i the iterval a 1 such that b a f b f a b a f a f Questio 14 cotiues o page 17 16

18 Questio 14 (cotiued) (c) Let f be ay smooth cotiuous fuctio defied over the iterval : a b, with a real root i ad f 0 Defie A f f Let 1,, 3, be a sequece of umbers defied by the recurrece k k 1 A, where k 1,, 3,, with startig poit 0 (i) Use the result i (b) (iii) to show that there eists k i such that k k 1 A k (ii) Hece show that if 0 1 ad k 0, 1,, 3,, the k 1 as k A k for all Questio 14 cotiues o page 18 17

19 Questio 14 (cotiued) (d) Let f 1 l, which has e as a solutio Newto s Method is used to approimate e usig the startig poit 0 71 (i) Let A be defied as i part (c) 1 Show that if 3, the 1 1 A 3 (ii) Use part (c) to fid how may applicatios of Newto s Method 3 are eeded to obtai a approimatio of e that is guarateed to be correct to at least 016 decimal places Ed of Eam 18

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