2011 HSC Mathematics Extension 1 Solutions

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1 0 HSC Mathmatics Etsio Solutios Qustio, (a) A B 9, (b) : 9, P 5 0, 5 5 7, si cos si d d by th quotit ul si (c) 0 si cos si si cos si 0 0 () I u du d u cos d u.du cos (f) f l Now 0 fo all l l fo all Rag is f (d) 0, I Lt d u du d d du 0 u Qustio (a) P a Now P a 7 9a 9a a P P th maid is Cambidg Uivsity Pss - -

2 (b) f cos (c) f si f 0 0 f A gal tm of 0 0 cos si (d..) is k k k k k k k Fo th tm i : k k k th tm i 5 th cofficit is: (d) f cos f cos Domai: f Rag: 0 0 f y () (i) # aagmts = 0! 5 - (ii) # aagmts of th favouits =! Qustio (a) # aagmts of oths = (0 )! =7! total aagmts =!7! d (i) Acos t Bsi t d Asi t Bcos t d A cos t B si t Acos tbsi t as quid (ii) t 0, 0, d Substitutig ito : Acos t Bsi t 0 Acos0 Bsi0 A 0 Substitutig ito d : d Asi tbcos t Asi0Bcos0 B B A0 ad B (iii) > 0 th aticl movs to th ight d Now cos t d Th aticl stos wh 0 cost 0 cos t 0 t t Cambidg Uivsity Pss - -

3 (iv) sit Wh t : si si amlitud = (b) (i) Tim fo o oscillatio distac tavlld = = y dy d dy At Ptt, : t d tagt at P: yt t t yt tt y t t as quid (ii) Tagt at Q: y t t (iii) Tagts mt wh t t t t t t t t t t t tt Substitutig ito th tagt at P y t t tt, R tt (iv) Th locus of R is th li But tt tt t Qustio th maimum valu of tt i.. tt This maimum occus wh t But PQ t tt th locus of R is th at of th li wh y (a) f f (i) (ii) Th maimum will occu wh f o But 0 has o solutio l l l Now f l th maimum tuig oit occus at l, (iii) f l 0 l l (iv) As, 0 ad 0 f 0 (v) Wh 0 0 0: f 0 th y-itct is Cambidg Uivsity Pss - -

4 (vi) y 0.5 SP But 0 y0 Q cos, si l l cos ad si y cos SP si (b)(i) AOC Agls at th ct ad cicumfc stadig o th ac AC. (ii) DAB ABD (bas agls of isos. ABD) CDA (tio agl of ABD) Now AOC CDA ad ths stad o th sam sid of AC ACDO is cyclic (covs of agls i th sam sgmt) (iii) I th cicl with ct O: OM AC (joi of ct to midoit of chod) I th cicl with ct P: PM AC (joi of ct to midoit of chod) (ii) (iii) cos cos si si si si cos si si cos si cos si cos si cos cos sc ta as quid NOQ (adjact agls) NOQ is isoscls with ON OQ ONQ OQN ONQ (agl sum of NOQ) ONQ ONQ SNP (sam agl) Qustio 5 O, P ad M a all o th dicula to AC though M O, P ad M a collia. (a) (i) Giv TQN SPN TQ TN (matchig sids of simila s) SP SN 0 y0 SP SP (iv) I SNP : ta SNP cos ta si cos si sc ta fom (ii) (v) sc ta cos si fom (ii) ta fom (iv) Cambidg Uivsity Pss - -

5 Now 0 (b) (i) T 5 5 kt (ii) Wh t =, T = 0 k k 5 5 k 5 k l 5 5 k l kt 5 T AB ; k l ; A Lt 0 am b wh t = 0 ad T = B B T kt Wh T 7 : 7 kt 5 kt 5 kt 5 kt l kt l 5 5 l t l 5 l 5 t 5 l... tim of day was 0 am. hous i.. aoimatly : am Qustio (a) To ov: Tst : LHS 5 5 RHS 5 5 LHS RHS Tu fo Lt k b a valu fo which th sult is tu. Th k k k k k is tu (*) Tst k : To ov: k k5 kkk5 k k kk k k kk k k k k kk k 5 k k 9k 0 k k k 5 LHS by (*) RHS Th sult is tu by mathmatical iductio (b) (i) Th ball stiks th goud wh y 0 h gt 0 (ii) gt t h h g t h g as t 0 vt y h gt v y gt y Uo imact: ta 5 gt v v gt But vt gt Cambidg Uivsity Pss - 5 -

6 h Wh t, d g h d g g d h as quid (c)(i) I Gam, Lt Phit ad X umb of hits obabilitis a giv by th tms of Fo Dacy to wi h must hit th tagt at last oc: X X P Dacy wis P P 0 0 (ii) I Gam, Now obabilitis a giv by th tms of Dacy ds to hit th tagt at last twic PDacy wis PX PX PX (iii) Pwi Gam Pwi Gam 0as 0 ad 0 Pwi Gam > Pwi Gam h is mo likly to wi Gam (iv) P wi Gam Pwi Gam Qustio o 7 7 but dl (a) 0 ad h (smi-vtical agl is 5) (i) Th volum of th at of th u co which is immsd is giv by V R H wh R H l Vmovd l Th volum maiig i th low co is giv by V Vlow co V h l h l movd as quid dv (ii) Now l dl l dv dv dl But dl dv l 0 0 l 0 wh l 0 th volum is dcasig at 0 cm /s Cambidg Uivsity Pss - -

7 (ii) Wh V movd h Th l h l h l h dv 0 l 0 5 h th volum dcass at h 5 h cm /s 0 (i)... 0 (b) Difftiatig both sids with sct to :... Multilyig both sids by :... as quid... Difftiatig both sids with sct to :... () (ii) Fom Cambidg Uivsity Pss - 7 -

8 Substitut : i.. as quid () (iii) Substitut ito th sult lablld () 0 () () (): But is v ad ad so th th tm of ach asio is ositiv... Dividig though by givs:... as quid Ed of solutios Cambidg Uivsity Pss - -

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