( ) ( ) ( ) 2011 HSC Mathematics Solutions ( 6) ( ) ( ) ( ) π π. αβ = = 2. α β αβ. Question 1. (iii) 1 1 β + (a) (4 sig. fig.

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Godwin Simpson
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1 HS Mathmatics Solutios Qustio ( sig. fig.) (b) (c) ( )( + ) d d (d) l ( ) () 8 6 (f) ( ) ( ) (iii) β + + α α β αβ 6 (b) si π si π π π +,π π π, (c) y + dy + d 8+ At : y + (,) dy 8( + ) d 8 m 8 ( ) tagt is y 8 + y 88 y 87 (g) Numb dfctiv. 8 6 Qustio (d) y dy ( ) + ( ) by th poduct ul d + (i) 6+ ( 6) α + β 6 αβ () d d + + ambidg Uivsity ss - -

2 Qustio $, $., $, is a aithmtic squc with a ad d. (i) T a ( ) d + wh T +. th th floo will cost $ millio Now ml m l l l A+ By+ (iii) d A + B wh A, B, ad, y, S a d wh 9 S + ( ) ( ) th floos will cost $7. millio (b) S, ad th dicti is y th paabola is cocav up y S V y a 6 a V, (, ) (iv) (v) + d + OE ( OE AB) OB fom (i) OB OE + BE by ythagoas 9 + BE BE BE ( BE > ) Aa BOE BE OE 9.6 (c)(i) y-itcpt of + y occus wh : y y m l m l B (,) A usig m B th aa is.6 uit Qustio d si ( si ) ( )( cos ) d si by th quotit ul si cos si ambidg Uivsity ss - -

3 (b) d [ l ] l l l l l but l dy d 6 y + y + sub (, ) : ( ) ( ) + + y (c) 6 though (, ) (d) (i) () y 9 ( 9 ) ( 9 ) ( ) y dy d 9 Qustio by th chai ul 6 d 6 d Th gio blow y is y< Th gio blow y is y > Th shadd gio is wh y< ad y > 7,, 8, is a gomtic sis with a ad (i) ad T a T 7 96 th a 96 mmbs o Day Now T > a > 7 > > 7 ( ) l > l ( 7 ) l ( 7 ) > as l > l ( 7 ) l > + l > it will occu o Day (b)(i) ( d o Moday) (c) As th a d shits ad oly yllow, th colou wo o days must b d. ( sam colou o days) (iii) Th a possibl scaios: RYR o YRY ot sam colou o coscutiv days ( RYR) ( YRY) t v wight v dt th jogg tavls appoimatly 8 m i.. about.8 km ( sig. fig) ambidg Uivsity ss - -

4 Qustio 6 (iii) A B (iv) Aa A+ B+ (ot Aa B > ) A 8 (b) (i) E D D 6 (tio gula ptago) D 7 D 7 (tio gula ptago) D ( 8 7 ) ( sum of D) 6 Now D ED (ABDE gula) ED ED (bas agls of isoscls tiagl EOD) ED 7 (tio of ED) ED 6 D (fom abov) E is isoscls ( qual agls) Now A + B + + y + + y y y + y + y + + y + ( ) + y 6 Which is a cicl with ct, ad adius. (v) π π Qustio 7 cos d Aa B + Aa A + f + f (i) f ( ) 6 Statioay poits occu wh f ( ) ± (, ) 6 a mi f (, ) 6 a ma At : f + f > At : + ( ) f <, is a local miimum ad, is a local maimum (-, ) y (c)(i) At, y cos (, ) π π cos d si si si π (, ) ambidg Uivsity ss - -

5 t (b) 88 (i) Wh t : 8 8 it is iitially statioay t ( 8 ) t 6 t Now > fo all t > t 6 > fo all t > th acclatio is always positiv fo all t >. (iii) Th paticl stats at st ad is actd upo by a acclatio of 6 ms -. This movs it i th positiv dictio. As th subsqut acclatio is always positiv, th paticl will cotiu to mov i this dictio with v icasig vlocity. t (iv) As t, t 88 8 i.. th vlocity appoachs 8 (v) 8 v (b) (i) V π h dy h y dy as y h π y π h π π h V h H h OH cylid π wh ; y h h At : H cylid h π h paaboloid cylid π h V π h h π h V V h as > (b) (i) 6% pa.% p moth. Qustio 8 (i) + + cos 6 (cos ul) Usig th quadatic fomula: ± 8 ± 76 but > as it is a distac t Th fist $ gows to $. Th scod $ gows to 9 $. Th thid $ gows to $(. ) 8... Th last $ gows to $(. ) (.) + (. ) (.) a( ) wh a (.). ( ) $ $ 8 to th ast dolla th distac is km to th ast kilomt ambidg Uivsity ss - -

6 () A M. (.) A A + M + + A ( 9 7 M)(.) M (.) M 9 7. M.. A M () A M ( ) ( ).(. ) + M... M M M..8 ( dp) sh ds to ivst $.8 ach moth Qustio 9 (i) I AB ad ADE AB. ( B is th midpoit) AD. A is commo A. ( is th midpoit) AE AB ADE ( sids i popotio ad th icludd agl qual) I BF ad DEF. BF FDE (altat B DE). BF DFE (vtically opposit) BF EDF (quiagula) B BF ED EF BF EF BF : FE : (b)(i) (c) (matchig sids of simila tiagls) t Flow A Flow B + + t+ t+ t + t+ t+ t+ + t t + t+ t t + t( + t) t + t th flow at diffs by t lits p miut Th chag i volum is dv dt V t dt t 6 8 t th is 8L mo liquid A tha liquid B y AB A BD E B DE (giv) (atios a psvd) O - ambidg Uivsity ss - 6 -

7 (d) (i) Qustio ( ) ( ) + ( ) + ( ) ( ) + ( 99 )... 9 I.L (i) Now L.( ) I ( sig. fig.) th itsity is about 6. 8 watt/m (iii) Lt th oigial loudss b K. Th th oigial itsity is.k I Wh th itsity doubls,.k I W d th valu of L at this itsity ( K) ( K)..L ( θ )..L l l.l.k.l.k.l.k ( L K) l.l.k l l. l L K. L K l 7 th icas is about 7 dcibls (b)(i) Lt th lgth of th ac b l. l θ l+ θ + + as quid 9 I > 8..L 9 > 8..L > 8..L > 8.L l > l 8.L l > l 8 but l.l > l 8 l 8 L >. L > A θ But ( θ + ) θ + θ A A as quid about 9 dcibls ambidg Uivsity ss - 7 -

8 (iii) da d d A d da Ma/mi occus wh d d A Th < a ma occus d th aa is maimisd wh ombiig ths two stictios givs < < π + as quid Ed of solutios (iv) Now θ wh : θ th agl is adias (v) Fo th aa to ist, θ > > > > < But i a cicl, th gatst possibl agl is π < π < π + < ( π + ) < ( π + ) > π + ambidg Uivsity ss - 8 -

2011 HSC Mathematics Extension 1 Solutions

2011 HSC Mathematics Extension 1 Solutions 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