MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS SEMESTER TWO 2014 WEEK 11 WRITTEN EXAMINATION 1 SOLUTIONS
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1 MASTER CLASS PROGRAM UNIT SPECIALIST MATHEMATICS SEMESTER TWO WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT QUESTION () Lt p ( z) z z z If z i z ( is fctor of p( z ) thn p( p ( ( ( ( ( ( i ( ( ( [ ] ( ( ( ) As p(, thrfor z i is fctor of z z z s rquird. (b) Sinc th cofficints of z z z r ll rl, it follows from th conjugt root thorm tht z i is lso linr fctor. Thrfor ( z ( z ( z ) z z is qudrtic fctor. Thrfor z z z ( z z )(z ). Thrfor z is rl linr fctor. Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg
2 QUESTION log log log du Substitut u log : du u du u C (log ) C Substitut whn : ( log ) ( ) C C log C C (log ) ± Sinc whn, tht is, (). Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg
3 QUESTION Product rul: d ( ) Chin rul: ( ) d d d Diffrntit both sids of with rspct to : At th point P(, ): ( ) ( ) m ( ) Gnrl qution of th norml t (, ) ( ) 5 ( ) ( ). m is: Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg
4 Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg QUESTION ) ( 5 Tn ) ( ) ( 5 Tn () Tn Tn 8 Tn Tn
5 QUESTION 5, n n h () Eulr s rul (givn on th VCAA formul sht) h f ( ) Lt f ( ) with ( ) n, hf log log 8 n n n log, nd h.5. ( ) log f ( ) whr ( ) n, hf 8 7 log log ( ) log f (.5) whr (.5) f f (b) log c log : Sub ( ) log log c c log log log ( ) log log log whn : ( ) 5 log Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg 5
6 QUESTION () Vrticl smptots: f ( ) sc() is undfind whn cos( ) cos() cos( ), ± ± Y-intrcpt: sc( ) X-intrcpts: sc( ), cos( ) ± ± Endpoints: ± sc( ± ),,, (, (, ) -), Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg
7 (b) f ( ) sc() : sc( ) cos( ) ± 5 V V V whr: V Volum of clindr of rdius nd hight V volum of solid of rvolution formd whn th r nclosd b th curv sc( ), th -is nd th lins / sc () [ tn() ] / / / ± is rottd bout th -is / / tn tn Thrfor V cubic units. QUESTION 7 () t cos, t t cos () t sin, t t sin () Eqution () Eqution () : ( ) ( ) ( ) t cos t sin Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg 7
8 (b) t t t t (c) Motion is in clock-wis dirction. Dirction of motion t n tim t is givn b th dirction of t t vt () sin i cos j dt d ~ v( t) : ~ dt At t : v sin i cos j i j v Unit vctor in dirction of motion of objct t t : v i j i j v Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg 8
9 QUESTION 8 () for,,, nd t th vlus,,, nd. (b) sin c Sub whn : c sin sin c c sin Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg 9
10 QUESTION 9 () rg( iz ) rgiz rg( rg z rg( rg( z rg( z i i Not: Whn multipling compl numbrs, dd th ngls. Thrfor: rg( iz ) rg( z rg( z (b) Arg( iz ) < < Arg( iz ) < < Arg( z < < Arg( z < Boundris: rg( z is r from (but NOT including) th point corrsponding to z i t n ngl msurd nti-clockwis from th horizontl. rg( z rg( z is r from (but NOT including) th point corrsponding to z i in th dirction of th imginr is. Not tht z i is not includd bcus rg( ) is not dfind. Shdd rgion rquird. Im z R z Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg
11 QUESTION Forcs (msurd in Nwton) in dirction prpndiculr to slop: R T sin 5g cos 5 T 5g R () Forcs (msurd in Nwton) in dirction prlll to slop: T cos 5g sin 5 5g T g T 5g Nwton. () Substitut T 5g into qution (): R T T sin T T cos T 5 g cos g 5g R 5 5 g sin kg wt 5g 5g 5g( ) R Nwton Th School For Ecllnc Mstr Clss Spcilist Mthmtics Wk Em Pg
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