3cd trial calculatorassumed 2010 solutions

8/16/2019 3CD Trial CalculatorAssumed 2010 SOLUTIONS

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Hale School

MATHEMATICS

SPECIALIST3CD

Semester Two Examination 2010

MARI!" E# an$ S%L&TI%!S

Section Two

Calc'lator(Ass'me$


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MAS 3CD Semester 2 2010 Section 2 : Calculator-Assumed [80 marks]MARI!" E# an$ S%L&TI%!S

)'estion * +* mar,s-

Evaluate te !ollo"in# de!inite inte#rals exactl. usin# Calculus and al#e$raic tecni%ues :

a&∫ 2

2

1

' - ( d(

)ut ( * 2 sin u

[' marks]

Solution

∫ 2

2

1

' - ( d( *

π

π ∫

22

+

' - 'sin & 2cos u duu

*

π

π

∫ 2

+

2 ' cos u du

*

π

π

∫ 2

+

2,cos 2u 1. du

*[ ]

π

π 2

+

sin 2u 2u *

π 2 3 -

3 2 Specific Behaviours

Can#e te limits o! inte#ration E(/ress inte#rand in terms o! u usin# te CSE D45E A65E result

Anti-di!!erentiate correctl7

Determines te e(act value

$&

∫ ln 2 (

(

0

e d(

e 2

[' marks]

Solution

∫

ln 2 (

(

0

e d(

e 2 *

( ) ln2

(

0ln e 2

* ln ,eln2 2. - ln ,1 2.

* ln ' ln 3

* ln,'93.

Specific Behaviours

ln

Anti-derivative evaluate eln2 * 2 and e0 * 1

e(/ress as an e(act value usin# natural lo#aritm values

See ne(t /a#e


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)'estion / +1 mar,s-

Consider te !ollo"in# trans!ormation matrices in te co-ordinate /lane :

rotate ;0o clock"ise a$out te ori#inD dilate verticall7 a$out te ori#in "it dilation !actor 1&<

S ori=ontal sear /arallel to te ( a(is "it sear !actor -1&

,a. 6ive matrices > D and S&[3 marks]

Solution

*

0 1

-1 0 D *

1&< 0

0 1&< S *

1 -1

0 1

Specific Behaviours

ne mark !or eac correct matri(

?e 3 dia#rams $elo" so" a rectan#le&

Dia#ram 1 Dia#ram 2

Dra" te ima#e o! tis rectan#le under te action o! trans!ormation :

,$. D ,on Dia#ram 1.[1 mark]

,c. S ,on Dia#ram 2.[2 marks]

,d. ten S ,on Dia#ram 3.

[2 marks]

See ne(t /a#e

–

enlar#ed rectan#le

/arallelo#ram dra"n sear to te 5E@?

rectan#le rotated correct /arallelo#ram sear e!!ect


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Dia#ram 3

)'estion / +1 mar,s-

,e. ! te rectan#le is trans!ormed $7 matri( ten S> "at matri( "ill return te resultant

ima#e $ack to te ori#inal rectan#le [3 marks]

Solution

?e trans!ormation "as S *

1 -1

0 1

0 1

-1 0 *

1 1

-1 0 &

∴ e%uire trans!ormation ,S.-1 *

0 -1

1 1

Specific Behaviours

Determine matri( S

eco#nise inverse matri( re%uired

Determine te inverse matri(

A student o$serves a can#e in te area o! te ima#e "en "orkin# "it matri( D& ?estudent "rites a conBecture a$out te area o! te resultant ima#e> "ere n is te num$er o!times te ori#inal rectan#le as $een dilated $7 matri( D&

,!. Su##est a conBecture !or te Area,ma#e. "en te ori#inal rectan#le as $eendilated n times $7 matri( D&

Also test 7our conBecture&[3 marks]

Solution

ConBecture : Area,ma#e. * Area,$Bect. ( ,2&2


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ence te conBecture is correct&

Specific Behaviours

se o! determinant to !ind area ratio

Correct conBecture

?est o! conBecture usin# a kno"n result

)'estion 10 + mar,s-

?ree rival su/ermarkets com/ete !or customers in te su$ur$ o! nnaloo& Market researc

surve7s so" tat #iven a customer so/s at a su/ermarket one "eek> tere is a cance

tat te7 "ill can#e and so/ at anoter su/ermarket te ne(t "eek& ?e transition matri(

? so"n $elo" summarises tis :

? *

0&D 0&2 0&1

0&2 0&+ 0&1

0&1 0&2 0&8

,a. E(/lain in "ords "at element t23 o! matri( ? re/resents&

[1 mark]

Solution t23 is te /ro$a$ilit7 tat #iven a customer so/s at su/ermarket C te7 "ill $e

so//in# at su/ermarket 4 ne(t "eek&

Specific Behaviours

Correct descri/tion ,#iven so//in# at C ten so/ at 4 ne(t "eek.

,$. 6iven tat a customer is so//in# at su/ermarket 4 tis "eek> #ive te /ro$a$ilities

tat te customer "ill $e so//in# at eac o! A> 4 or C in 2 "eeks time&

[2 marks]

Solution

See ne(t /a#e

Current Su/ermarket

C4 A

Su/ermarket !orE? "eek

A

4

C


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Require matrix T 2 *

0&

?en te /ro$a$ilities o! :

So//in# at A * 0&28

So//in# at 4 * 0&'2

So//in# at C * 0&3

Specific Behaviours

eco#nises tat second column o! ?2 is re%uired

Correct /ro$a$ilities #iven ,all 3.

)'estion 10 + mar,s-

,c. ! tere are currentl7 '0F so//in# at su/ermarket A> 30F at 4 and 30F at C> #ive

te /ro/ortions so//in# at eac su/ermarket in ' "eeks time&

[2 marks]

Solution

e%uire matri( ?'

0&'

0&3

0&3*

0&3308;

0&2D1;30&3;D18

So in ' "eeks time> tere are a//ro(& :

33&1F so//in# at A

2&2F so//in# at 4

'0&0F so//in# at C&

Specific Behaviours

eco#nises tat ?' ( ),0. is re%uired

Correct /ercenta#es ,or /ro$a$ilities.

See ne(t /a#e


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O

C (2,8,0)

B (6,6,0)

(4,0,0) A

(4,0,8) D

E (6,6,4)

F (2,8,12)

z

y

x


,d. Determine te lon# run /ro/ortions o! customers so//in# at eac o! te

su/ermarkets A> 4 and C&

[2 marks]

Solution

Consider ?n (

0&'

0&3

0&3as n #ets lar#e

ence in te lon# run tere "ill $e a//ro(&

31&+F so//in# at A

2+&3F so//in# at 4

'2&1F so//in# at C&

solve !or ) ,3 ( 1 matri(. suc tat

? ( ) * )

Specific Behaviours

Consider $eaviour as n is lar#e te matri( ) tat is uncan#ed $7 ?

Correct /ercenta#es ,or /ro$a$ilities.

)'estion 11 +11 mar,s-

?e end o! a solid trian#ular /rism is sa"n o!! so tat its to/ !ace G DE@ is ? /arallel to

te $ottom !ace G A4C& )oint is te ori#in and i> and , are unit vectors in te direction

o! te /ositive (> /ositive 7 and /ositive = directions res/ectivel7&

See ne(t /a#e


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,a. Determine te unit vector in te direction o! vector E@ &

[2 marks]

Solution

Hector E@ *

−

'

2

8 ∴ nit vector *

( )1

-'i 2B 8k8'

Specific Behaviours Determine vector E@

Divide $7 ma#nitude o! E@ to o$tain te ? vector

,$. 6ive te vector e%uation !or te line containin# /oints E and @&

[2 marks]

Solution

r *

+

+

' + λ

−

'

2

8 *

λ

λ

λ

+ - '

+ 2

' 8

Specific Behaviours

@orm o! vector e%uation usin# /arameter

se o! te direction vector E@


,c. Determine te interce/t o! te line containin# /oints E and @ "it te (7 /lane&

[2 marks]

Solution

r *

λ

λ

λ

+ - '

+ 2

' 8 (7 /lane as e%uation = * 0

∴ ' 8I * 0 #ives I * -0&<

ence interce/t "it (7 /lane is ,8> 0.

Specific Behaviours

See ne(t /a#e


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solve !or /arameter I to #ive = * 0

#ive co-ordinates !or interce/t

,d. 6ive te vector e%uation !or /lane DE@&

[2 marks]

Solution

sin# CAS :

E%uation /lane is 2( = * 1+

Hector e%uation r & ,2i 0 1,. * 1+

vector e%uation !orm

,e. State te normal vector !or /lane DE@&

[1 mark]

Solution

ormal vector is n * ,2i 0 1,.

Specific Behaviours

se vector e%uation to deduce normal vector


,!. 6ive te an#le> correct to te nearest de#ree> $et"een /lane DE@ and te (7 /lane&

[2 marks]

Solution

eed to !ind te an#le $et"een te normals o! eac /lane&

ormal vector !or (7 /lane is n * ,0i 0 1,.

See ne(t /a#e


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∴ 2,0. 0,0. 1,1. * √


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?e #ra/ o! 7 *

− ÷

1 (sin2 is so"n $elo"> "ere -2 K ( K 2&

,a. Determine te co-ordinates !or /oints A and 4&

[2 marks]

Solution

)oint A ( * 1> 7 * sin-1,192. * π9+ ence A ,1> π9+.

)oint 4 ( * 2> 7 * sin-1,1. * π92 ence 4 ,2> π92.

Specific Behaviours

Evaluates inverse sine !unction correctl7 in terms o! π to !ind 7 values

6ives co-ordinates !or A and 4

,$. @ind te e%uation o! te tan#ent to te curve at /oint A&

[' marks]

Solution

6ra/ is #iven $7 7 * sin-1,(92.

i&e& (92 * sin 7

∴

1 d7 * cos 7 &

2 d( At /oint A 7 * π9+ ∴ m * 19√3

See ne(t /a#e

4

A


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E%uation o! tan#ent 7 - π9+ * 19√3,( 1.

i&e& 7 * (9√3 ,π9+ - 19√3.

Specific Behaviours

E(/ress curve in terms o! sin 7

Di!!erentiatte im/licitl7 to !ind d79d(

@ind slo/e o! tan#ent

Determines e%uation o! tan#ent


,$.

Solution

CAS solution :

At /oint A ( * 1 ∴ m * 19√3 * 0&


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,c. So"> 'sin the anti($eri4ati4e o5 the sine 5'nction> tat te e(act area o! te

re#ion saded is e%ual to

π < - 3

+ s%uare units&

[' marks]

Solution

sin# ori=ontal slices : Area * 1,π9+.

π

π

∫

2

+

,2 - 2 sin 7. d7

See ne(t /a#e

dA * ,di!!erence in ( values.& d7


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15/21


,a. 6ive te acceleration o! te c7clist at t * < seconds&

[1 mark]

Solution

Acceleration is #iven $7 te slo/e o! te tan#ent * vL,


16/21


Specific Behaviours

e%uire G( * 1


17/21


eiθ - e-iθ * cis,θ. - cis,-θ.

* cos θ i sin θ - ,cosθ - i sin θ. * 2i sin θ

Specific Behaviours

ses /olar !orm to e(/ress cis,θ. and cis,-θ. correctl7 conBu#ates

ii& ence /rove tat : 1+ sin


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,$. i& Solve> in e(act /olar !orm> te com/le( e%uation = 2> 3> '

= *

π ÷

cis< >

π ÷

3cis

< >

π ÷

<cis

< >

π ÷

Dcis

< >

π ÷

;cis

<

= *

π ÷

cis< >

π ÷

3cis

< > -1 >

π − ÷

3cis

< >

π ÷

cis -<

Specific Behaviours

E(/ression !or


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Specific Behaviours

se te SM o! te roots is PE

o$tain 2 multi/les o! cos,π9


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1000 dN = 0.5 dt

N(1000 - N)∫ ∫

Hence :

= 0.5 dt1 1

+ dN

N 1000 - N

÷ ∫ ∫ ln - ln,1000 . * 0&


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t is considered tat Qever7one in te to"n as eard te rumour "en ;;F o! te to"nLs/o/ulation ave eard it&

,d. Determine o" lon# ,correct te nearest minute. it takes !or Qever7one to ave eardte rumour

[2 marks]

Solution

e%uire ,t. * ;;0

@rom CAS> t * 1;&+8 J rs

i&e& "ill take 1; ours ' minutes !or Qever7one to ave eard te rumour&

Specific Behaviours

E%uation to solve !or t usin# ;;0 /eo/le

Solve !or t correct to nearest minute

3cd trial calculatorassumed 2010 solutions

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