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Specialist Mathematics Notes Solutions

EngageEducationFoundation

Specialist Mathematics The Engage Education Foundation

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Question 1

(a) 6

(b) 243

4

(c) 3

4(d ) c 3

Question 2

(a) 2 5cis(2

3)

(b) 14cis(3

4)

(c) 49

10 5cis( )

(d ) 49

10 5i

Question 3

(2-5i)

The Engage Education Foundation Specialist Mathematics

www.engageeducation.org.au Page 2

Question 4 P and w are shown.

Question 5

z 3 5i, z 1

Question 6a

i

1

2(1 i)



Question 6b

(2 4i)2 41 (3 6i)

416i 1612 24i

8i

Question 6c

z (2 4i) 8i

2

1 2i 2 2i

2

1 2i 2 1

2(1 i)

1 2i (1 i)

3i,2 i

Question 7a Circle, centre (3,-4), radius 5

Question 7b

Question 7c

z

3

4(1 17)

i

4(5 17)



Question 8



Question 9

w 1cis where

n5

, n is an integer which is NOT a multiple of 5

Question 10 2

2 2 2 2

( )

( ) (2 )

a cz bz a

b c z ac b z a

Question 11 Proof

Question 12

1(4 2 )

7AK t u

Question 13a 38°

Question 13b

11

14(%i 2

%j 3

%k)

Question 14 (a) show

2 2( ) (1 4 ) 1

( ) it is the ABC logo!

b y x x

c

Question 15

The portion of 3( 6) 1y x from (6, -1) to (7,0)

Question 16



Question 17

Intersect at (12,29)and(

2

5,29

5)

Question 18

either : (x 2)2 9 x 5

x 5,2

or :(x 2)2 9 x 5

x 5,0

overall : x 5,0,2

Question 19



Question 20

Question 21 ( ) 2cos 1

( ) 2cot

a

b x

Question 22

2

5( )

6( ) 1

7( )

11

9( )

31

( ) 2

( ) 1

2 6( )

71

( ) 3

a

b

c

ad

e

f

g

h



Question 23

(a) y 50sin( x

100)

y 100

sin1(

x

50)

(b)

69 25

km / h

Question 24 (a) show

( ) , ,6 4 3

b x

Question 25

Curves both pass through (2

,1),(0,0),(2

,1)



Question 26 4

5

2

3 32 21 1 1

2 2 2

2

1

2

2

45( )

4

405 5( ) 5 45 45 405ln 9

9 2

10 10( 9)( ) 9 : 5( 9). 90 90( 9)

3 35

( ) ln( 9)25

( ) tan ( )3 35 1 1 5 3

( ) ln6 3 3 6 3

( )5 9

5( ) ln 9

2

ta t c

tb t dt t t c

t

u tc let u t u u du u c t c

d t c

te c

tf dt c

t t t

g t c

h t

1

2

6

2 2 2 4

( )5sin ( )3

1 cos(18 ) 5 5sin(18 )( )5

2 2 365sin(9 ) 5

( ) ln cos(9 )cos(9 ) 9

9( ) 9sec (5 ) tan(5 )

5

sin (9 )( )

54

c( ) sin(5 ).sin (5 ).cos (5 ) sin(5 ).cos (5 ) sin(5 ).cos (5 )

c

ti c

x x xj dx c

xk dx x c

x

l x dx x c

xm c

n x x x dx x x x x dx

3 5

6

21

os (5 ) cos (5 )

15 25

tan (9 )( )

54

1 1 (ln(9 ))( ) (ln(9 )) .

5 10

x xc

xo c

xp x dx c

x

Question 27 2

3 33 2

32 22

cos , 1 sin , sin .

cos sin. sin . cos . (sin 1)cos . sin

sin 3

(1 )1

3

x u x u dx u du

u uu du u du u u du u c

u

xx c



Question 28

y '' a(x 2)

y ' a(x2

2 2x b)

y '(1) 0 b 5

2

y ' a(x2

2 2x

5

2)

y a(x3

6 x2

5x

2 c)

y(2) 49, y(1) 5

a 6,c 1

2y x3 6x2 15x 3

Question 29a

dy

ey e2x dx

e y e2x

2 c,c

3

2

y loge(3 e2x

2)

Question 29b

(b) Domain = (,loge

3)

Question 30

3061000

⇒1000

30000 6



10006

630000 6

10006

|30000 6 |

02000 ⇒ ln |

(a) 5000 3000

30 ⇒ 2494, 2494 /

(b) 4000 ⇒ 183.1 (c) Sugar poured out = initial sugar + sugar in - sugar left

2000 183.1 30 4000

3493

Question 31

(a)

V=kh|t+1|+c

00⇒ 0

102000

⇒200011

⇒ | 1|

(b) 1000 → 2.3166

(c) 5 → 1494.4

Question 32

(a)

dV

dt kV 2 t



(b)

dV

V 2 k t dt

1

V2kt

32

3 c,c

1

56given : t 9,V 28

k 1

5618

1

V

1

56(1

t3

2

27)

t 9,1

V

1

56(1 27)

1

2V 2

Question 33a

Question 33b



Question 34 (a) H (b) I (c) C (d) F (e) A

Question 35a

tan ( 1

641.5 ⇒ tan 0.7972, 1.1956

⇒ 38.56° 50.09°

Question 35b

tan2

1

700.6 ⇒ ,



Question 36

0 40 cos 7°

0 40 sin 7° 0.09

⇒ 12.76

0 40 cos 7°

0 40 sin 7° 0.09

⇒ 82.79

⇒ ∈ 12.76,82.79



Question 37

01.510

2

⇒1.520

0 40 cos 7° sin 7°

401.520

cos 7° sin 7°

Solve for A: A = 88.93N

Question 38

A = 400-20x

20, 2 /

0 40 cos 400 2 sin

40 400 2 cos 40 sin 0.09

⇒ (364.72cos 428 sin 20 cos 0.18 sin )

12

⇒ 364.72 cos 428 sin )x + 20 cos 0.18 sin ) + c



When 02⇒ 0

202⇒ 21.124°

Question 39 (a)

(b) ∑ 0 10000 cos 6°

10000 10000 sin 6° 0.1

⇒ sin 6° 0.1 cos 6°10000

497.474810000

(c) .

10000ln|497.4748 |

00

| .

.

2543.6

⇒ 0.01509

(d) 0 ⇒.

32967 /

Question 40 (a)

(b) 1

⇒ 1

⇒ 1



(c) 1

⇒1

⇒1

(d) | 1| 2| | 1 2√3 tan√

00⇒

√318

(e) 0.99 ⇒ 1.4159

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Engage Education Foundationengageeducation.org.au/engagedownloads/Seminar Doc… · · 2017-04-11Specialist Mathematics The Engage Education Foundation ... dV dt kV 2 t. ... Microsoft