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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)
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Question 4 P and w are shown.
Question 5
z 3 5i, z 1
Question 6a
i
1
2(1 i)
Specialist Mathematics The Engage Education Foundation
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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)
Specialist Mathematics The Engage Education Foundation
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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
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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
Specialist Mathematics The Engage Education Foundation
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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
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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)
Specialist Mathematics The Engage Education Foundation
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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
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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
Specialist Mathematics The Engage Education Foundation
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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
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(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
Specialist Mathematics The Engage Education Foundation
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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 ⇒ ,
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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
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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
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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