Answers
Exercise 1A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Exercise 2A
1 x 6
x 2 or x 3
1 __ x 1
__
√3 x 1 or 1 x √3
3
x _2 or 0 x 1
x 1 or 0 x 2
x 2 or 1 x 1 or x 2
1 x 2
x≠0
or 1 x 0 or 0 x 2
14
x 4 or x __
3
2 x 5 or x 8.5
5 x 0 or x _31
x 0 or 2 x 5
x 2 or 0 x 1
x 3 or 1 x 1
a _31 x _21
But x ≠ 0
_31 x 0 or 0 x _21
2
n(n 3)
______________
4(n 1)(n 2)
3
1 ______
1
a __
2r (r 2)
4
1
1 _____
1
_____
a ____________
(r 2)(r 3) r 2 r 3
n(3n 5)
b ______________
4(n 1)(n 2)
n
b ________
3(n 3)
5
3
2 _____
1 _____
a __
r r1 r2
7
n(n 2)
________
(n 1)2
Exercise 3A
ALL ANSWERS FOR CHAPTER 3 TO FOLLOW . . .
b 1 x _31 or x _21
Exercise 1B
1
x _76
2
1 √13
1 √13
_________
t _________
3
4
5
6
7 x 2 √7 or 2 √7 x 3
x 2 or x 1
x 3 or x 1
x _31 or x 1
7
8
1 x _31
__
x 1__ √3
__
or √2 x 1 √3
b 2 x 1
or 0 x 2
b x a or x a _21
9
10
2
___
___
__
2
__
Mixed exercise 1C
1
2
3
4
5
6
7
8
9
1x5
3 x 3
x _72
__
__
x 1 √3 or x √3 1
0 x _23 or 3 x 4
x 1 or 1 x 11
b x _94
__
__
c x √7 1 or 0 x __
1 or x √7 __
1
c x 3 or 1 x √6 1 or x √6 1
109
ANSWERS
110
ANSWERS
Review exercise 1
15
1
2
3
4
5
6
7
8
x 0, x 4
x 4, 1 x 2
x 0, 2 x 4
3 x 0, x 4
_21 x 0, x 3
x 4k, 2k x 0, 2k x
b x2
b x _31 a
9
a
a
8
6
B
yx
O
1
x
1
2
O
y
y 冷3x 2冷
1
1 _____
1
a _____________
_____
(x 1)(x 2) x 1 x 2
18
2
1 _____
1
a ____________
_____
(r 1)(r 3) r 1 r 3
19
n
b ________
2(n 2)
20
1 _____
2 _____
1
a f(x) _____
x1 x2 x3
21
2
22
2
3
5
x
12
a 2 x 3
b 2 x _21
13
14
_31 a x _71 a
a The curve meets the x-axis at (2, 0) and (4, 0)
The line meets the x-axis at (3, 0)
b The coordinates of A are ( _27, _43 )
The coordinates of B are (5, 3)
x 3_21, x 5
2r 1
a ________
r2(r 1)2
2ln(n 1)
23
1 __
1 _______
1
a _______
r(r 2) 2r 2(r 2)
24
20
b ____
861
a A 24, B 2
c 194 380
n(n2 1)
b _________
2(n 1)
25
b x _23, x _47
5 x _31
x
1 _____
1 _____
1
b __
6 n2 n3
y 冷x 5冷
11
4
a _25, _47 and 1
17
5
O
3
b x _25, _47 x 1
( _31, _31 ) and (1, 1)
c x _31, x 1
a
2
b 2 √2 and 4 √2
c 2 √2 x 4 √2
16
10
A
y
y 冷2x 1冷
b
y
27
28
29
2r 3 __
3 _____
1
a _______
r(r 1) r r 1
1
b 1 _________
3n(n 1)
a
y
3
2
1
O
x
b 3
111
ANSWERS
30
32
33
35
13
, ____
5
__
9
, ____
, ___
, ___
17
___
(
47
)
10 10
2 10 10 10
The image of |z| 1 under T is the imaginary axis.
a
i sin __
2 cos __
(
3__ i 3 ___
3__
b z ___
√2
√2
a
y
(
z cos i sin , where
)
1
)
3
3
9
17
25
33
__
__
___
___
___
_1
_1
_1
_1
_1
a z 22 ei20, 22 ei20 , 22 ei 20 , 22 ei 20 , 22 ei 20
b
y
O
z2
x
1
c
v
z3
z1
x
O
z4
36
a
u
O
1
2
z5
y
48
a
y
P
O
x
1
z
2
z 2i
z2
2 C
3
Q
2
x
O
__
38
40
41
42
43
44
46
b maximum value of |z| is 3 √__
5
minimum value of |z| is 3 √5
b 0.809, 0.309 1
(5 i)z 2i
b ____________
z
5
, __
, ___
7
a ___
9 9 9
3
, 1.209 (3 d.p.) and 1.932 (3 d.p.)
, ___
b 0, __
4 4
1
__
(cos 6 6 cos 4 15 cos 2 10)
b 32
__
b √2
49
b The image of |z| 1 in the z-plane is
|w i| |w 1|
in the w-plane.
c, d
y
P
5
___
1
__
10 __
c 32
2 32
b The solutions of z6 z3 √2 1 0 are cos i sin ,
3
, 7
3
, ___
, 7
, ___
, ___
___
___
where ___
4
12 12 12 12 4
a
y
O
1
x
v
P
3
2
O
3
Q
1
O
112
x
(
1
1 , 1
2
2
)
u
ANSWERS
50
a z 1 i, 1 i
b a 0, b 2, d 2i
3
x c where c is constant
y __
3
3
y
d centre ( 0, _34 ), radius _32
y
Exercise 4A
1
y
2
x2
3
y
x2
3 1
y
x2
3 2
4
y x2 3
3
y x2 2
2
1
y x2 1
2 x
O
1
2
3
y x2
2
y x2 1
2
1
y x2 2
3
O
x
1
4
4
y ln Ax
y
2
4
y Ae where A is constant (A ec)
y
x
3
1
y
y x2 c where c is constant
y
2
2
x2
3 3
y 3e
x
2
y 2ex
1
y ex
6
4
2
3
O
1
2
6 x
4
2
3
2
4
1
x
O
5
y Ax2
y
1
y 2x2
y x2
2
3
y
y ex
y 3ex
y 2ex
1
2
x2
x
O
y 12 x2
y x2
y 2x2
113
ANSWERS
6
y2 x2 2c
9
y sin x c
y2 x2 4
y x 16
2
2
y
y2 x2 1
y sin x 1
y2 x2 0
y
x
y sin x
O
4
3
y sin x 1
2
y sin x 2
1
4 3 2 1 O
1
1
2
3
10
4 x
2
4
3
3
y x 1
2
y2 x2 16
O
y 3 sin x
3
y sin x
4
1
y ln ________
(x c)
1
y ln x
11
y ln
y ln (11 x)
π x
π
2
1
2
y2 x2 4
y 2 sin x
t __
x tan t c for __
2
1
1x
2
y
6
y
(21 x )
y 3 sin x
1
2
y ln
y 2 sin x
y sin x
2
4
7
y A sin x
y
y ln
4
1
2x
4
3
x tan t 1
x tan t 2
2
2
x tan t
1
6 5 4 3 2 1 O
1
1
2π
2 x
π
2
O
2
x tan t 2
2
3
x tan t 1
4
8
4
6
Ax , x 0
y _____
x1
12
y
t
Ae
x _______
t
1 Ae
x
4
y
3
y x 3x
1
2
y
2x
x1
1
y x x 1
x
3et
1 3et
0.5
O
114
x
t
e
x1
et
1
4x
x1
x
1 t
e
2
1 t
e
2
1
O
t
x
ANSWERS
13
a
y
7
c
1 __
a y 1 __
2
x
x
b i
ii
2
(
)
1
y ( __3 __1 __
4 2 x)
1
y 1 ___
2x
2
5
1 __
iii y 1 __
2
x
x
x
O
y
y2 x
y1
1
x
5
x2
y2 4x
y1
1
x
1
4x2
y1
1
x
1
x2
y2 9x
y 16x
2
14
y1
b y2 9x
a y2 5x2 45
b
y
(2, 34 )
4
12
3
2
O
1
x
A(x 1)
ln ________
1
3
O
8
3 x
a y
(x 2)
__________
ln x
(x 1)
16 _______
___
3
(
x 2)
b ln _________
ln x
2
3
4
Exercise 4B
1
c
1
__
y __
x sin x x
2
3
4
y xe2x e2x cex
y 3x cosec x c cosec x
y xex cx
5
c
y ln __1 __
2 x2
___________
6
y
[
]
√( 6__1 x 2___cx )
2
Exercise 4C
y _31 ex ce2x
y cot x c cosec x
y xecos x cecos x
y e2x cex
x2 c cos x
1 __
5 y __
1
2
3
4
(2 2
6
)
c
1
__
y __
x ln x x
y x ln (x 2) cx
_1
y _41 x cx3
9 y (x 2) ln (x 2) c (x 2)
7
8
10
c
1 e x __
1 e x __
y __
x4
x3
x4
11
1 x __
1 x __
1
y __
x e x2 e x2
12
13
1 cx
y ___
3x2
4x
1 ___
y ___
3
3x2
c
a y _31 (x2 1)2 _______
(x2 1)
2
b y _31 (x2 1)2 _________
3(x2 1)
14
c
a y 1 ____________
sec x tan x
cos x
1
b y 1 ____________
or y 1 _________
sec x tan x
1 sin x
Exercise 4D
1
y2 2x2 (ln x c)
115
ANSWERS
2
3
4
5
6
7
8
y3 3x2 (ln x c)
x
y _______
ln x c
y3 x3 (A x 1)
______
cos x
y √______
4x c
_1
x (1 ce6 t )2
4x
y ____________
(x 1)4 4c
______
xc
y ______
1 x2
10
t
__
b Ec ke Rc
c
c q ___________
(cos pt Rpc sin pt) k1e Rc ,
(1 R2p2c2)
k
where k1 _________
is constant
1 R2p2c2
k ex ceax
a y _____
a
b y (kx c)eax
t
__
3
11
kx
c y ______
eax ceax
n1
1
y ____________
1 cx ln x
n1
√
12
Mixed exercise 4E
1
t
__
a q ke Rc
_____
_1
(x2 16)2 c
y
13
cx
_____
y √cos
x
y
x2 16 2
14
2y2 (ln y c) x2 0
y
x2 16
15
2x
16
6
y
4
x 16 4
2
17
18
y __1
1 ln x c
___
ln y __
2
2
3x2 c
ln y ___
2y2
1 x
y x_____
c
2x
1 Ae
y x 2 ________
2x
1 Ae
x
O
Exercise 5A
2
_1 x2
2
1
Exercise 5B
_1 x2
b y 2e 2
c
y 3e 2 x2
1
y
y 2e 2 x2
1
y e 2 x2
1
3
x
O
3
1 [(g ku)ekt g]
v __
k
v
y 2 sin x c cos x
_1
y 5 c (1 x2)2
x __c
1 __
6 y __
22 x
y __2 x2 __xc
_3
5
y _21 cex
______
9 y 2x c x√1 x2
8
116
y (A Bx)e5x
y (A Bx)e9x
y (A Bx)ex
y (A Bx)e4x
y (A Bx)e7x
_1
y (A Bx)e4x
_1x
y (A Bx)e 2
_5
y (A Bx)e2x
_3x
y (A Bx)e4 __
y (A Bx)e√3 x
Exercise 5C
4
5
7
1
2
3
4
5
6
7
8
9
10
t
O
g
k
y Ae3x Be2x
y Ae2x Be6x
y Ae5x Be3x
y Ae7x Be4x
y Ae4x Be3x
y A Be5x
_1
y Ae3x Be2x
_41x
y Ae Be2x
_2
_1
y Ae 3x Be2x
_2x
_51x
y Ae Be3
1
2
3
4
5
6
7
8
9
10
a y e2
2
1
2
3
4
5
6
7
8
9
y A cos 5x B sin 5x
y A cos 9x B sin 9x
y A cos x B sin x
y A cos _34 x B sin _34 x
y e2x(A cos 3x B sin 3x)
y e4x(A cos x B sin x)
y e2x(A cos x B sin x)
y e10x(A cos 3x B sin 3x)
_1
y e 3x(A cos _32 x B sin _32 x)
10
y e 2 x(A cos _23 x B sin _23 x)
__
√3
__
ANSWERS
Exercise 5D
1
2
3
4
5
6
7
8
9
10
11
y Aex Be5x 2
y Ae6x Be2x 2 3x
y Ae4x Be3x 2e2x
y Ae5x Be3x _31
y (A Bx)e4x 1 _21 x
y (A Bx)ex 4 sin 2x 3 cos 2x
y A cos 9x B sin 9x _61 e3x
y A cos 2x B sin 2x _31 sin x
y e2x(A cos 2x B sin 2x) 3 8x 5x2
1 x
__
y ex(A cos 5x B sin 5x) 25
e
a _21
b y (A Bx _21 x2) ex
9
10
11
y 2 cos 2t sin 2t cos 3t
y Aex Be2x 3 2x xe2x
_1
y e4 x sin _21 x x 3
12
13
y _65 e2x _61 (cos 3x sin 3x)
x 3 cos t 2et sin 3t
14
A __
B _1 ln x _3
a y __
2
4
2
15
9
4 ___
_1
_3
b y __
x 4x2 2 ln x 4
y _21 cos (sin x) _25 sin (sin x) _21 esin x
3
4
5
6
7
8
9
10
a 2e
6
7
A __
B 4x
y __
x xe
2
3
4
5
8
9
y
x
x B sin x 1)
__
y Ae√2 sin x Be
c ex xex
2 e2 8e
3
x
f(n)(x)
2x
2ne2x
n(n 1)(n 2) n!
(1 x)n 3
ex (ex xex) 2ex (ex xex) nex xex
d (1 x)1
2ex xex
3ex xex
(1 x)2
(1)(2)
(1)n 1 (n 1)!
(1 x)3 (1 x)n
2(1 x)3
2
dny
a ____n 3n e2 3x 3ny
dx
(ddxy)
6
b
2
____
6 ln _1
(9)
e e2
36 __
36
a 3 sin 6x
d2y
d3y
b ____2 18 cos 6x, ____3 108 sin 6x,
dx
dx
d4y
____
648 cos 6x
dx4
(ddxy) 648 cos 648
dy
dy d y
2{ y
3 dx dx }
dx
4
c
____
4 __
6
3
____
2
___
6
b
7
d f(0) 1
f (0) 0|
f(0) 1
__
√2
sin x
4
sin2 x
5
Mixed exercise 5G
__
__
√3
√3
y e 2 x (A cos __
x B sin __
x)
2
2
6x
2 y (A Bx)e
3 y A Be4x
_1
1
2 e 4e
2x
____
3
2
Exercise 6B
e (A cos
___
2
x
f (x)
2 2x
(1 x)n 2
Exercise 5F
A __
B
y __
x4 x
1
y (A B ln x) __
x2
A __
B
y __
x2 x3
A Bx4
y __
x7
B
y Ax7 __
x2
1
y __
x [A cos ln x B sin ln x]
f (x)
2x
b n(1 x)n 1 n(n 1)
3
1
f (x)
1
y e5x e2x ex
y 2 _23 e2x _23 e2x
y _61 e6x _61 e7x _31
y cos 3x _32 sin 3x 2 sin x
_1
y sin x (1 e2 x)
x et e2t t
x e3t e3t sin t
x (t _21 t3)e2t
x _21 (cos _56 t sin _56 t 1)
x et sin t 1 2t t2 or x et sin t (1 t)2
x
Exercise 6A
Exercise 5E
1
2
x
1 sin kx
cos kx __
4
y
5
y e x sin 3x
6
7
8
y e2x (A cos 3x B sin 3x) _91 e2x
y Aex Bex 2xex
a y (A Bx)e2x
k
b ??????
c y (A Bx 2x2)e2x
8
a e 2.718 (3 d.p.)
b ln( _56 ) 0.182 (3 d.p.)
9x ___
9x ___
27 x4 …
a 1 3x ___
2
2
8
3
8
x
___
2
4
b 2x 2x 4x …
3
(8) 4
2 x2 0x3 _____
c 0 0x __
x …
2!
4!
3
_
b 2
2
3
Exercise 6C
1
x __
x … valid for all values of x
a 1 x __
2
6
2
3
32x … valid for all values of x
b 1 4x 8x2 _____
3
3
x __
x …
c e 1 x __
2
6
{
2
3
}
valid for all values of x
117
ANSWERS
(ddxy) 43 321 1283
2
x __
x __
x … 1 x 1
d x __
4
2
3
2
3
4
____
2
x ___
x3 _____
x5 _______
x7 … valid for all values
e __
2 48 3840 645120
of x
3x ___
9x2 ___
9x3 … _2 x _2
ln 2 ___
3
3
2
8
8
f
2
4
5
b
(x
x __
x … , 1 __
3
5
)
5
3
c 0.4055 (4 d.p.)
21
b __
2
5x2 ___
7x3 _____
17x4 …, _1 x _1 (smaller
a x ___
2
2
4
2
3
interval)
2
3
6
8
4
3
3 x 3
81
9
162
4
6
2
x
4
x
2
___
___
____
2
a 1 2x x8 …
45 315
3
x4 ___
2x6 ____
1 x8 …
b x2 __
45 315
3
a 1 2x 3x2 4x3 …
3
9
10
11
1
x __
x …
y 1 __x x2 __
2
x …
y x __
3
x …
y 2 x x2 __
4
y 1 2x _21 x2 _32 x3 _81 x4 …
5
y 1 (x 1) _25 (x 1)2 _35 (x 1)3 …
6
y 1 x x2 _21 x4 …
7
b y 1 2x _25 x2 _38 x3 …
8
____
y √2 √2 ( x __
) 3 2 ( x __
) …
4
9
3
x __
x ___
x ____
x …
a 1 __
2
8
48 384
b 1.711 (3 d.p.)
13
b q 2
p _32
k __
2
4
6
8
1
2
1
__
a 1 _21 (x 1) _81 (x 1)2 16
(x 1)3
3
5
___
(x 1)4 …
128
2
b 1.095 (3 d.p.)
(x e) (x e)2
a 1 _______ _______
…
2e2
e
__
__
4√3 x __
2 40 x __
3…
b √3 4( x __
( 3 ___
3
3 (
3
(cos1)
(sin1)
______
______
2
c cos 1 sin1 (x 1) (x 1) 2
6
(cos1)
______
3
4
(x 1) (x 1) …
24
)
)
__
1 4
__
x …}
{ x _21 x2 _61 x3 24
2
1 2
1 3
1
__
___
____
ln5 _51 x 50
x 375
x 2500
x4 …
ii
__
__
1 √3 x
iii sin( x __
3
2
{
)
__
√3
___ x4 …
4!
b 1.64866 (6 s.f.)
__
√3
1 x3
___ x2 __
2!
3!
}
6
7
8
10
11
12
13
14
4
b e1{ 1 _21 (x 1)2 _31 (x 1)3 _81 (x 1)4 … }
5
11
(x 1)3 _41 (x 1)4 …
(x 1) _25 (x 1)2 __
6
a
5
√2
___
1
a i
25
75 2
__
__
_43 16
x 64
x …
7
√3
___
1
__
2
__
8
a
( x __6 )
___
__
__
__
√
__
2
4
4
2
d4y
d3y
dy ____
d2y
____
____
___
b
2y 3 6 2 0
dx dx
dx4
dx
( x __4 ) cot x ( x __4 ) 2( x __4 )
2
2( x __
4
x __
x …
c ln2 __
2
8
x0
32 4
256 6
a 1 8x2 __
x ___
x …
45
3
2
a y 2x _23 x2 _21 x3 …
b 0.2155
3x2 2x3 …
y 2 4x x2 _32 x3 …
x …
y x __
3
6
y 2(x 1) _21 (x 1)2 _21 (x 1)3 …
x ___
5 x4 …
a 1 __
24
2
x3 ___
2 x5 …
b x __
15
3
2
13 3
1 x 4x2 __
x …
3
x …
a y 2 x x2 __
6
1
2 _____
f(x) (1 x)
2(1 x)ln(1 x)
1x
(1 x){1 2ln(1 x)}
3
)
(
2
2 x __
__
3(
6
x _23 x2 _31 x3 1 …
3
)
15
…
( ddxy ) 21 81 161
3
118
4
3
6
2
f(x) _____
1x
( x __6 ) √3 ( x __6 )
√3
___
3
__
3
6
2
f (x) (1 x) _____
{1 2ln(1 x)}
1x
3 2ln(1 x)
b 0.4059 (4 d.p.)
6
6
16 4
128 6
x ___
x …
b 4x2 __
45
3
)
3
4
3
Mixed exercise 6F
Exercise 6D
1
3
2
c y 1 x x2 _34 x3 _67 x4 …
b 1
c 1 x x2 _32 x3 _61 x4 …
2
____
3 (x 3)2 …
1
1 ___
1 (x 3) ____
_______
b y ________
__
256
√(1 x) 2 16
__
17x _____
11x …
x 2x2 _____
6
___
Exercise 6E
x1
x ___
x …,
2x __
2x ____
b 2 ln 3 ___
__
3
___
16
x __
x ___
x …
a x __
2
6
12
b 0.116 (3 d.p.)
2
3
4
x __
x …
b 1 x __
2
2
2
3
3
)
…
ANSWERS
17
19
{ (
)}
1 dy d2y
d3y
a ____3 __ ___ 3 ____2 1
dx
y dx dx
5x3 …
b y 1 x x2 ___
6
c The approximation is best for small values of x
(close to 0): x 0.2, therefore, would be
acceptable, but not x 50.
x2(ln 3)2 x3(ln 3)3
b 1 x ln 3 ________ ________ …
2
6
c 1.73 (3 s.f.)
__
20
2
y
a
line x ⫽ 2
y
b
3
__
__
__
3√2 x __
11√2 x __
____
2 _____
3 …
b √2 √2 ( x __
4
4
4
2 (
6 (
)
)
x
2
y⫽3
)
x
Exercise 7A
1
2
a
c
e
a
c
e
(13, 67.4°)
(13, 112.6°)
(2, 30°)
__
(3√3 ,__3) __
(3√2 , 3√2 )
(2, 0)
b (13,
112.6°)
___
d (√13 , 56.3°)
y
c
__
b (3√3 ,__3) __
d (5√2 , 5√2 )
4
3
Exercise 7B
1
2
b x3
2
4
c y5
x
a x2 4ay or y ___
4a
b x2 y2 2ax or (x a)2 y2 a2
3a 2 ___
9a2
c x2 y2 3ay or x2 y ___
4
2
2
(
3
a
c
a
c
_3
a r 2 cos 6
3__ sec __
c r ___
( 4 )
√2
a arctan 2
3
y
a
)
a
_3
(x2 y2)2 8y2
x2 1 or x 1
r4
r2 sin 2
5
b (x2 y2)2 2x2
2
b r2 8 cosec 2
x
a
Circle centre (0, 2 ) radius a
2
4
b r2 _________
1 sin 2
a cosec
b r __
2
(
b
a
__
3)
c r tan sec a sec θ0
2a
Exercise 7C
1
x
4
a x y 4
2
a
a
π
2
c
Circle radius 6
6
θ 5π
θ 6π
6
b
θ0
5π
4
Half-line
(half of y ⫽ x)
π
7π
6
θ 6
3π
2
c
π
4
Half-line
(half of y ⫽ ⫺x)
119
ANSWERS
4
a
6
a
π
2a
a
3a
4π
2π
θ⫽0
θ⫽0
2a
2π
b
b
6a
a
7a
5c
θ⫽0
θ⫽0
6a
c
c
4a
a
5
7a
θ⫽0
θ⫽0
a
Exercise 7D
3a
1
3
2a
θ⫽0
2a
a
5
7
b
9
7a
11
6a
6a
θ⫽0
2
3
c
7a
4c
4a
a
θ⫽0
2
3
a
_____
4
a
__
6
8
10
2
4
2
4
2a2
3
_____
3
a2 (2p2 q2)
____________
2
__
3√3
a2 __
____
__
4 4
16
[
]
a, ___
a, _____
2
, __
2
(2a, 0) and __
2 3
2 3
a (9.15, 1.11)
b (212, 2.68)
__
a√6
2a
___
a
, 0.421
b r ____ cosec 3
9
(
)(
(
4
(
r cos 3
6
( 2a, __
4 )
)
r cos 1
Mixed exercise 7F
1
2
9
a
_____
8
)
)
7_21a, 1.32
5
2
120
2
8
(
2)a2
_________
48 __
a2 ln √2 or ______
a2 ln 2
_______
4
2
a2 (11
24)
__
4
a2
____
12
5
___
4
Exercise 7E
1
5a
a
____
__
√5 1
cos _______
2
r 3 sec r sec ANSWERS
θ ⫽ 3π
3
Review exercise 2
C
y x2 x __
x4
x3 C x
2 y __
1
R1
P
2
2a
θ⫽0
3a
R2
x ln x C
y ____________
(x 1)2
4
y _21 (e2x 3) cos x
2 sin3 x ______
C
y _______
3 sin 2x sin 2x
xex ________
5ex ________
ex
6 y ________
4(1 x) 2(1 x) 4(1 x)
Ce2x 2x 1
9e2x 2x 1
7 a y _____________
b y _____________
5
5
___
4
3
___
__
4
( a√ 23 , 6 ) and ( 0, 2 )
5
a
√ __
___
__
4
θ ⫽ 4π
θ⫽0
4
1
4
8
b 8.77 m s
9
1
b v 2 _______
ln x c
10
x e ____
e C
a ______
2
2
11
θ ⫽ ⫺ 4π
b 2
2 x2
(3 s.f.)
x2
ex ____
ex __
C
b y ____
2x
2x3 x3
2
2
a y sin x cos x c cos x
c
y
1
2
6
a
O
2a
π
2
π
3π
2
θ⫽0
2π
x
a
7
12
(2a, )
a
π
4
x __1 C e2x
a y __
2 4
c
y
b
( _21 ln 5, _41 ln 5 )
π
⫺ 12
( 1 In 5, 1 In 5)
2
θ⫽0
r ⫽ 2 cos ⫺ θ
1
⫺ 4π
__
8
√3
___
b __
6
8
a
O
r ⫽ 4 cos θ
θ⫽0
4
2
x⫽2
b
9
10
a
(
4)
3 a, __
__
(2
3
)
a y x 1
2
2
__
( 2√2 , __
4 )
(120 t)2
120 t _________
a S _______
4
600
15
C
b 3v2 8v 3 __
2
16
b u _31 ex C e2x
17
18
3 e2t cos t
a k 12
19
5
a2
b ___
8
b y
1
___
2x
20
21
x
⫺1
4
13
2
2
__
2√2 , __
4
b 9 _83 kg
x
2
2
1 __
1 ex2 __
2 e2x2
c __
3
y2 3
sin 2x 3x sin 2x
b y 2 cos 2x __
4
a a 5, b 1
b y e2x (3 2x) 5 x
a y e2x (A cos x B sin x) sin 2x 8 cos 2x
a y et (A cos t B sin t) 2 et
b y et (2 sin t cos t) 2 et
121
ANSWERS
22
a x et (A cos 2t B sin 2t)
b x et (cos 2t sin 2t)
c y
30
31
32
1
_1 dy
a 2t 2 ___
dt
2
2
2
c y A e x B e4x _61 e2x
1
1__
__________
c x ___________
d ___
√2
√(cos t 3)
dy
a t ___
dt
x
d y e3e (A cos(ex) B sin(ex)) _61 sin(2 ex) _31 cos(2 ex)
35
21
71
A 1, B 2, C __
, D __
2
3
36
8 3
4 2
__
__
x 81
x …
a _31 _92 x 27
8 4
4 3
__
__
b _32 _94 x2 27
x 81
x …
O
3π
8
37
t
7π
8
38
23
_1
a y A e2 t B e3t t2 t 1
_1
b y _54 ( e2 t e3t ) t2 t 1
24
c
a
b
c
d
39
1.45 (3 s.f.)
2
y A cos 3 x B sin 3x 2x cos 3x
y cos 3 x 2x cos 3x (1 2x) cos 3x
40
y
1
π
2
π
6
O
5π
6
x ___
x …
a __
2
12
__
4 x __
a 2( x __
4
4
3(
b 0.416147 (6 d.p.)
2
4
)
)
41
42
)
__
__
2 x __
2 4√3 x __
a ln 2 √3 ( x __
( 6 ____
6
3 (
6
b 0.735166 (6 d.p.)
dy
a ___ sec2 x
dx
d2y
____
2 sec2 x tan x
dx2
d3y
____
4 sec2 x tan2 x 2 sec4 x
dx3
)
)
3
)
…
2 x __
2 8
3
b 1 2( x __
( 4 3__( x __4 …
4
d3y
a ____3 1
b 2 x 2x2 _61 x3 …
dx
)
x
x2 ___
x4 …
b __
2
12
3
5…
4 x __
___
4
15 (
dy
b 8 ___
dx
)
2
)
( ) 2(4y 1) ddxy …
2
____
2
c _21 x _23 x2 _34 x3 …
25
b y e (A Bt) 2t e (A Bt 2t )e
c y (3 8t 2t2)e3t
y
d
3t
2 3t
2
3t
43
44
45
O
1
2
1
( 6__5, 9__1 e )
_5
26
27
t
46
29
122
dy ____
d2y
1 ___
a __
y dx 3 dx2 1
(
)
b 1 x x2 _65 x3 …
c The series expansion up to and including the term
in x3 can be used to estimate y if x is small. So it
would be sensible to use it at x 0.2 but not at
x 50.
a 1 _23 x2 2x3 _45 x4 …
b 1.08 (2 d.p.)
47
c ln 2 x2 _61 x4 …
_1
48
5 4
__
x …
a 1 x x2 _32 x3 12
b i 1.1107 (4 d.p.)
ii 1.2460 (4 d.p.)
dy
___
c
≈ 1.222 (3 d.p.)
dx
2
dy
____
≈ 2.45 (2 d.p.)
dx2
49
50
_1 a2
2
a r2
a x A e2 t B e2t t 2
b x e2t t 2
a A _21
a k3
b y A sin x 3x
3
sin x
d y 3x ___
2
2
__
b v A cos 3x B sin 3x _91 x2 81
c y
1 x 2x2 2x3 …
1.12 (2 d.p.)
.
1.5 0.8x 0.208x2 0.131982 x3 …
1.578 (3 d.p.)
2
b x ( 1 t _21 t2 )et
28
a
b
a
b
2
__
Ax cos 3x Bx sin 3x _91 x3 81
x
b r 3 sec __
52
c r 2√3 sec __
6
a2 _89 _89 a2
(
)
ANSWERS
__
__
53
θ 4π
a
60
O
1
a _23 a2
2
, B: __
2
1 a, ___
1 a, ___
b A: __
2 3
2
3
c _49 a
(
initial line
) (
B
b (0.667, 0.421) and (0.667, 0.421)
a
θ 2π
61
27√3
d _____ a2
8
e 113 cm2 (3 s.f.)
a
θ 2π
A
r a (1 cos θ)
ra
a
O
O
55
b At __
3
__
3√3 a
_____
r
cosec 4
__
3__
3√3 a
_____
r
cosec 4
At , θ 6π
a
initial line
62
θ 6π
a2 __
___
a2
b __
4
3 12
a (x 3)2 y2 9
__
x √3 y 6
b
θπ
63
A
(
) (
(
64
2 3
3 a, __
3 a, __
, B: __
a A: __
2
3
2 3
a2 [72√3__ 32
] (9√3__ 4
)a2
c __
8
d 9.07 cm2 (3 s.f.)
a A:(5a, 0), B:(3a, 0)
) (
a
P
P
2a
3a
initial line
6
Q
initial line
Q
3 a, __
3 a, __
, Q: __
b P: __
2 3
2
3
__
2
a
___
√
c
(4
9 3 )
16
(
57
5
, ___
a ___
12 12
58
a P:(4a, 1.107) Q:(4a, 1.107)
59
5√5
b ____ m
4
a
__
√3
___
b ___
12 16
) (
)
Examination style paper
__
1
θ 6π
2
3
3
θ 4π
C
O
( 3, __
3 )
O
)
D
O
θ 4π
)
5
, D: 4a, __
b C: 4a, ___
3
3
2
c
2a
initial line
b (0.943, 0.615)
a
O
56
)
__
π
θ 4
54
2√6
c ____
3
3
__
b 32
(2
3√3 )
A
___
( 1.93, 12
)
9x2 ___
5x3 …
b 1 4x ___
2
3
1
1
a _______
_______
2(r 1) 2(r 3)
c 0.043 (2 s.f.)
initial line
4
3
1
__
__
x 40
cos t 40
sin t e4t (A cos 3t B sin 3t)
5
a
__
3
i sin ___
3
( 4√2 ( cos ___
4
4 ))
3
, ____
35
19
, ____
11
, ____
27
, ____
b ___
20 20 20 20 20
123
ANSWERS
z2
c
8
a
y
20
z1
z3
α
10
z5
z4
6
7
Ax
b v _______
1 Ax2
Ax
c y _______
1 Ax2
2
3
⫺6
a x y 8x 4y 14 0
b
2
c x _58
8
6
4
2
5
O
x
2
4
c
y
8
6
4
2
10
5
O
2
4
d 1 7i and 1 3i
124
⫺2
b _58
y
10
⫺4
2
2
x
__
x 3 √7
4
6
8
10 x