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1
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������������ ���������������� -��������������������������������
(1) (2) 217 (2) (3) { }1R − −
(3) (2) 2
(4) (1 ) sec 1x <
(5) (1) 0
(6) (1) 45
(7) (3) 1
(8) (4) 4
(9) (2) ( )1, 2−
(10) (1) 3
(11) (4) 25
(12) (2) 60˚
(13) (2) ˆˆ ˆ2i j k− −
(14) (4) ���� ��� (15) (3) 0
(16) (3) 0
(17) (1) 6
(18) (2) log1
x
x +
(19) (4) 3 52
3log 2
x
c+
+
(20) (4) 2
3
(21) y = cos x
(22) log log logx y z
ky z z x x y
= = =− − −
��.
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2
log x = ( ) ( )k y zk y z x e
−− ⇒ = ... (1)
log y = ( ) ( )k z xk z x y e
−− ⇒ = ... (2)
log z = ( ) ( )k x yk x y z e
−− ⇒ = ... (3)
( ) ( ) ( )1 2 3× ×
xyz = ( ) ( ) ( )k y z k z x k x ye e e
− − −× ×
= ( )k y z z x x ye
− + − + −
= 0 1e = xyz = 1
(23) ( )tan 45 A° − = tan 45 tan
1 tan 45 tan
A
A
°−+ °
= 1 tan
1 tan
A
A
−+
(24) ������� ������������� ����� ����� = 6!
2!
= 6 5 4 3 2!
2!
× × × ×
= 360
(25) S = 4 7 10
15 25 125
+ + + +L
a = 1
1, 3,5
d r= =
s∞ = ( )21 1
a dr
r r+
− −
= 2
13
1 5 3 25 3551 4 5 16 1611 15 5
×+ = + × = − −
(26) OAuuur
= ˆˆ ˆ2 3 5i j k+ −
OBuuur
= ˆˆ ˆ3 2i j k+ −
OCuuur
= ˆˆ ˆ6 5 7i j k− +
ABuuur
= ˆˆ ˆ2 3OB OA i j k− = − +uuur uuur
BCuuur
= ˆˆ ˆ3 6 9OC OB i j k− = − +uuur uuur
= ( )ˆˆ ˆ3 2 3i j k− +
BCuuur
= 3ABuuur
���, BCuuur
�!"# ABuuur
��$��� %��&'�(. )��� B ��� %������ �(�
)�����
A,B,C *�+ ���,-� .���� )/#.
(27) ( )2 16
4
xf x
x
−=
+
( )f x ��� 4x = − -�� ��+�"����&1��.
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3
���, ( )f x ��� { }4x R∈ − − , �2 *3%��4 �(�5# %��&'67��/#.
(28) y = 10 10log log logx x e
e=
dy
dx = 10
1loge
x
dy
dx = 10loge
x
(29) sin
1 cos
xdxx+∫
= logdu
u cu
−= − +∫
= log 1 cos x c− + +
(30) 4 2
1A
x
= −
( )( )2 3A I A I− − = 4 2 2 0 4 2 3 0
1 0 2 1 0 3x x
− × − − −
= 2 2 1 2
1 2 1 3x x
− − − −
( )( )2 3A I A I− − = 0 �� %�������,�(8�
2 2 1 2
1 2 1 3x x
− − − −
= 0 0
0 0
( )
( ) ( )( )2 2 4 2 3
1 2 2 2 3
x
x x x
− + − − − − − + − −
= 0 0
0 0
2 2x − = 0
x = 1
(31) .���� a S∈ -�/# ( , )a a R∈ ����� R )�� �!9,&�/#.
( , )b a R∉ ������� .��5��" ( , )a b R∈ � :� ;"��# ���. ��"
1����� <=��� *3%��4 ;"�� ( , )a b R∈ -�/# ( , )b a R∈ �� <2 >-5#. �����
R )�� ��6?+�/#.
( , )a c R∉ @��" ( , )a b �!"# ( , )b c )A� �4;"����8 R –� ��$ ����.
����� R)�� “�&�� .�” ��� ;�����". ��� “ R )�� �&�� )/#”
�2 <!" ;�����/#. ��� R )�� �&�� )/#.
����� R )�� �!9,�. ��6?'. �&��� %��&'� )/#. ��� %�������,&
%��&'� ����� %��&'��/#.
(32) �.�.
= cot(180 ).sin(90 ) cos( )
sin(270 ). tan( ).cosec(360 )
θ θ θθ θ θ+ − −
+ − +
= cot cos cos
( cos ) ( tan ) cosec
θ θ θθ θ θ× ×
− × − ×
1 cos x u+ = ��. sin x dx du− =
sin xdx du= −
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4
= 2cos cos sincot cos
sinsin
cos
θ θ θθ θ
θθθ
×× =
= �.�.
��� CDE����,&�
(33) 9 ��(1��� 5 ��(1�F�/ 1�&������& ����#. .��
2 ��(1�F�/ ��-���� 1�&���� ����#
��� 1�&�����!���
����� ����� = 37C
= 7 6 5
3 2 1
× ×× ×
= 35 .
(34) 2
3
1, , 10a x b n
x= = =
1rT + = n r r
rnC a b− ×
= 2 10
3
110 ( )
r
r
rC xx
− ×
= 20 2 310 r r
rC x x− −×
1rT + = 20 510 r
rC x −
20 5r− = 15
5r− = 5−
1r =
1 1T +∴ = 20 5
110C x −
= 15
110C x
��� 15x -� /$�# 10 .
(35) ��+�&���4:�,
1m = tan 60 3° =
2m = tan120 tan(180 60 )° = − °
= 3−
∴ ������� �������(
y = 1m x c+ ,
3 7y x= + , 3 7y x= − +
(36) �.�. 2 2 2
1 1 1
x y z
x y z
=
60°
30°
120°
O
y
x
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5
2 2 1c c c→ −
3 3 1c c c→ −
=2 2 2 2 2
1 0 0
x y x z x
x y x z x
− −
− −
= 2
1 0 0
( )( ) 1 1y x z x x
x y x z x
− −
+ +
( )y x− �!"# ( )z x− G 2c �!"#
3c -�4:� >�2�� ���
= ( )( )[ ]y x z x z x y x− − + − −
= ( )( )( )x y y z z x− − −
=�.�.
(37) | | 3 | | 4 | | 5a b c= = =rr r
| |a b c+ +rr r
= 0 �� %�������,�(8���
2| |a b c+ +rr r
= 2 2 2| | | | | | 2( )a b c a b b c c a+ + + ⋅ + ⋅ + ⋅r r rr r r r r r
0 = 9 16 25 2( )a b b c c a+ + + ⋅ + ⋅ + ⋅r rr r r r
50− = 2( )a b b c c a⋅ + ⋅ + ⋅r rr r r r
25 .a b b c c a− = ⋅ + ⋅ + ⋅r rr r r r
(38) logx x dx∫
logu x= dv∫ = x dx∫
du =1dx
x v =
2
2
x
udv∫ = uv vdu− ∫
logx x dx∫ = 2 21log | |
2 2
x xx dx
x− ×∫
= 2 2
log | |2 4
x xx c− +
(39) ( )P A = 3
8
1( )
8P B =
( )P A = 1 ( )P A−
= 3 5
18 8− =
( )P A B∪ = ( ) ( )P A P B+ ( AQ �!"# B *��2%���" 1�/# C�H67�( ( ) 0P A B∩ = )
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6
= 3 1 4 1
8 8 8 2+ = =
( )P A B∩ = ( ) ( )P B P A B− ∩
= 1 1
08 8− = .
(40) 0
2 2 2 2lim
2 2x
x x
x x→
+ − + +×
+ +
=( )0
2 2 1 1lim
2 2 2 22 2x
x
x x→
+ −= =
++ + .
(41) (a) | |x =
;
;
x
x
−
0x ≤
0x > I�
I�
( )f x =
x x
x x
− ++
0x ≤
0x > I�
I�
( )f x =
0
2x
0x ≤
0x > I�
I�
( )g x =
x x
x x
− −−
0x ≤
0x > I�
I�
( )g x =
2
0
x−
0x ≤
0x > I�
I�
0x ≤ @#%��J�
( )( )f g xo = [ ( )] ( 2 )f g x f x= −
0x > @#%��J� ( )f g xo = [ ( ) (0) 0f g x f= =
0x > @#%��J�
( )( )g f xo = [ ( )] [2 ]g f x g x= =0
0x ≤ @#%��J�
( )( )g f xo = [ ( )] (0) 0g f x g= =
(b) 3
3
x y
x y
+ ≥
+ =
2 5
2 5
x y
x y
− ≤
− =
2 3
2 3
x y
x y
− + ≤
− + =
x 0 3 x 0 2.5 x 0 -3
y 3 0 y -5 0 y 1.5 0
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7
(0,3), (3,0)A B ��� 3x y+ = �2 ���,-K(8 �4 �(��(. (0,0) , 3x y+ ≥
�2 .�����,-� L'M� �/�N� ���.
(0, 5), (2.5,0)C D− ��� 2 5x y− = �2 ���,-K(8 �4 �(��(. (0,0) , 2 5x y− ≤
�2 .�����,-� L'M� �/�N� ;(8�.
(0,1.5), ( 3,0)E F − ��� 2 3x y− + = �2 ���,-K(8 �4 �(��(. (0,0) ,
2 3x y− + ≤ �2 .�����,-� L'M� �/�N� ;(8�.
���∆ GHI ��� L'M �$��/#.
(42) A B C π+ + = ��%�������,�(8�.
(a) �.� cos cos cosA B C= + +
= 2cos cos cos2 2
A B A BC
+ − ⋅ +
2 2 2
A B Cπ+ = − Q
= 2cos cos cos2 2 2 2
C A BC
π − ⋅ − +
= 22sin cos 1 2sin2 2 2 2
C A B C ⋅ − + −
= 1 2sin cos sin2 2 2 2
C A B C + − −
= 1 2sin cos cos2 2 2 2 2
C A B Cπ + − − −
= 1 2sin cos cos2 2 2 2 2
C A B A B + − − +
= 1 2sin 2sin sin2 2 2
C B A + ⋅ ⋅
= 1 4sin sin sin2 2 2
A B C + ⋅ ⋅
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8
=�.�.
(b) �.�. cos cos cosa A b B c C= + +
= 2 sin cos 2 sin cos 2 sin cosR A A R B B R C C+ +
sin
a
AQ = 2
sin sin
b cR
B C= = .
= [sin 2 sin 2 sin 2 ]R A B C+ +
= 2 2 2 2
2sin cos sin 22 2
A B A BR C + − ⋅ +
= [ ]2sin( ) cos( ) sin 2R A B A B C+ ⋅ − +
= [ ]2sin( ) cos( ) sin 2R c A B Cπ − − +
[2sin .cos( ) 2sin cos ]R C A B C C= − +
= 2 sin [cos( ) cos ]R C A B C− +
= 2 sin [cos( ) cos[ ( )]R C A B A Bπ− + − +
= 2 sin [cos( ) cos( )]R C A B A B× − − +
= 2 sin 2sin sinR C A B×
= 2 sin 2sin sinR A B C
= 2 sin sina B C
�.�. = �.�.
��� CDE����,&�.
(43) (a) 2( ) 2 4 6 2P n n n n= + + + + = +L n N∀ ∈ ��.
�-�-�-�- (1)
n = 1 ��.
(1)P = 22 1 1 1× = +
2 = 1
(1)P∴ ��� ;���.
�-�-�-�- (2)
n = k ��.
( )P k = 22 4 2k k k+ + + = +L ��� ;��� ��
�-�-�-�- (3)
n = 1k + ��.
( 1)P k + = 22 4 2( 1) ( 1) ( 1) ( 1)( 2)k k k k k+ + + + = + + + = + +L
�.�. = 2 4 2 2( 1)k k+ + + + +L
= ( 1) 2( 1)k k k+ + +
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9
= ( 1)( 2)k k+ +
( 1)P k∴ + ��� ;���.
��� ���� %��/��=��� O# ( )P n ��� n N∈ -� �� ���E!/#
;���.
(b) ���P���1� )+#� ����� 30a = ��.
*3%��4 �� �Q+># ���P���1� ����� �+�� �&R������.
���P���1� �����N� %��&'>�2
1 230,30 2 ,30 2× × L
�� *4 %�4�/� %��&'>�2��/#. ��
2 �� �Q+ >-1� = 30 2 1202× =
4 �� �Q+ >-1� = 430 2 480× =
��� n�� �� �Q+ >-1� = 30 2n× .
(44) (a) �.�.
log log log
log 2 log 2 log 2
log3 log3 log3
x y z
x y z
x y z
2 2 1 3 3 1,R R R R R R→ → → −
log log log
log 2 log 2 log 2
log3 log3 log3
x y z
=
log log log
log 2 log3 1 1 1
1 1 1
x y z
× 2 3R R∴ ≅
= 0
�.�. = �.�.
��� CDE����,&�.
(b) ar = ˆˆ ˆ2 3i j k− +
br = ˆˆ ˆ2 3 4i j k− + −
cr = ˆˆ 2j k− +
ar = lb mc+
r r
ˆˆ ˆ2 3i j k− + = ˆ ˆˆ ˆ ˆ( 2 3 4 ) ( 2 )l i j k m j k− + − + − +
1 = 2l− 1
2l⇒ = −
2− = 3l m− ... (2)
3 = 4 2l m− + ... (3)
(1) �!"# (2)G L'��
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10
1
32
m − −
= 2= −
3
22
− + = m
1
2m =
l �!"# m -� ����� ������ (3)-� E+�N&
1 1
4 22 2
− ×− + × = 3
2 1+ = 3
��� *4 %��&' )�� �!2 �4 %��&'��� *4 �-���� �������/#.
��� �3%��&'�( *4 �8 %��&'�8�/#.
(45) (a) ( )h x �2 ��'� ( , )R = −∞ ∞ -�� ;(8 ��� �(���K#
��+�"����,�(8�.
0
lim ( )x x
h x→
=0
1lim sinx x
xx→
= 0 0
0
1( )
sinx h x
x=
0x = 0 ���!/1
( ) sinh x xx
=
x− ≤ 1
sinx xx≤
( )g x = ,x− 1 1
( ) sin , ( )f x h x xx x
= =
0
lim ( )x
g x→
= 0 , 0
lim ( ) 0x
h x→
= �!"# 0
1lim sin 0x
xx→=
��&U,�� ��!2����-
0
limx→
1
sinxx
= 0 (0)h=
( )h x∴ ��� �:� >J ��&%��NK# %��&'67����
(b) sin y = sin( )x a y+
x = sin
sin( )
y
a y+
���N&
1 = 2
sin( ) cos sin cos( )
sin ( )
dy dya y y y a y
dx dx
a y
+ ⋅ ⋅ − × + ⋅
+
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11
2sin ( )a y+ = [sin( ) cos cos( ) sin ]dy
a y y a y ydx
+ ⋅ − + ⋅
= [sin( )]dy
a y ydx
+ −
2sin ( )
sin
a y dy
a dx
+=
(46) (a) I = 6
2 1
xdx
x⋅
+∫ .
= 2 1
62
tt dt
t
−×∫
= 3 2( 1)t dt−∫
= 3
33
tt
−
= 3 3
33
t tc
−+
= 2( 3)t t c− +
= 2 1(2 1 3)x x c+ + − +
= 2 1(2 2)x x c+ − +
= 2( 1) 2 1x x c− + +
(b) 1A �!"#
2A ��� %��=��8'�( I �!"# II %�V5# ������ /=�/#
C�H67�( ��.
B ��� ���N� E�W C�H��!��� C�H67 ��.
1( )P A = 60
100 1
3( / ) 0.03
100P B A = =
2( )P A = 40
100 2
4( / ) 0.04
100P B A = =
��UX-� ��!2����-,
1( / )P A B = 1 1
1 1 2 2
( ) ( / )
( ) ( / ) ( ) ( / )
P A P B A
P A P B A P A P B A
⋅⋅ + ⋅
=
60 3 18
100 100 100060 3 40 4 18 16
100 100 100 100 1000 1000
×=
× + × +
= 18 1000 9
1000 34 17× =
2 1x + = t ��
2 1x + = 2t
2dx = 2tdt
��%��J� x = 2 1
2
t −
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12
2( / )P A B =
40 4
100 10034
1000
×
= 16 1000 8
1000 34 17× =
1( / )P A B > 2( / )P A B
��� %��=N��8' I ��=�W���!��� C�H��M .����� �4�����
���N� Y!�,& ���2 .�' ��W��4��� <�#.
(47) (a) ,a v �!"# s ��� >���#. ������# �!"# Z+# ��.
v = ds
dt
a = 28 /dv
m sdt= −
dv∫ = 8 dt−∫
1
8v t c= − +
E�+���� �������#���� 0t = , 24 /v m s=
24 = 18 0 c− × +
1c = 24
v∴ = 8 24t− +
ds
dt = 8 24t− +
ds∴ = ( 8 24)t dt− +∫
s = 2
2
824
2
tt c− + +
s = 2
24 24t t c− + +
t = 0 @#%��J�, 0s =
0 = 2c
s∴ = 24 24t t− +
���,&�' ���A�8 C"��#%��J� , v = 0
8 24t− + = 0
3t =
t = 3 @#%��J� 24 3 24 3s = − × + ×
= 36 72− +
s = 36 [,&'
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13
���,&�' ���A( �����/ 4 [,&4�/ >��� C"�����#.
(b) 2 22 3 6 19 20 0x xy y x y− − − + − = .
2 22 3x xy y− − = (2 3 )( )x y x y− +
2 22 3 6 19 20x xy y x y− − − + − = (2 3 )( )x y l x y m− + + +
;x y %�J���85# �!"# ��=���85# ��� %�V�
6− = 2l m+ ... (1)
19 = 3l m− ... (2)
20− = lm ... (3)
(1), (2)G L'��,
2l m+ = 6−
3l m− = 19
5m = 25−
5m = −
( 10)l∴ + − = 6−
l = 4
��� �I�� �������(
2 3 4x y− + = 0
5x y+ − = 0
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