高中数学知识整理

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1){0}á0NL?è5àY02?,XL? ü.

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B2$L?,X?Bü

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? G,f??L? üAF?YAA?

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3)AB???BC?\@AC?.

B3$L?,XtD

L? ü1,23{,,,,,}()knAaaaaakn=≤LL\@è

1)L? üA??$L?tD:2n

2)L? üA??$L?
ù ??êáù ??kXQ(Mn2?,X$L?,XtD:2nk?

3)L? üA??$L?
7??ù ?kXQ(Mn2?
,X?t2?,X$L?,XtD:22nnk??

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?XKL? ü,ì1,X5ê

AB???BA?\@A,B,ì1èA?AB=.

(2)L? ü,X$è1k

?JL?

íL? üA,Bè2bL? üA?
2bL? üB??2?X?,XL? ü/?L? üA?#B??JL?,A?0

ABU.

{|ABxxA=∈U?
}xB∈#￡E?"?

?xL?

fL? üA,Bèp 2bL? üAè àê ?2bL? üB??2?X?,XL? ü/?L? üA?#L? üB??

xL?,A?0ABI.

{|ABxxA=∈I??}xB∈#￡E?"?

?fy

L? üA,B"uYE à2?êè GAB=?Ipè/?L? üA?#Bfy.

??L?

íbL? üA,Bè+2bA~á2bB??2?X?,XL? ü/?AfB???L?,A?0AB?.

{|ABxxA?=∈??}xB?#￡E?"?

?-L??>9L??

9L??,A?

0cA.

{|cAxxU=∈??}xA?#￡E?"?

?L? üE¤1k,X?Bü

?-1
B1AA?=U,A?=?I,cAA=?I,cAAU=U

B2()ccAA=,cU?=,cU=?

B3AB?=?\@AB?

B4ABA=I\@AB?,ABA=U\@AB?

B5AB???BC?\@AC?

B6AB?\@ccAB?

B7()()ABBA??=?U\@AB=

(3)L? ü,XE¤1k"?í
`2?,XtD

?L? ü,XE¤1k"?í

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?B1x6_

ABBA=UU,ABBA=II

B24§ ü_

()()ABCABC=UUUU

()()ABCABC=IIII

B3úG!_

()()()ABCABAC=UIUIU

()()()ABCABAC=IUIUI

B4

_

()AABA=UI,()AABA=IU

B5?=
_

()cccABAB=IU,()cccABAB=UI

B6 ún_

()ccAA=,{()}ccccAA=

2.
QNl

(1)
QNl,XEúè úèEú ú

?
QNl,XEú??á?Eú ú

s
QNlpq→

qp→Xa s
QNl,XEú
QNl

~~pq→Xa s
QNl,X ú
QNl

~~qp→Xa s
QNl,XEú ú
QNl

WàKè,XG2?V ?ò/

!8ê,??
QNlaJEú ú
QNl1?.

?
QNl,XEú
QNlè ú
QNlèEú ú
QNl,X,óè

B1
QNlpq→ ?JEú ú
QNl~~qp→1?.


QNlpq→Xa,óêèJEú ú
QNl~~qp→?o,ó.


QNlpq→XaêèJEú ú
QNl~~qp→?o.

B2
QNlpq→??Eú
QNla ú
QNlEú úG2?è′!81?
QNl.

(2)ú??U5ê

???U5êèú5ê


QNlpq→Xa,óê,?
?ú??U5ê

pq???qp?pèp?q??ú??U5êèA?0pq?.!8ê,q3p??ú??U

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5ê.

???U5ê,ú5ê
`,ó)úL? ü

A?5êp,q??,ó)úL? üú?P,Q

B1p?q??ú5êPQ??

B2p?q????U5êQP??

B3p?q??ú??U5êPQ?=

3.rD

(1)rD,XE¤1k

?D2?

B1Y)úD:0m≠p,6VúDnm(m,nXaf2?,XHD)

B2′)úD:′L$á~)??D

B3ù ?G2?

7?''
DL?N,HDL?Z,Y)úDL?Q,′)úDL?I,rDL?Rè}Kù ?G2?V?.

NZQR???,IR?

QIR=U,QI=?I

?rD$è1k,X
??Bü

?-1rDa,b,c

B1x6_:abba+=+,abba=

B24§ ü_:()()abcabc++=++,()()abcabc=

B3úG!)[()abcabac+=+,()abcacbc+=+

(2)rD,X??G2?

?rD,X
??Bü

B1í??rDaè??YNM
Yè ?Y?t?0?.

0a>,0a=,0a<

B20a>,0b>\@0ab+>,0ab>

B30a>\@0a?<,0a<\@0a?>

B420a≥

?GbrD??,X
??Bü

?-1YrDa,b,c

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B1ab>,bc>\@ac>.

B2ab>\@acbc+>+,acbc?>?.

B3ab>,0c>\@acbc>.

B4ab>,0c<\@acbc<.

?rD,X4±í

B14±í:rDaüDEHTíh,X&? s&?,XC±/?è/?rDa??4±íèA?0a.

B24±í,Xn

íbrDa??4±ínV?.

(0)

(0)

aaa

aa

≥?=

??
B34±ík?Bü

??4±í,Xn ?-1.

1)0a≥

2)aa=?

3)22aa=

4)abab=

5)aabb=(??0b≠)

?rD,X?Bü

frDa,bY

B12200,0abab+=?==

B200,0abab+=?==

B32200,0abab+=?==

4.áD

(1)áD

?áD,XG?

B1,? a1b1???DA?i

21i=?(??1i=?)

!8ê,?iXa<.D )!.

B2f??,X!7Da?T

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1)aai?=

2)a???G?
ai±

(2)áD,XE¤1k

?áD,ì1,X5ê

a,b,c,dXarDê

B1,abicdiacbd+=+?==

B200,0abiab+=?==

?EEAáD

B1f-áDabi+(a,bXarD)??<.DF?ú,X0ú ?6?Bó ?è/?áDabi?XaEEAáD,A?

abi+. G,abiabi+=?

B2áDaJEEAáD
`?rD.

?áD,XˉíE¤1k

áD,Xt"?è￡"?ú<.D )!i,??0ú ?è''
arDF?úè -E?F?úú?Eˉ>t￡E¤1k.

áD,X,"?E?/?
èyü21i=?Eˉ>Au1kè5àáD,XL8"?íú$ú!? àê,1ú!?,XEEA

áDEˉ>Au1k.

?-1rDa,b,c,d

B1t"?:()()()()abicdiacbdi+++=+++

B2￡"?:()()()()abicdiacbdi+?+=?+?

B3,"?:()()()()abicdiacbdadbci++=?++

B4L8"?:2222()()()()abiabicdiacbdbcadicdicdicdicdcd++?+?==+++?++/,0cdi+≠)

?áD,XE¤1k"?í

?-1áDL? üC?f??,XY2?1z,2z,3z

B1?Ká:12zzC+∈,12zzC∈

B2x6_:1221zzzz+=+,1221zzzz=

B34§ ü_:123123()()zzzzzz++=++,123123()()zzzzzz=

B4úG!_:1231213()zzzzzzz+=+,1231323()zzzzzzz+=+

?i??,?

1i=?,21i=?,31i=?,41i=,5ii=,61i=?,L

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?áD??,X?Bü

B10a<,0b
L8Z0a<,0b<,X??èüJa???aa,b??0ú ?′GF?Yabab=

B20a>,0b
L8Z0a>,0b
?NM?,X$è1k

? )NM?a?NM?

1) )NM?: ?YD+
`0ú ?,/?,XNM

2)?NM?:+ )NM?ê )NM?,X
`4??,X>
?X5?NM?,ìG,XVé

??NM?


1)￠?:?4???NM?,X ?t )NM?

2)DNM:áù ?‘(Mn0ú ?’??NM

3)(MnNM,X2?D:?(MnNM
èL8 ?‘(Mn0ú ?’=-,XF?ú

4)(MnNM,X?D:(MnNM,X,/?
‘(Mn0ú ?’,ì,,XtD

5)?NM?,X?D: ?NM?D
?P?,X?D

??NM?,Xt"?è￡"?è,"?

1)?NM?$è1k,X?Bü

A,B,C>
?Tx6_èúG!)[è4§ ü_?0?.

2)?NM?,Xt"?è￡"?è,"?

?NM?,Xt"?è￡"?:''?à ?è üJ à2ONMEˉ>H)ú.

?NM?,X,"?:yüúG!)[)? a üJ à2ONM.

,"?@?

?
?,X,"?@?

1)()mabmamb+=+,()mabmamb?=?

2)222()2abaabb+=++,222()2abaabb?=?+

3)()2()2ababababab+=+?=?+,22()()4ababab?=+?

4)22()()ababab+?=?

5)2()()()xbxaxabxab++=+++

6)2()()()axbcxdacxadbcxbd++=+++

7)32()()()()()xaxbxcxabcxabbccaxabc+++=+++++++

8)333()3()abababab+=+++,333()3()abababab?=???

9)333()3()abababab+=+?+,333()3()abababab?=???

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10)2233()()abaabbab+?+=+,2233()()abaabbab?++=?

11)22222()abcabcabbcca++=+++++

12)222()abcabcabbcca++=++?++

13)222333())3abcabcabbccaabcabc++++???=++?

14)2222424()aabbaabbaabb++?+=++

1.′?ú?·,X
?@?

(1)()mambmab+=+,()mambmab?=?

(2)2222()aabbab++=+,2222()aabbab?+=?

(3)22()()ababab?=+?

(4)2()()()xabxabxaxb+++=++

(5)2()()()acxadbcxbdaxbcxd+++=++

(6)32()()()()()xabcxabbccaxabcxaxbxc+++++++=+++

(7)3223333()aababbab+++=+,3223333()aababbab?+?=?

(8)322()()ababaabb+=+?+,322()()ababaabb?=?++

(9)2()()abcabbccaabc+++++=++

(10)333222abcabcabcabcabbcca++?=++++???

2221(){()()()}

2abcabbcca=++?+?+?

(11)4242222()()aabbaabbaabb++=++?+

01.Y)ú? ?JE¤1k

1.Y)ú?èú?

(1)Y)ú?:?NM?,AB??!¨, G6VúDAB(0B≠)???$

(2)úD?:á?NM?,XY)ú?

2.Y)ú?,X?Bü

?-1?NM?,,ABC(0B≠,0C≠)

(1)AACBBC×=×

(2)AACBBC÷=÷

3.Y)ú?,XˉíE¤1k

?-1?NM?,,,ABCD

(1)ABABCCC±±=,ABADBCCDCD±±=(J
è0C≠,0D≠)

(2)ACACBDBD×=(J
0B≠,0D≠),ACADADBDBCBC÷=×=(J
0B≠,0C≠,0D≠)

02.F?úú?
`4ú?

1.F?úú?R??′?ú?·

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(1)1111()ABBAAB=??

(2)1111()ABCCAABBC=??

2.?Nú?

(1)

A

AB

CBC=

(2)AACBB

C

=

(3)

A

ADB

CBC

D

=

(4)AADCBDC

BD

=+

+

03.!¨_?,X?Bü

1.!¨_?,X?Bü

!¨_?::abcd=(?acbd=?adbc=?0adbc?=)?0?êèí

(1)abcdbd++=

(2)abcdbd??=

(3)abcdabcd++=??

(4)acbd=??

2.?W!¨?Bü

!¨_?:::abcdef==?0?êèí

ReaceacePaQc

bdfbdfPbQdRf

++++====

++++(J
,0bdf++≠,0PbQdRf++≠)

04.′)ú? ?J$è1k

1.′)ú?


?Y,X>
2.G?
?
`0??
?

(1)G?
?è0??
?:f??,XrDêJ4§prD,X>
B12||AA=

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B233AA=

(2)G?
,XE¤1k"?í:0,a>0b>p,

B1abab=

B22abab=

B3aabb=

B42aabb=

> sè]+B10,a≤0b≤\@abab=?

??B30,a≥0b<\@aabb=?

3.ú!?Y)úê

?-1!7Da,b(J
ab≠)

(1)

1ab

abab=?±

m

(2)

33223

33

1aabb

abab

+=

±±

m

4.?JG?
?

(1)0,a≥0b≥p,2ababab++=+

(2)0ab≥≥p,2ababab+?=?

05.′)úD,ì1,X5ê

1.′)úD,ì1,X5ê

,,,abcdXaY)úDè,mnXa′)úDê,

(1)0abm+=?0,0ab==

(2),abmcdmacbd+=+?==

(3),ambnabmn+=+?==

01.?/??axb=???·

1.?/??axb=???·

(1)0a≠p,???/??axb=??,X
bxa=.

(2)0a=p,?é0b≠\@+b0(0)xb×=≠?′
.

?é0b=\@+b00x×=??T′Dt?·.

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02.`??/?,X?·

1.`??/?,X?·

[?"?1]yü′?ú?·,X?·"?

f-`??/?′?ú?·?6V()()0(0,0)axbcxdac??=≠≠6?ê,k?t


bx

a=?


dx

c=

[?"?2]yü`<G??,X?·"?

f-`??/?ê?`<G??2()xAB?=èk?t


xAB=+?
xAB=?

[?"?3]yü"
@?,X?·"?

?-1`??/?20axbxc++=??"
@?

24

2

bbacx

a

?+?=?
24

2

bbacx

a

???=

(M?
,x??2?DJDê G2bb′=p

2bbac

xa′′?+?=?


2bbac

xa′′???=

03.`??/?,Xhü

1.`??,X′?ú?·

`??/?20axbxc++=???t
ú?α,βpè`??2axbxc++z?′?ú?·

2()()axbxcaxxαβ++=??

2.`??/?
b2?D,XG2?

A?`??/?20axbxc++=???
ú?,αβèí

b

aαβ+=?,

c

aαβ=

3.
B?/?,X?·.Bn`??/?

?-1D,αβ?TAαβ+=,Bαβ=?0?èí ?1.Bn`?NM2?D1??t-1 ?G￡x??`??/?

20xAxB?+=

J
,αβXa??/?,X?t


4.`??/?EEA
,XH)ú

(1)Y2?DY)úD,X`??/?,X
abm+(,abQ∈,mI∈)\@abm?3A1?/?,X




(2)Y2?DrD,X`??/?,X
abi+(,abR∈,1i=?)\@abi?3A1?/?,X
.

04.`??/?,X???

1.`??/?
,X???

2?DrD,X`??/?20axbxc++=?fè???24Dbac=?\@

(1)0D>?Y?tá à,Xr


(2)0D=?G?


(3)0D

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2.???,Xhü

[2O
_1]2?D
rDèt-1G￡,xy??`??/?(,)0fxy=
81Y,xy??rD,X5êê,\@ú

A1?/??B5?Gbx(?
y)??`??/?,''
ayü???0D≥k?K'',XGby(?
x)??

L$
5ê.

[2O
_2]`??2axbxc++(,,abcXarDè0a≠)ê?`<G??,Xú??U5ê

`??/?20axbxc++=?????0D=.?<

2240()

2

bDbacaxbxcax

a=?=?++=+

[2O
_3]2?D
rDèt-1G￡,xy??`??(,)fxy?f ?1ú?·????,/?6?,Xú??U5

êú?/??(,)0fxy=?B5?Gbx(?
y)??`??/?"?·???D?Gby(?


x)??`<G??`??/?0D=?????D′?Y0D′=

01.P???/??

1.P???/?,X"?·

[2O
_1]′?ú?· a"?·–íbP???/?()0fx=f-()fx′?ú?· ayü1?,X?BüúJE@ê

"??/?Eˉ>"?·.

[2O
_2]E?E? ?6
`6Eˉ>"?·–f-> ?6è6 a
B[2O
_1]Eˉ>"?·.

2.Y??/?31x=??<.
ω???Bü

321(1)(1)0xxxx=??++=???t<.
ωèí+bω?210xx++=??


13

2

iω?+=?
13

2

iω??=

B131w=,210ww++=

B21ww+=?,1ww=`

B311ww+=?

B42ww=,2()ww=→Y??/?31x=??Yt
21,,xωω=.

B5Y??/?33xa=??Yt
2,,xaaaωω=.

02.?NM??/?,X?Bü

1.Y??/?
a2?D,XG2?

??Y??/?320axbxcxd+++=??Yt
ú?,,αβγ

,baαβγ++=?,caαββγγα++=daαβγ=?

2..BnY??/?

?-1YtD,,αβγ?TAαβγ++=,Bαββγγα++=,Cαβγ=\@

?1.BnY?NM2?D1,XGbx??Y??/?

320xAxBxC?+?=

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J
,,αβγXaA1?/?,XYt
.

3.?NM??/?EEA
,XH)ú

(1)Y)úD2?D,X?NM??/?Y′)ú
êè?YEEA,X′)ú
.

(2)rD2?D,X?NM??/?Yá
ê,?YEEAá
.

4.?NM??/?,X ?6

?é?-1?/?()0fx=??
èí?"1()0,()0,()0fxcfcxfx?===??


03.?/?4?,X"?·(1)

1.4???/?4?,X"?·

yüt￡"?,·9"?,1"?4y￡t-1G￡,XtDèE?E? à?· ?6 a"?·.

2.?/?4?,Xánaá6?

Gb,xy??4???/?4?0,axbyc++=0axbyc′′′++=(0a′≠,0b′≠,0c′≠)

(1)abcabc==′′′??T′0K?t?·.(án)

(2)abcabc=≠′′′?′?·.(á6?)

04.?/?4?,X"?·(2)

1.`?/?4?,X"?·

[2O
_1]6V?/?4?

??

?

=

=

0

0

?J???+

?????+???6

f-4???/?H)ú?¤t-1G￡,X>
[2O
_2]6V?/?4?

??

?

=

=

0

0

?J???+

?J???+???6

(1)B1Vp¤`??/? ?1′?ú?·èí′?ú?·??t???/?.

B2f-???/?ú?a`??/?,ì4§ üY[2O
_1]Eˉ>"?·.

(2)B1f-`??/?Eˉ>t￡E¤1kè#\ ?`?NM|????/?.

B2f-|?,X???/?
`?t`??/?
?)?t,ì6(0?èX?[2O
_1]???/?4?Eˉ>"

?·.

(3)B1f-?t`??/?Eˉ>t￡E¤1kè#\ ?DNM ?? ?1′?ú?·,X`??/?.

B2￡B1?"?·.

[2O
_3]6V?/?4?{xyAxyB+==???6

yü,ABXE??tGb,xy??`??/?Eˉ>"?·.

05.E à


1.|μE à
,X?"?

t-1 ?G￡x???NM??/?()0,()0fxgx==??E à


[?"?1]ú?"??t?/?,X?·L?èü"?t?·L?,XxL?.?<,

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1.á1?axb>???·

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B3Gby??í/?/?| aèkò6,X?/?(,)0fxy?=

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1)ín
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2222cosbcacaB=+?

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