数学知识手册

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⁚᦮ᦪ᦮ᦪ(Z)ᒹ᦮ᦪ()ஹ᦮ᦪ(Z-)ᐸ᦮ᦪ0᦮ᦪ.ᯠᦪ(N)ᒹ᦮ᦪ0,ᨬḄᯠᦪ0.ஹ⌕Ḅᦪ1.᜻ᦪᏔᦪপ᜻ᦪ#$%&2᦮◀Ḅ᦮ᦪ()⊤+2k+l,,᦮ᦪ.ফᏔᦪ#%&2᦮◀Ḅ᦮ᦪᒹ0,()⊤+2&,%᦮ᦪ.(3)Ꮤᦪ᜻ᦪ/0ឋ2#Ꮤᦪ3Ꮤᦪ=Ꮤᦪ᜻ᦪ3᜻ᦪ=Ꮤᦪ᜻ᦪ5Ꮤᦪ=᜻ᦪ᜻ᦪX᜻ᦪ=᜻ᦪ᜻ᦪXᏔᦪ=ᏔᦪᏔᦪXᏔᦪ=Ꮤᦪ(4)678᦮ᦪ9᜻Ꮤ᜻ᦪ:Ꮤᦪ;᜻ᦪ<᜻ᦪḄ=>᜻ᦪᏔᦪ<᜻ᦪ=>Ꮤᦪ;᜻ᦪḄ᦮ᦪ?@>᜻ᦪᏔᦪḄ᦮ᦪ?@>Ꮤᦪ.2.2ᦪᔠᦪ(1)2ᦪ#᝞C<ᜧE1Ḅ᦮ᦪF%&1GHI᦮◀JKL<᦮ᦪMN2ᦪ(2ᦪOPᦪ).᝞2,3,5,...(2)ᔠᦪ#<᦮ᦪ◀Q%&1GHI᦮◀᜜S%&ᐸTḄ᦮ᦪ᦮◀L<᦮ᦪMNᔠᦪ.᝞4,6,8,...(3)⌕ឋ21

1জ2ᦪᔠᦪVᙠ᦮ᦪYZᨵ\ᦪ]<1_$>2ᦪO$>ᔠᦪ;ঝ2>bḄ_>2ᦪc>ᏔᦪḄ᦮ᦪᓽ>bḄᏔ2ᦪ.ᜧE2Ḅ2ᦪ9᜻ᦪ2ᦪFᨵ<Ꮤᦪ2,ᨬḄ2ᦪ2;ঞᨬḄᔠᦪ4,hiᔠᦪV()ᑖkl<2ᦪḄ:%ᑏᡂl<2ᦪḄ:Ḅ᦮ᦪo>ᔠᦪ.(4)p2ᦪ#qrᦪFᨵ1Ḅ6<ᦪp2ᦪ᝞49.ஹ᦮ᦪ᦮◀Ḅᱯt᦮ᦪ4◀)᦮ᦪb,ᖪw>᦮ᦪx\yᦪzᑣ஺%&}᦮◀ᡈ6%᦮◀4t᦮ᦪ஺◀)᦮ᦪb,ᖪ᦮ᦪyᦪr$0z᦮◀.ᐸ#a\b=cr,ᓽa=b-c+r(04r0ᡈὅ5Ḅᦪ%&5᦮◀(6)%&6᦮◀Ḅᦪ9%&2,3᦮◀ஹᨬᜧᦪᨬᦪḄᐵ(1)ta%&b᦮◀z஺>bḄᦪ஺>aḄrᦪ.(2)l<ᦪqᨵḄrᦪMNLl<ᦪḄqrᦪ;ᐸᨬᜧḄ<,MNLl<ᦪḄᨬᜧqrᦪ.l<ᦪqᨵḄᦪMNLl<2

2ᦪḄqᦪ;ᐸᨬḄ<MNLl<ᦪḄᨬqᦪ.3ᨬqᦪḄ⊤+#ᦪ5ᵨ⊤+᝞15,20ᓽ1520Ḅᨬqᦪ.ᨬᜧqrᦪḄ⊤+#ᦪ5ᵨ⊤+᝞15,20ᓽ1520Ḅᨬᜧqrᦪ._a,b\=,ᱯta=1,ᑣ஺=ab.4ᨬqᦪ¡#ᑖk2¢ᦪ¡£◀¡;q¡¤᝞12,18,20,/ᵨ£◀¡(¥12,18,20=2x3x2xlx3x5=1802|1218203|69102|2310135
©⁚«ᦪ᦮ᦪᨵᳮᦪ஺ᨵ▲ᦪᡈ\▲¯°ᦪ«ᦪR\ᳮᦪ²ᳮᦪ²▲¯°ᦪ\ᳮᦪஹᑖᦪஹᦪஹ³ᑖᦪ1.ᑖᦪ´ᓫ“1”¸ᙳᑖᡂº»¼⊤+L᪵Ḅ¼ᡈl¼ḄᦪMN3

3ᑖᦪ.¾ᑖᦪ#ᑖ¿(ᑖÀᑖᦪᎷᑖᦪ#ᑖ¿ÂᑖÀ2.ᦪᨵ▲ᦪ#Ã᝞0.21ᦪ¯°ᦪ\▲ᦪį°ᦪ#0.312$¯°ᦪ#23.ᦪᑖᦪpᓄ(1)ᨵ▲ᦪᓄᑖᦪ21ᵨ10,100,1000ÆNᑖÀ᝞0.21=*)0(2)ȯ°ᦪᓄᑖᦪ⌕ᵨ9,99,999ÆL᪵ḄᦪNᑖÀᐸ“9”Ḅ<ᦪÆE<¯°⁚ᦪḄ<ᦪ;<¯°⁚ḄᦪᡠÊᡂḄᦪo>L<ᑖᦪḄᑖ¿.ab-g217q(⊤+#O.ab-g=-----------,᝞0.21=——=9999933¯°⁚ᦪ9Ḅ<ᦪ¯°⁚ᦪ(3)į°ᦪᓄᑖᦪᑖÀ⌕ᵨ90,ᐸ“9”Ḅ<ᦪÆE<¯°⁚ᦪḄ<ᦪ“0”Ḅ<ᦪÆE$¯°Ḅᦪ<ᦪ;ᑖ¿>$¯°Ḅᦪ<¯°⁚ḄᦪᡠÊᡂḄᦪÍÎÏ$¯°Ḅᦪ.312-3309103Ã᝞#().312=990990-330=4

44.³ᑖᦪ⊤+<ᦪ>Ð<ᦪḄ³ᑖlḄᦪMѳᑖᦪÒᵨ"Ô”ᩭ⊤+©ஹ«ᦪḄ᦮ᦪÖᑖᦪÖᑖ×EhØ«ᦪx,ᵨÙ⊤+$ÚÛxḄᨬᜧ᦮ᦪ(ÜᦪÝ5Þ[àáâᙠXḄ,ã);ä{x}=x[-x],ಘ>XḄ᦮ᦪÖᑖ{x}>XḄᦪÖᑖ.ᵫçè(¥éêᑡìᓫឋ2#(l)x=[x]{%+}(2)02{%}<1¤᝞3.8Ḅ᦮ᦪÖᑖᦪÖᑖ#[3.8]=3,{3.8}=0.8.íḄ᦮ᦪÖᑖᦪÖᑖ#["]=3,{77-}=775-.ஹᨵᳮᦪ\ᳮᦪ/0(1)ᨵᳮᦪ(îÎï◀)ᨵᳮᦪ=ᨵᳮᦪ(2)ᨵᳮᦪ(îÎ)\ᳮᦪ=\ᳮᦪ(3)ᨵᳮᦪ(0)(ï◀)\ᳮᦪ=\ᳮᦪ(4)0ï◀\ᳮᦪ=ᨵᳮᦪ(0)(5)\ᳮᦪ(îÎï◀)\ᳮᦪ=ᨵᳮᦪᡈ\ᳮᦪ(6)hiᨵᳮᦪV()ᑏᡂ.(6,neZ,Zð*()),\ᳮᦪ\¡⊤+ᡂᑖ¿ᑖÀV>᦮ᦪḄᑖᦪ.(7)ñḄò\ᳮᦪ5

577-=3.14,-e--2.7182=…ñ\ᳮᦪó$ôḄ᪷#᝞2/ö$ôḄ×ᦪ#᝞log233ஹ4ᦪḄ561.ï/0(1)t«ᦪ஺÷஺z/=1,(2)«ᦪḄ᜻ᦪ?øᦪ;«ᦪḄᏔᦪ?øᦪ.2.ó/0(1)ᙠ«ᦪYᑁ«ᦪ\Ꮤ?᪷;0ḄᏔ?᪷>0;«ᦪḄᏔ?᪷ᨵ6<Gúp7ûᦪᐸḄᏔ?᪷0ü᪷.᝞#tz஺Ḅ¸᪷>5”"ᐸ஻>«ᦪ4Ḅ0ü¸᪷.(2)ᙠ/0ᨵØèḄþÿ
01s⁚778——ஹ7(1)ᦪa,b◀ᦪ஺bḄ
a:bᐸ
aḄ⚗
ᓃḄ⚗.a◀஻Ḅᖪᦇ
ᑣa:஺Ḅ!.

66

7(2)Ḅ&'ឋ)*জa:b=k=a=kbঝa:b-ma:mb{m0)*1ஹ3᜜⚗ஹ3ᑁ⚗ஹ3⚗a:hc:dbd᝞78Ḅ!9
:aஹஹcஹᡂ3
a:b=c:dbd,db
ᡈJ=Jᐸ
஺83᜜⚗
8C3ᑁ⚗.A=B
6a8dḄ3⚗
EᯠAGb,dᙳIᦪB
JK஺84ḄMNOᙳ!.Pஹ3Ḅឋ)(1)9RSᳮ*a.b-c\dnad-be.(2)USᳮ*40%=.বWSᳮ*X=/='.aca+bc+d(4)ᔠSᳮ]=^dhdaca-bc-d(5)ᑖSᳮ*==----------•bclaca±mcY»=ia±c(6)ᔠᑖ7<ᳮ>
=—bdb±mdb±d7

8acea+c+ea9Sᳮ*b=——=------------bdfb+d+f3ஹ?@7(1)y=Ar(Z0,kklᦪ)
ᑣyxᡂI
n3oᦪ.pq*rsKx8ytBuᜧᡈwxyI.᝞Az<0B
xuᜧB
yW|wx.k(2)}=b1(0,%*lᦪ)
ᑣyxᡂW
%3oᦪ.8

9ABᦪC
⁚ᦪDEFGaN0ᦪ஺Ḅ!S*"1=-baᡈa>b.(7)£ឋ)*1•44¤I9

10H1,Q>0,\a\L!.=£.=,ᓽLLLᨵ¦§ᨵ!iᡈὅ-i.a\a\-1,a<0.a\a\ஹEFGIJKCপtb©44+ª4+x|ᐸ
¬9ᡂ®Ḅᩩ°:ab<0¦a2b±¬9ᡂ®Ḅᩩ°*ab>Q.^11-111+|<||+||0(2),,ᐸ
¬9ᡂ®Ḅᩩ°*Y±¬9ᡂ®Ḅᩩ°*ab>.ஹEFGḄLMNO(1)|x-஺|⊤³ᙠᦪ´^xµᑮaµḄ·¸!.(2)|x-a\+\x-b\⊤³ᙠᦪ´^xµᑮaµbµḄ·¸8.|¹b஺|+|ᡀb|Ḅᨬx!|஺b|,½ᨬᜧ!
Axᙠ஻JB
¾ᨬx!.(3)1%-஺I-IX-I⊤³ᙠᦪ´^XµᑮaµbµḄ·¸¿.\x-a\-\x-h\Ḅᨬx!b|஺b|,ᨬᜧ!|a-h\Axᙠf஺J᜜B
ᑖÀ¾ᨬx!8ᨬᜧ!
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b,cµ

11Ḅ·¸8.Áa

12\a-c\,½ᨬᜧ!
Axᙠ஺c
¦x=bB
¾ᨬx!.(5)᜻ᦪ!ÄᙠŵᜐÇᑮᨬx!YᏔᦪ!Äᙠŵ¡ᑁÇᑮᨬx!.⁚᦮C
ஹPQCপÉᐰO8(¿)ËR*(a±Z?)2=a2±2ab+b1(2)Éᐰ®8(¿)ËR*(a±Z?)3=a3±3a2/?+3ab1+ᔵ(3)O¿ËR*஺2b.=3+஺)ÐbÑ(4)®8(¿)ËR:஺3±஻3=3±Ñ(஺2+b2)(5)ÒÓËRḄÔÕ*জ1-M=(1-a)(l+a+cr++a"~')2221222+b+c±ab±bc±acs—(ab)±(Øc)i(c+a)±2ঞ(a+h+c)2=a2+b2+c2+2ab+2bc+2ac(6)Ú⚗R9(Sᳮ)*Ú⚗R9ᑣᐸÛÜᦪoᦪ9
Ú⚗Rq¾!Ú⚗RḄ!9.ஹRS◀Q<ᳮqoᦪÚ⚗R/(½)
g(x)[g(x)sKÅÚ⚗R],~11

13SáᙠÚ⚗R᜛(X),r(x),äå/(x)=q(x)g(x)çèᡂ®+,é«èÅÚ⚗Rᡈ(X)ḄÜᦪxg(x)ḄÜᦪ
¦q(x)8G)KìbḄ.ᡃèg(x)◀î(x)ᡠåᖪR
r(x)g(x)◀î(x)ᡠåðR.ஹTC<ᳮ/(x)ñò(x-஻)᦮◀ox-<=>/(x)=(x-a)-g(x)o/(x)öᨵ÷RG-a)<=>f(61)=0஺K/=(%)0Ḅ᪷./G)ñò(or-ᓃ)᦮◀="-)
(])=/(x)=(ax-b)g(x)=/(x)öᨵ÷R(Ë-b)b2o/=0/=(x)0Ḅ᪷.aa3ஹSC<ᳮᵫðRḄÜᦪ⌕x◀R
ᡠA◀RbÜ⊤ÇRB
ðR:lᦪ.পᵨbÜÚ⚗RX-஺◀Ú⚗R/(X),¢IJ/(x)=q(x)(xa)-r,+9R¬üx=aýᐭ
ᑣðRr=/(a).ᓽ/(x)=q(x)(ÿa)~f(a+).ফᵨ⚗ax◀⚗f(x),ᑣ12

14f{x}-q{x){axb)-r,+x=ᐭᑣr=f(b/a).ᓽ/(x)=q(x)(axb)-f(b-^d).#ஹ%ᓝ'()*1.ᓝ'()*ᵨ,ᑖ.abx1+(bp+aq)x+pq/Ḅ01234⚗Ḅᱯ673⚗Ḅ8ᦪஹ:ᦪ⚗;ᦪḄ<=⚗8ᦪ3⚗8ᦪḄ>ᦪ?:ᦪ⚗8ᦪḄ>ᦪ)<Ḅ@•ᑖ.Babx2+(bp+aq)x+pq-(ax+p)(bx+q)2.%ᓝ'()*C⍗ᑮ3F⚗ax2+bxy+cy1+dx+ey+/Gᵨ%ᓝ'()*HI>ᑖ.ᐸKL7(1)ᵨᓝ'()*ᑖ.ax1+bxy+cy1,Mᑮ;ᓝ'()N(ᨵᑡ)=(2)Q:ᦪ⚗/ᑖ.ᡂ;>SᙠU4ᑡV⌕XU3ஹU4ᑡ᪀ᡂḄᓝ'Z[\<Ḅ@,]^ḄᑖUஹU4ᑡ᪀ᡂḄᓝ'Z[\<Ḅ@,]^Ḅ_.ᑣax2+bxy+c)r+dx+ey+f-(«ixc+iy/+))(42xc+zyf+2)•13

15a,x3fa\ai=a,c\aGfg=/,=erac+ac=b,cfc+/e=,aja+f'd=122i1221122114

16AUᦪ
ஹVᔠ1.iᔠḄkl?⊤n(1)iᔠḄᭆp7iᔠ^Ḅᐗrᐹᨵtuឋஹwxឋ@yzឋ.(2):ᵨᦪi{ᐸ|*7N⊤n}ᯠᦪiN*ᡈN+⊤n᦮ᦪiZ⊤n᦮ᦪi஺⊤nᨵᳮᦪiR⊤nᦪi.2.iᔠḄᐵ8|lឋnNপAUAফ0GAবAG3AQBA^ḄBQC,ᑣAGC0iᡈᐗrᡈB2A,BভAG88GA,ᑣA=815

17(1)0uA(A¡AQB,AUB*0B^¢0i)©i(ᡈBnᨵᐗফAu8)*r,A)BUC,ᑣAu஺A**A^ḄᐗriᔠA=B,B,B(1)ACB(^Ḅ(2)8GAᐗr,A¦§iᔠAᨵ஻(஻w1);ᐗrᑣªᨵ2";0iªᨵ2"-1;0iªᨵ2"-1;¡¢0iªᨵ2"-2¡¢0i.3.iᔠḄ®|lឋnN(1)AA=A[x\xeA.ফA0-0交集ABX£B}(3)ABQAABQB16

18(I)AA=A{x\xeA,(2)A0=A并集AB•ᡈX£3}(3)AB^AAB?BCu(AB)=(CuA)(CuB)Cu(AB)补CA{x|xsU,集=(C(/A)(CuB)%uA}\.A(C(M)=02.A(CuA)=U4.iᔠ®^:ᵨ³´পAGBoAB=A,=AB=B.ফA8=A8,AB=AB.3ஹᐗ3·ᦪ{ᐸN¸1.ᐗ3·ᦪḄ¹প᪗»7y=ax1+bx+cb24஻c-b2(2)¼½7y=a(x+2a)+4a17

19(3)¾67y=a(x-x\)(x-xi)xi,X2⊤nᐗ3·ᦪ?x¿Ḅ;Z6.2.3·ᦪNÀḄឋপ3·ᦪ/(x)=or2t)+c(q*0)ḄNÀᩩÄᱥÆÇ+x¿½Èx=9:,⚔6ᙶ᪗(A%3Ì),y¿ÍÎ72a2a4a(2)C஺Ï0GÄᱥÆÐÑᔣV·ᦪᙠ(-8,-2ÓV⌴Õᙠ2a[-V+)V⌴×CØ=-2_Gf(x)=4ac~b~Cmin;2a2a4஺஺Ý0GÄᱥÆÐÑᔣÞ·ᦪᙠ(-%ß4]V⌴×ᙠ[--*,+)V⌴ÕCᑍ=---âG/ax(X)=~~~~2a2a4a(3)f(x)-ax+bx+c(a*0)ã=Z?4-acO>3·ᦪ2C2G,A/ANÀ?X¿ᨵ;Z6M(XI,0),M2(X2,0),\MlM21=1II-X2|=—.|a|4ஹᢣᦪ?ᢣᦪìḄ®1.ᑖᦪᢣᦪíḄ®ឋ(1)î•᮱=î+s(஻0,rÏ,s/?)G18

20(2)(/>=ars(a>0,r,sGR)(3)(ab)r-arb'(a0,>bQ,>rRe)(4)/I//t(a0,>r,s᧤2.ᢣᦪ·ᦪ{ᐸឋ·ᦪᢣᦪ·ᦪul·ᦪy=/(a0ுa*1)òóᢣᦪ·ᦪa>10

21ax>1(x>0)<1(x>0)·ᦪ÷Ḅ6TV=1(X0)=/=1(x0)=üᓄþÿ<1(x<0)ax>1(x<0)a
ᓄᙠ▲ᑁaᜧᙠ▲Ḅᑁ஺ᜧ.ûஹÇᦪ?Çᦪ®1.⌕Ḅᦪឤlogo1=0,logoa=1,logoab=b.2.ᦪḄឋ"᝞$aு0,a*l,Mு0,N>0,'(জ*+,log஺M+logoN=log(MN)fZ_Mঝ.+,logaM-logoN=loga-Nঞᦪ0,nlog«M=log”M"(஻R)eটaᔩN=NঠlogoM"=4logMS0,n/?*)ea,ablog/,No,>5.bডᣚ67,10gaN=(h1)*log/7a20

223.ᦪ8ᦪ9ᐸឋ"8ᦪ;<ᦪ8ᦪ=>8ᦪy=log஻x(஺0@aA1)BCᦪ8ᦪa>10E(0,+8)GERH=IH=I(1,0),ᓽKx=1Ly=0.ᓫNឋᙠ(0,+00)OPQ8ᦪᙠ(0,+8)OP.8ᦪ10gaX>0(X>1)logoX<0(x>1)8ᦪGḄlogox=Q(x1)=logox=0(x1)=
ᓄRSlogax<0(00(0

23WஹᱯY8ᦪ1.ᨬG8ᦪmax⊤_ᨬᜧG8ᦪmax{x,y,z}⊤_x,y,zdᨬᜧḄᦪ.஻ef⊤_ᨬgG8ᦪmin{x,y,z}⊤_x,y,zdᨬgḄᦪ.2.jG8ᦪ(l)y=|oxᓃ+ᐜry=acb+ḄuvwxxyzḄu{ᑮxxOz.(2)L+by\=c⊤_ᩩḄ7+by±^c,@ὅᐵI<.(3)|Ax-4+B.y-bC,KA=BL⊤_zKL2C⊤_☢s=-2.AB(4)\xy\i-ab=a\x\b+\y||nx|ᡈ|y\a,⊤=_H=,y஻ḄVᩩᡠᡂḄ☢S=4|a஻|.3.ᑖ¡8ᦪᨵ£8ᦪᐸ=>EᑁḄ¤
¥xḄ¦§G¦¨ᵨªḄ«᪆⊤_eP⌕ᵨᡈOḄ®⊤_¯°8ᦪ<ᑖ¡8ᦪ.ᑖ¡8ᦪ⊤_¦§Ḅ±G³¦§Ḅ⊤´.22

244.µ¶·8ᦪµ¶y="
k¸ᦪk*0·8XᦪkḄk>0k<0¹ºbJIuXrᡠᙠஹ»▲ஹV▲▲23

253AWX
ஹ
ᐗZWXḄ[Q1.½¾z+᝞(X+஺)2=D>0)ḄzÁÂᵨ½¾z+«ý¾zÄx+஺ᡈὅx+a=x=a+yjb.ÅÆ,Ç஻<0,zÁÉ«.2.Êz+ᵨÊz+«ᐗÌzÁax1+bx+c=0(a0)*Ḅ-ÎÏÐ.(1)Ì⚗Òᦪᓄ1,zÁÃÓ◀Ì⚗Òᦪ(2)Õ⚗,ÖzÁ×ÃÌ⚗ØÌ⚗Ùøᦪ⚗(3)Êz,zÁÃÓ*OÌ⚗ÒᦪÚḄzÛzÁᓄ(x+m)2=N0)Ḅ(4)ᵨ½¾z+«
ÝḄzÁ.3.7+ᐗÌzÁax1+bx+c^0(a0)*᪷Ḅᑨà,á=4-â(1)\>0ozÁᨵ¦ãḄä᪷,x=åæ'è4”`2a(é-4ac20)=/(x)ḄuØxxᨵëI.24

26(2)A=0ozÁᨵãḄä᪷o/(x)ḄuØxxᨵëI.(3)A<0=zÁÉä᪷=/(A-)ḄuØxxìᨵëI.4.íᑖ«+îHíᑖ«ÛzÁ
a(x-m)(jc-஻)=0,ᑣᨵx=mᡈx=n.ஹ
ᐗZWXḄ᪷ᦪḄᐵ(bc<ᳮ)1.ñzÁ2+bx+c=0(Q*0)Ḅ᪷XI,XI.ᑣᨵhcx+x=FX\X2=]2

aa2.ô9õ´=ᳮḄ÷9³ᵨ(1)X124-X22=(X1+X2)2-2X1X20ஹ11X+XXXxx12I2(3)ø+X23=(xi+X2)(x12-X\X2+X22)(4)|x-xj=/(x1)22X-XJ(=ix)2+4xx-I25

27dAJKCஹ¦Ḅûüឋ"1.<ឋᡈµýឋ,a>bcbb,b>c^>a>c.3.ឋ
a>b^a+c>b^c,ᑣ⚗ᑣᔣ
a>h,od^a+ob+d,a>byc-d)=>a-c>b-d.4.ឋ
a>h,c>0=ac>be;a>b,c<0^>acb>0,c>d>O=ac>bd,a>b>0,d>c>0(1/c>l/d>0)^a/c>b/d.5.ஹឋ
a>b>(OnsN)oan>bn>0,n/4>0.6.᎔ᦪឋ
a>b,ab>O^~a

28ஹeEFGḄJKCḄ[Q|x\0){x\-aa(a>0)ox+bᡂ᦮"#ᓄᡂ\ax+b|c(c>0)\x\a(a>0)%ᩭ'ஹ
ᐗZJKCfᐸ[Q1.ᐗ)*+᱐+c>0(a>0)᪗/%.12ax2+bx+c<0(a<0)Ḅ45ᑭᵨ89-1ᩭ:᪗/%(⌕=>?ᔣ@⌕ᦋ:)#ᡠ5DEF'ax2+bx+c>0(஺>0).5ᑭᵨ)*HᦪḄIJKL)*HᦪM)*ḄὶOPQRS12ᐗ)*Ḅ.27

292.ᐗ)*ḄᑨUA>0A=0A<0A=Z)2-Aac)*Hᦪ1/JJy=ax1+bx+c{a>0)c*0ḄI[-b±yjb2-4acbᐗ)*^2aX.=x2=---2aax2+Zzr+c=0(t7>0)rn᪷eᐸghix2kᨵ8'nḄ᪷᪷ᨵ8mn᪷{xIXVhᡈax~+Zzx4-c>0(f7>0)R2aḄX>x2}ax1+Zzx+c<0(«>0){xIXj

30a+b}}a=b"=”?.⊤
ᙳᦪNᙳᦪN2ᙳᦪNᙳᦪ.2.}Xl,X2.....5"ᦪ#ḄᙳḄ2ᙳ#ᓽᑅ“+£+
¤”2□...X”(>0஺=1#…,")#}n}Xl=X2…=¨஻=,?ᡂ.᝞ªb,c«nᦪ#¬஻+/?+l^i^Ttbc,}}a=b=c"=”?.3প“2+°Nlab-,(2)ab(±²;(3©2µ9±2"(4)-a+b~^2(abeR+)dஹᑖCJKCᑖḄºKL⚗᦮ᳮᡂ᪗/%.f(x)>°/(x)ᡈ<0,h¾ᓄᡂ᦮ᩭ.g(x)g(x)■>0=/(8)•g(x)>0g(x)29

31/(x)ঝ------<0o/(x)•g(x)<0g(x)/(x)N0o/(x)•g(x)20ঞ2——g(x)g(x)*of(x)ö÷#PåᑮæᑏS#IJᙠᦪʤá⊤ᜧâ#Ðá⊤â.f(x)=(%%-)(%X-)(-XX-)30

32ᨵ
⚗op`ᐸqo?⍗âß:?#▢üýᑖ/Q)>0Ḅ.31

33rAᦪᑡஹᦪᑡḄÿᭆ(1)ᦪᑡᢥ᯿Ḅ᣸ᑡᩭḄᦪ.(2)⚗ᦪᑡḄᦪ;✌⚗᣸ᙠḄᦪ!ᑏᡂa\,a?,...a%&'()*+,n-.᦮ᦪ.n(3)1ᦪᑡᔜ⚗345Ḅᦪᑡ.(4)ᦪᑡḄ7n⚗Ḅ8(9:S);<ᦪᑡ(a”*,@ᯠᨵnSn=+஺2+஻3++F஻=1HIa\=SKF஻N2HIa=Sn,ᓽnS/71=1a='().Sn-Sn-\(஻?2)Oஹ5QᦪᑡL5QᦪᑡḄS⚗T!U(஻஻*'5QᦪᑡI✌⚗'TQ'd,ᑣ=+(஻l)d,ᓽ=Y+(஻-q)d.2.5Q⚗Laஹbஹcᡂ5QᦪᑡIᑣ஺-aஹcḄ5Q⚗I\6="+23.5QᦪᑡḄឋ`abᵨ32

34(1)Ud+஻=஻+q=2wᑣ+a-cip+ciq=2anw(m,஻,p,qஹh3-.᦮ᦪ)ফUm,p,nᡂ5QᦪᑡIᑣa,a,ajᡂ5Qᦪᑡ(m,n,p3tnpn-.᦮ᦪ)(3)a-a+(n-m)d(m,஻3-.᦮ᦪ)nm(4)Uᦪᑡ(஻஻*ᡂ5QᦪᑡIᑣa=+pnq(pjqR)nমUᦪᑡ()*ᡂ5QᦪᑡIᑣᦪᑡI)+/?*(mn'1ᦪ)o'5Qᦪᑡ(6)U()*8(p*ᙳ'5QᦪᑡIᑣ()rs*j-5Qᦪᑡ4.5QᦪᑡḄ7஻⚗8T!஻(஻1+))n(77-1)5Ona=«(1n2125.5Qᦪᑡ7஻⚗8T!ឋ`প5QᦪᑡIvZ(xy2,ZeN+)⚗y8oᯠ-5Qᦪ{IJ,ᓽSk,Slk-Sk,S3A-S2k,S4k-S3kI…ᡂ5QᦪᑡI\TQ'k2d.(2)5Qᦪᑡ(஻*IU஻=஺n(tn=it),ᑣ᝞஻O0U33

35Sn=m,Sfnn?m=஻)Iᑣ(+mn)+(3)U()*8(p*ᙳ'5QᦪᑡI7஻⚗8ᑖ-S,87,ᑣᨵaS_n—2«-1bT~n2n-l6.5Qᦪᑡ7஻⚗8T!ᦪḄᐵ5Qᦪᑡ7஻⚗8T!S“=஻(஻_])na\---------1--------dᑏᡂ2d2ddd22Sn—A”?Bn+.ஹ5ᦪᑡ1.5ᦪᑡḄS⚗T!a=aq"-la=aqn--mnn19m•2.5⚗᝞ᦪx,G,yᡂ5ᦪᑡIG:x8>Ḅ5⚗,ᐸG?=IG=±V-3.5ᦪᑡḄឋ`(1)T'qḄ5ᦪᑡḄᔜ⚗¡'¢Ḅᦪm,ᡠ¤ᦪᑡo-5ᦪᑡITo'T34

36(2)Um+n=p+q,m,n,p,qeN+,ᑣaa=aa.mnpq*ḄT'4I(3)U5ᦪᑡ(஺ᑣ1_-¥'TḄ5ᦪa"qᑡ.ভU()*(஺“*ᙳ'5ᦪᑡIᑣ(᝞s*j'5ᦪᑡ.4.5ᦪᑡ7஻⚗8S⚗T!§5ᦪᑡ(஺”*Ḅ7n⚗8'S,,,ᑣS=a\+஺2+...+annপF

37´ஹ⌴¶T!᝞)+1ᡈ¸-1Ḅᐵ!&'⌴¶T!IU¹ºᦪᑡḄ⌴¶ᐵ!a✌⚗Iᑏ»ᐸY⚗I¼½⌴¶T!-¾ᦪᑡḄ¿À⌕m!I⌴¶T!Ḅ1ᵨÂÃLᑡÄÅSÆ⌴¶T!Çᑮ7ÉᐗËᦪÌḄÍÎIᩭᑨÐÑ☢ᐗËḄᦪÌ.ᐜᑡÄ7☢UÔ⚗IÕÇÍÎI-ᕜ×ØÙḄÍÎ.2.ÚdÅ;<᝞a+\-a+/(஻)ᡈ¸+1-a-/(஻)I&'Û5QᦪnnnᑡIᑏ»UÔ⚗IᯠÑÜᔜ⚗4d.3.ÚÅa+1;<᝞•)+1=஺"/(஻=)ᡈan/(஻)I&'Û5ᦪᑡIᑏ»UÔ⚗IᯠÑÜᔜ⚗4.4.᪀⌼ᦪᑡÜàáᑖâᡂãᦪᑡIãᦪᑡäᔠ5Qᡈ5ᦪᑡIæ»ãᦪᑡÑIçæèᦪᑡ.পUãᦪᑡéêë+11ᦪIᑣâᡂ5Qᦪᑡᑖ᪆b-(2)Uãᦪᑡéê*11ᦪIᑣâᡂ5ᦪᑡᑖ᪆.íᐸ36

38a=qad+!ḄᦪᑡISÆîᑖ1ᦪIïᡂnaᔣ+c=qñQ஻có+Ḅ!Iç᪀⌼5ᦪᑡæô.37

39tALM
⁚u☢LMஹõLõḄឋ`(1)ö¬÷øy8ᜧ<øIö¬÷øyQs<øᓽa-b

40(2)CDB஺7ᑣ☢GS=4$4.#H(1)ឋJ2#H@K#7@K:LᡂNOᓽ/A=A',zB=B',NC=C,ABACBCAB+AC+BCA'B'A'C'B'C'A'B'+A'C'+B'C'CAR2(2)CA48CDA4'B'C',ᑣ}—,RS#HKJM'B'C'A*B'ḄTஹ:ஹ4ᑖ:ஹᕜDḄN!@K:LḄN.5.ᐰ%&Y#Z7ᜧ\#7ᑣ]%ὅᐰ._`ab(SSS),(ASA),(SAS),(A4S)ᩭᑨe.6.Yᑨe(1),2./0ᳮᡈὅᨵ;&B90°(2)Ὺ,2(Nij1:1:\ᡎὅᨵ%#Ḅ,(3)2#ᡈὅ&ᑁ#ᡈὅlmᔠ;(4)Ὺ2ᨵ%#Ḅ39

417.Ḅlmlm0pqrᱯtᑁmᑮz{#ᑁᑗᑁmᙊḄ4ᑖ:S=}(஺b+c)+,SB☢G7rḄwx2ᙊmBᑁᑗᙊ<᜜mᑮ&⚔xz{#7,᜜Ḅ᜜mᙠ6Ḅx7᜜᜜mᙊḄᚖ:Ḅwᙊmxᙊ

429.0ᳮᙠABC7AD,BE,CF#w!Z;x஺7S^ABO:SMCO=BD:DC.¡¢2£B¤AC஺T7ᦑSMBD2SAACD=5O:£)C,£BBODCOD,WS&BODSSACOD=BD:DC,¦§ᨵ(SAAB஺-S&BOD)2(5AACD-S^COD)=BD:DC,ᓽSMBO:S^ACO=BD:DC10.¨©0ᳮᙠRtABC7zACB=90,CD6AB9ḄT7ᨵ᝞«0ᳮ2(1)CD2=ADDB-(2)BC1=BDBA•(3)AC2=ADAB-41

43J4KAC-BC=AB-COᙳ_ᑭᵨ#H¡¢°☢GS=M,ᕜDL=2abK,+@:D41²஻.+3.´☢GS=&L¶7q:D=·¸.224.¹lDᙳB“7`aB»ḄTB¼☢GBS=H?J.=஻,212ᐸ/1,/2ᑖÀB@:ḄD7¹%@:#ᚖ,4ᑖ7ᕜDL=4a5.≈0ᳮপÄÅlḄNOᐵÇSSSI4QD1+S4==.£ÈS]•S3=S?•S&S2S3OBS2+S342

44DSABজ——l----=Si,S3=S2,S4S2S3OBhSa2ঝ—=—#H°S3b2ঞS2+S3=S3+S4ns2=S4Êᔠ`97Ë;NOÌᑮ2Si:S3:S2-S4=a1:b2:ab:abஹᙊᡧ{পḄÍÎ2ÏᙊÍDÎ)<ḄN]B@KᙊmḄÍÎ.43

45180TT1ÍÎ=",1=180ÍÎ.(2)ÑᙊḄ

46(3)ᐰ☢GSᐰ=2TTrh2TTI2+.Û2ᙊã×Ḅß☢äåæ;&DB2çè7éBêḄë.Øãß☢äåæìë7Øãß☢G25=»☢ᕜD•T7Øã×GV=»☢G•Tஹᳫ×ᳫḄ

47⁚[᪆LM;ஹ%xx%xP%,ú)¤ûx,y)Ḅxᙶ᪗BA+஽73+>12I>I222}ஹ%xÿ
Ḅᙶ᪗P1()P2(X2,y2)Pl2Ḅ
|1ᑣ1#2W(X2~-A-1)2~&'2-y\)2(ஹ*᳛(1),*-./01X234ᔣᡠᡂḄᜳ-9,*-a.⌕=<7€[0,77].(2)DE.Z=tana,a*42L(3)NOPi(xi,yi),P2(X2,”)Ḅ/0Ḅ*᳛k=y2f,X2~XIXI*X2;(4)VW,*-*᳛46

48TT77TT77a06432.X=tana0160033ஹᑮḄC/0/Ḅ4ZAr+5)+C=0,P(xo,yo),ᑣPᑮ/0Ḅ
.1\Ax+Bv+CI().()IVA*2+B2dஹWXḄd⊤c{C(1)*c.de/0/Ḅ*᳛fX,஺f/0/ᙠy2iḄcᑣjkl/0Ḅ*c4Zm=n஺+.(2)*.oe/0/Ḅ*᳛fk,pqNOP](xi,r)ᑣjkl/0Ḅ*4Zy-yi=k(xf).ব.de/0/NOPl(XI,yi),Pl(X2,>,2)>(XIuX2qyi*”)ᑣjkl/0Ḅ4Z)'VI-X-X\y-yx-x212I47

49ভc.dex2iḄca,y2iḄcb,ᑣjkl/0Ḅc4Zᨵ1.ম.Ar+gy+C=O
A,Bᐰyஹᩩ/0Ḅ{|ᐵ~ᩩ*c/0Ḅ{1\:y=k\xb\■+•/j:a\x+b\y+ci=0|ᐵh\y=kixb2+2.஺2x+ᓃ2y+C2=0~abck\-kiibi*11Q2blC2ahki*k?1+1a2bibabcᔠki=ki,b\2=1=11=42blC2aaᚖ/k•k='-L---=1=஺஺2+b\b2=012bbl2./11/2Ḅᜳ-8,ᑣtan8.,_h1+M|II2|ᑮ-./0/1⌮┐4ᔣᑮ1/2ᔠᡠḄ-48

500,ᑣtan0=2τ.1+k\k2ஹ/0Ḅ
ᩩ/0.Ar+B)+஺=01/2.Ax+By+Cz=0Q-Q|Ḅ
.d=VA2B2+ஹ9⚪1.ᐵ9⚪(1)ᐵ9M(xo,yo)ᐵP(a,b)Ḅ9f(2a-xo,2/?-yo).(ᵫᙶ᪗kᑮ).(2)0ᐵ9deIḄ4Z.Ar+By+C-O(A2+B2*0)P(xo,yo),=/ᐵḄ9/04Z.P(x,y)f9/0/i¡ᐵP(xo,yo)Ḅ9(2xo-x,2yo-y)ᙠ/0Ii,¢ᐭkA(2xo-x)+B(2yo-y)+C=O.¤/0ᓽᡠ=9/0.2.ᐵ09⚪(1)ᐵ0949

51deM(xo,yo),/0/:Ar+By+C=0(4B*0),Mᐵ/0ெḄ9N(஻?,஻),ᑣᵫZMN•Q=1kᑮªᐵ஻Ḅ4Z¬0MNḄᙠ/0/ikᑮ¯ªᐵ/»,"Ḅ4Z²4Z³>´µxo+niyo+nA--^—+B~~^+C^0j=¶N(m,n).(2)0ᐵ09জ/0Ḅ9hill,¸/i1/ᐵ/9,ᵫ9Ḅឋº᧕eZi//h,q/ᑮ/i1hḄ
d\1di¼./)Ḅ4Zax+by+ci=0,IḄ4Zax++c=0,ᑣj(2Ḅ4Zax+by+C2=0,᪷¿
¼kᑮ/2Ḅ4Z.6++(2c-ci)=0.ঝ/0Ḅ9Â.ᵫ41-P,j=¶ᙶ᪗.ÃĶi(P◀

5250

53᜜ᐵ/9ḄḄᙶ᪗Çᵨᐵ/09Ḅ4ÂÃ᪷¿=¶/0hḄ4Z.ÂÉ.ᵫ9ឋjeZiᑮIḄ-1/ᑮbḄ-¼q/2O1/ḄP,ᵫᑮ-=¶/2Ḅ*᳛Ã=¶PḄᙶ᪗Êjᵫ*=k/0hḄ4Z.
3ᙊᐵ09Ì◤=¶ᙊÎᐵ/0Ḅ9ÃᵫÏÐÑ=¶ᙊḄ4Z.3.ᱯÓḄ9/094ஹp(xo,yo)ÔÕI:ax+by+c=0ᐵX2p\x,-y)ll:ax-by+c=0Öyᣚᡂ900-yᐵy2p'(%,y)r:-ax+by+c=0ÖXᣚᡂ900-X51

54Öxᣚᡂ-ᐵÙP'(-X0)10)r:ax+by-c=0X,Öy9ᣚᡂ-yᐵp'(y,Û)r:aybx+c=0Öx1y900ᣚÖXᣚᡂ-ᐵy=-xp’(-yo,-xo)r:ay+bx-c=0yÖy9ᣚᡂ-xஹᙊḄWX1.ᙊḄ᪗Ý4ZÞᙊÎCÇa,஺ÏÐrᙊḄ᪗Ý4Z(X-1)2+(y-1)2=”.2.ᙊḄ4Zx1+y2+Dx+Ey+F-Q"2+E2=D2+£2-4Fß4kx+—y+-----------224প/1VḄ~ᦪ¼L(2)D2+£2-4F>0L(3)ᙊÎC(--,ÏÐr22ᓝஹᙊḄᐵ52

55P(Xp)ᙊ(x-X0)2+(y-yo)2=J,/ᙠᙊ᜜ᓝ
ஹᙊḄᐵ1.éêᑨDÂ/0/:Ar+5y+C=0,ᙊ(x-஺ì+(y-ঢ2=JḄÏÐᙊÎM(a,b)ᑮ/0I
d/A஺+8"Cï+mপ/0/1ᙊM=dr.2.¢ᦪᑨDÂAr+By+C=0ᵫòᐗ,kᑮᐗÉ4ZḄᑨõ(x-a)2+(y-b)2=i2ᑣ(J)A>O/01ᙊLফ=0/01ᙊᑗLব\<()஺/01ᙊ
.ᓝஹᙊᙊḄᐵᙊC:(x-a)2+(y/?)2=÷2ḄᙊÎ஺31,ᔊ)ÏÐrjL53

56ᙊC2>(X-42)2+(y-ú)2=û2ḄᙊÎC2(஺2/2)`ÏÐ1~1,ᙊḄᙊÎ
d=|C1C2I.পᙊCi1ᙊC2ri-F2|ri+f2ᐳᑁᑗ02ᩩᐳ᜜ᑗ02ᩩ.মᙊC1ᑁÿᙠᙊC2ᑁO04d<ᐳᑁᑗ0ᩩᐳ᜜ᑗ0ᩩ.ᓝஹᨬ⚪(1)!"aḄᨬ.%?"஺'=*+ᓄ-./(a,6)ᑮ3/(x,y)Ḅ6᳛Ḅ9.ফax+byḄᨬ.%ar+by=c,ᓽy=+-X£,+ᓄ-3ABCḄᨬ.bb54

57(3)(x-a)2+(y-b)2Ḅᨬ.%d2=(x-“)2+(y-b)2,&=J(x-a)2+(y-by,+ᓄ-./(a,b)ᑮ3/(x,y)ḄCEḄᨬ.(4)ᙊFḄ/ᑮACEḄᨬ.GᙊHᑮAḄCEI᪷KᙊLAḄMNᐵPQ.RSCETUVᡈCEXUVSᐸᨬ.ম[ᙊFḄ/ḄCEḄᨬ.GᙊHCIXUVᡈTUVᓽ\.(6)+ᓄ-ᐗ"^_ᦪᨬ.(7)LᙊᨵᐵḄᨬ⚪bbLᑗᡈAVஹUVᨵᐵ.55

58Aᦪᑖ᪆
⁚ᦪᳮ—•ஹ[efghᳮ1.TjhᳮklmnoᡂqᨵnrsjtrsjuᨵvwxyḄzj,{|oᡂ}mnᐳᨵN=vm++…v+wxyḄzj.12n2.jhᳮklmnoᡂq◤⌕ᑖᡂ஻eteᑖᨵ/WwxyḄzj{|oᡂ}mnᐳᨵN=஻"•T2••T"wxyḄzj."ஹ᣸ᑡL᣸ᑡᦪ1.᣸ᑡḄ.஻exyᐗuG᪷(4஻)eᐗᢥ᯿.᣸ᡂᑡ-nexyᐗuGmeᐗḄe᣸ᑡ.2.᣸ᑡᦪ()()()(஻-T)Am=P"11n2-n-m..A-+஻=.ᐰ᣸ᑡA/=n(nl)(-n2)-21-஻=!(“!-nḄ▤).56

59◀¤¥᜜¦§.A*=1,0!=1.ஹᔠᔠᦪC1.¨ᔠḄ.஻exyᐗuGT(mV஻)eᐗª-¨-nexyᐗuGmeᐗḄe¨ᔠ.2.¨ᔠᦪAm="(w])(஻——m+])==n„A!"m\m\(n-m)

60᣸ᑡSᐜ¨ᔠI᣸ᑡkAm=C,n-A,n=Cn-m

61nnin¨ᔠᦪឋkG/"=G®-'"3ஹ⚗C<ᳮḄᭆ(a+b)n=C°a"+C'a"-'b++CJan-rbr++Cnnbn(neN*)¯nn}e°±l"⚗.ᳮ³´Ḅµ⚗±l(a+L”Ḅ"⚗·¸¯qᐳᨵ”+1⚗ᐸuCJ4-7/'±l"⚗·¸Ḅ¹⚗.·¸u"⚗PᦪḄឋk(1)Cf=CLফ஺"+Cm-'=C"1nnn+57

62বGi°+++Cn"=2"dஹtᜧWQ(1)»¼ᐗᡭᒹᣊÀj¯x»¼ᐗÁÂj¯»yᐗ◞Äj;ÆÇᐗzÈj¯ÉÊLxÉʯËÌᑡÌj¯Í☢Qj(2)ÏᵨḄᐗᑖÑÉÒÓÔ:neᐗmeÉteÉÒÖÂÉÒ\¥ÂÒ»y»yᑡÌjᑖr+ᑡÌj◞Äj»yxy◞ÄjC"Iiwln+/n-l`L1ᑖᚮjØÙxy»yᑖr+ᑖᚮjxyxyᐜᑖᚮI᣸zÈj(ᑖÚÛÜ)"Ý58

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