2021年全国高考甲卷数学（理）试题（答案版）

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2021
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24D44mx=3,᪷ឋ@{A(3,6),4(3,—G),

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v,%5,£=%•»᳝=஽¢_6LV,-Vi/ஹxy,y,»½¾¿-A4Ḅmy-y¸¸721^-%,ᓽ£=------+/u•z2-᳝y+%ᵫ-OMᑗ1,ᓄÁ2%+3Â_1%2_Ã+3=0¢ᳮᵫ-4E3OMᑗ2%+3%-1v7%+3=0Äm2yy+
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471%-Ù+3=0,ᑭᵨHᑮ-Z[ÚÛÜÝ6¸ÞmÁᓫàáâãäXa21.
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'!<=ᡡᨵ,-./ᑭᵨ>ᦪẆ@g(x)ḄᓫឋABᔠg(x)ḄDE/F᩽▲Iᑖ᪆g(x)ḄLMNOᑮ00,mw>ynr(x)=xa<=>xlna=«lnx<=>^^=^-^ᦪg(x)=^^,a'xaxᑣg'(x)=^ߟtg'(x)=0,x=e,ᙠ(0,e)ᑁg'(x)>0,g(x)ᓫ⌴;ᙠ(e,+oo)yg'(x)<(),g(x)ᓫ⌴•••g(x)3=g®=g(l)=(),mX+8ng(x)0,ᡠ!y=/(x)
'!y=1ᨵ*+ᨵ,-./ᓽ!y=g(x)
'!y=^~ᨵ,-./ḄᐙᑖR⌕ᩩ¡XVᓽX0

28ᵫy=/(x)
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y=1ᨵ*+ᨵ,-./0141ᙠ«¬((),+<»)ᑁᨵ,-7Ó8Ḅk.Ô4£=Õᡠ,Ö`ᦪalnx=xlna,ᓽlnx=2a×⚪014g(x)=Inx
p(x)=᝞@aaᨵ*+ᨵ,-./.জm0<“<1n5@<0,p(x)
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31

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p(x)=3ᨵ*+ᨵ,-./.aaeaër(஺)=3,/(஽)='«,t"5)=஺ᨵ஺=e.஻e(l,e),/z(஻)>€),//(஻)ᙠ«¬(l,e)ᑁᓫ⌴;஻€(0,+00),//(஺)<0,ê(஻)ᙠ«¬(d+8)ᑁᓫ⌴஻=en஻(஺)ᨬᜧI4g(e)=Lᡠmᨴ.e-ÈIn஺1஺wenᨵ0<---<.aeñyᡠò9ᦪ஺Ḅ`Ib4(l,e)u(e,+s).ô:'õ/(x)=(x>0),f(x)=öa7”=÷(a7஺(a')஺Ô4x>0,ᵫ/'(x)=Ox=3.InamOvavln/(øᙠ«¬(0,+8)ᑁᓫ⌴7äå⚪¿;mn>0,ᵫr(x)>0O=,y(x)ᙠ«¬(,+ooᑁᓫ⌴.InaIIn஺a>1,ᓽெInaJü“ஹᓽÔ4ù”3=0,*ú/)=0,ᡠ/

33a^(Ina)"a'naina>(ina)aalnn>ln«,,Ö`ᦪ1^—Jlna>In(lna),ᓽlna—l>ln(lna).tlna=f,ᑣf-l>lnr,t஻(x)=lnx-x+l,ᑣ/(x)=ý—1,ᡠ஻(x)ᙠ«¬(0,1)ᑁᓫ⌴ᙠX«¬(1,+0஺)ᑁᓫ⌴ᡠ஻(x)<஻প=0,ᡠZ—12hU,!—ḄᡠInawl,ᓽawe.ᦑᦪ஺Ḅ(l,e)u(e,+8).ூ᦮!"#௃

34%⚪ὃ(ᑭᵨ+ᦪẆ-.ᦪḄᓫ0ឋ2᪷456786Ḅ9":ᦪ;<ᦪḄ=>?⚪2@ABC⚪2DEFGH?⚪IJKLMᓄ2ᑖP<ᦪ2᪀⌼.ᦪ2ᑭᵨ+ᦪẆ-.ᦪḄᓫ0ឋ7ᨬ>2TU2ᑭᵨᦪVWᔠYZ;.DE[GH?⚪=\2᪀⌼].ᦪ2ᑭᵨ+ᦪẆ-.ᦪḄᓫ0ឋ7ᨬ>.DE^GH?⚪=\2ᑖᡂgaxc=lnxe஻agc=h0j:.ᦪ2Ẇ-\ᦪ.ᦪkl"Ḅᑗ6?⚪2Hᑗa6n᳛eFp.ᦪḄn᳛qArᑮWt.DEuG8v;+Ẇ-᩽>2ᓫ0ឋ2ᨬ>2rᑮWt.a[c⌱ὃ⚪Gᐳ10ᑖ.zὃ{ᙠ}22ஹ23⚪⌱F⚪.᝞2ᑣᢥᡠḄ}F⚪ᑖ.⌱4-4ᙶ᪗e<ᦪD10ᑖc22.ᙠ8ᙶ᪗X0Y2ᙶ᪗l"᩽"2x᩽᩽ᙶ᪗256CḄ᩽ᙶ᪗Dp=20cos6.a1cHCḄ᩽ᙶ᪗Dᓄ8ᙶ᪗Da2c"AḄ8ᙶ᪗a1,0c,MC£Ḅ¤"2"¥¦§¨=0©2ᑏ«PḄ®GḄ<ᦪD2°ᑨ²CeG³ᔲᨵ¶ᐳ".ூᫀ௃a1caX-u¹+¥=2a2cPḄ®CḄ<ᦪDºF»+2cos"a஺<ᦪc2ce\'y=2sin஺CÁᨵ¶ᐳ".ூ᪆௃ூᑖ᪆௃a1cH56CḄ᩽ᙶ᪗Dᓄ22=2ÃÄCOS஺2HÇ=285ÉÊ=2Ë11஺ÌÍÎrফDEFGPax,yc2Ma0+0cos஺,⊈sin஺c2᪷4ᔣÒᐵᓽÎ;r¥Ḅ®GḄ<ᦪD2;«jᙊᙊÕÖ27×Ø]qAÎr.ூÙ௃a1cᵫ56CḄ᩽ᙶ᪗DP=2&cos6Îr02=2WCOS஺1HÇ=PÜ5஺,¹=2411஺ÌᐭÎr/+y2=2à2ᓽaX—&J+Ê=2,ᓽ56CḄ8ᙶ᪗DaX—&â+>2=2Ga2cDEFூᨬä௃

35P(x,y),M(Ã+&cos8,Ãsin8)•••AP=y/2AM2/.(x-l,y)=V2(V2+V2cos^-l,^sin^j=(2+2cos6-0,2sine),x—1=2+2cos0—y/o,x=3—+2cos0ᑣ<^2ᓽ<,y=2sin^[y=2singᦑPḄ®GḄ<ᦪDAM=(%]—11)1),ᵫ©=ô©-ᓽ!x-i=0(᳝-1)[>=à1öøf+1r÷V2XᡠM(1(x-1)+1,ùy),Ìᐭ஺ḄDrK஺-1)+1_&2+(K¹)2=2,ᓄûr"PḄ®D³(x-3+0)2+V=4,⊤ýᙊÕJ(3-Ã20),×2Ḅᙊ:x=3-Ã+2cos஺ÿ…jᓄ<ᦪD³þ,஺ᦪ;y=2sin஺ICC|=|(3-&)-&|=3-2&<2-,ᡠᙊCᙊGᑁ#$ᨵ&ᐳ(.ூ᦮,(-௃/⚪123ὃ5ᑭᵨ8ᐵ(:;<(Ḅ>?@A3⚪,

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37y“3-4,x<—231g(x)=|2x+3|-|2x-l|=<4x+2,BB-2(2)/(X+Q)=|X+Q-2|,᝞wᙠBᙶ᪗L¡rE/(x),g(x)wxy=f(x+a)¢y=£ᓫ¥Oᑮᑣ⌕¨/(x+a)2g(x),◤Ry=/(x)ᔣᓽ஺©0,ªy=/(x+a)«|C+஺B2|=4,MO஻ᡈB3(°),\2)222ᑣᦪᔠO◤®¯Ry=/(x)ᔣyᓫ¥a2

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