2022年广东省高考数学综合能力测试试卷（三）（三模）（附答案详解）

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2022
ḕὃᦪᔠᔁஹᓫ⌱⚪ᜧ⚪ᐳ8⚪ᐳ40.
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318.᝞sRt┵P-ABCCḄv☢wxAB=2,AD=25/2,⚔Pᙠv☢4BCDḄzᢗ|ADḄ0.(1)g}Gm☢PAC1m☢P0Bh(2)Hm☢P4B~m☢PCDḄn1,PD=2,g~m☢P4CᡠᡂcḄᜧP.19.ᦪᑡaḄn⚗©=1,a>0,ᓽᓽ+1=4S“-1.nn(1)஺2ḄgaḄ⚗hn(2)i=(-l)naa,gᦪᑡ%Ḅn⚗7.nn+1n20.4Pp2021i,&“R”ḄU⚪ᙽᑣ᝞$Gᵨᡝ¡ᐭ“R”£ᐳ◤U⚪%¥¦¥¥§¨©ª«¬ᓛ®3~ᵨᡝ,U⚪¦¥U⚪¯°§᪷²U⚪³´Rᑖµ¶,ஹ·ஹ¸ஹR¹.✌¥Ḅ¶,¹q3ᑖ¶·ஹ¸¹ᙳq2ᑖ¶R¹q1ᑖh¶·¥Ḅ¶,¹q2ᑖᐸ½¹¾ᙳ¶4⚓ᐳ21⚓

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5¶6⚓ᐳ21⚓

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10᪷3⚣᳛67ᭆ᳛ᩩᭆ᳛:;ᓽ=>?.@⚪ὃCᵫ⚣᳛67ᭆ᳛ᩩᭆ᳛:;EFẠ⚪.10.ூIᫀ௃ABDூ?᪆௃?!M4NBDḄPQRM,STOM,ᑣOM1DB,ᡠW(Y+Y).[=2Y•[=0⌱⚗AMB,]^_PO`ᙊ0ME,F,ᑣ.=-\PA\\PC\=-\EP\\PF\=(|OE|-\PO\)(\OE\+|PO|)=\PO\2-\EO\2=-2,⌱⚗8MC,NACḄPQRN,STON,ᑣON1AC,OAOC=(ON+AM)-(ON+/VC)=2222.?ON-NC=ON(4—Y)=2ON—4,Y0WYwτYe=2ᦑᑐ•ᐗḄNhjkl-4,0⌱⚗C┯MD,m)AB-CD=(AP+7D)■(CP+~PD)=AP-CP+~PB-7D=-\AP\\CP\-\PB\\PD\=-2\EP\\PF\=-4-⌱⚗஺.ᦑ⌱!ABD.᪷3⚪]PḄᙊFnᳮ=ᑨqAC஺ḄN4cḄPQRN,ST஺N,ᑭᵨᔣuḄ_ឋwx=ᑨqBḄ.@⚪y⌕ὃC'{☢ᔣuᦪu~Ḅឋᐸwx⚪ὃC'ᦪᔠ`wx>?kP᫏⚪.11.ூIᫀ௃CDூ?᪆௃?!]f(x)=x+e%ᑣ/(x)ᙠRᓫ⌴v/(6)—/(/na)=b+eb-(Ina+eina)=a+Ina—(Ina+a)=0,■-b=Ina,]=t>0,ᑣa=bt,ᓽhia=b=ln(bt)=Inb+Int,•Int=b—Inb5]g(x)=%—Ex5%>0,g'(x)=1—!=mx£(0,1))g0,

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!ᑭᵨqrsᳮMNᓽiᑨPC*u⚪ὃwxᵨᔣKLyAz{|}⚪!~ὃwxḄBC!ᙠ=⌱⚗ᡠḄ>?!⚪.13.ூᫀ௃40ூy᪆௃yᵫ⚪3+1=M!ᑣ)an3ᦪᑡajḄ=27✌⚗!3l¢Ḅ£¢ᦪᑡ!ᑣS4=¥¦=40.ᦑᫀ40.ᵫ§¨ᔠ£¢ᦪᑡḄsª«¬l®ᓽi¬y.u⚪¯⌕ὃwx£¢ᦪᑡḄᑨP«£¢ᦪᑡḄ¬l®!±Ạ⚪.14.ூᫀ*´y᪆µy¶Mna=2,ᑣsin(2a4--)=—sin2a+—cos2a=—•'4!2222sinacosa+cos2a-sin2aV22tana+l-tan2a=-------------------sin2a+cos2a--2tan2a+l¿14⚓!ᐳ21⚓

14V22-2+1-22y[2--------=—f222+110ᦑᫀÂ.10ᑭᵨÃDḄrl®!Ä⌕¬Ḅ®ÅnÆ!ÇᑭᵨÈDcDdᦪḄ±uᐵG!ÊËDl®!¬fÌḄ¨Í.u⚪¯⌕ὃwÃDḄrl®!ÈDcDdᦪḄ±uᐵG!ÊËDl®Ḅᵨ!±Ạ⚪.15.ூᫀ௃^+-=1494ூy᪆௃yᵫ⚪;!ÏᙊḄᯖᙠyÓÔ!22!ÕÏᙊÖ×3+Ø=l(a>b>0),a1b2••ÛÓÜ4,•2b=4,ᓽb=2,ᵫÏᙊsª!|PF/+|PF2|=2Q,•I^II=6|PF|,2.,.ÝQ=7,22•••ÏᙊCḄÞ᪗ßÖ×ᓃ+±=1,494ᦑᫀ^+-=1.ᵫ⚪;iÕÏᙊÖ×â+ã=l(a>6>0),ᵫ|P&|+\PF\=2a|PFi|=6\PF\,22ifçè21=!ç&1=é!aêᱯì&íîᓽi.u⚪ὃwxÏᙊḄsªZïᓫឋò!±Ạ⚪.16.ூᫀ௃2ூy᪆௃y:(1)/(1+2x)-f(x)=1+210g2Q+2%)—1—210஺2(1+%)

15=2log2=2*2(2)/(m-1)=14-2log[l+(m-1)]=1+2logm22ff(n-2)=1+2log[l+(n-2)]=14-2log(n-1),22f(m—1)+f(n—2)=f(n)—1£ú1+2logm+1+2log(n-1)=210g2(1+n),22B[Jlog[2m(n-1)]=log(l4-n),22ᦑ2᪷({-1)=n+1,ᐸþm>0,n>1*ᡠ27n+2ÿ=*+2=3++2(1)23+4(n-1)=7,ᡂ1ᓽ=2,⊈=5ᡂn-l2ᦑm+ᨬ"#ᦑ$ᫀ&'(1)2((2)(1)ᑭᵨ+ᦪ-.ᓄ0ᓽ1((2)ᑭᵨ+ᦪ-.ᓄ213ᑮ2nl(1)=+1,ᐸ7m>0,n>18:ᑭᵨ;<=>?ᨬ"#ᓽ1.<⚪ὃCD+ᦪ-.E;<=>ḄGᵨHI7᫏⚪.17.ூ$ᫀ௃M'(1)ᙠ4ABC7ᵫSTUᳮE2asinA=(2b+c)sinB+(2c+b)sinC3:a2—b2—be=c2,ᵫ]TUᳮ3cosA==-i,2bc2_013a2=b2+c2+bc,ᵫ]TUᳮ1?cos4=-ᵫ4ey0,ᐔz13rḄ#.(2)4஺fAABCḄhiᑖkNB4D=^DAC=pᵫS-BC=S-BD+SASD1?1?☢.16⚓ᐳ21⚓

16<⚪⌕ὃCDSTUᳮ]TUᳮhḄ☢>ᙠMh7ḄᔠGᵨὃCD.rᓄHI;Ạ⚪.18.ூ$ᫀ௃y1'ᙠRtAABC7tan/ACB=ᙠRtAAOB7tan4B஺=ᑣ4cB=/.ABO,If4cB+@OBC=90°,ᡠc2C1BO,q&P01i☢rBCD,ᑣACIPO,APOCBO=0,ᡠc4cli☢P08,4Cui☢PAC,ᡠcP4C1i☢POB(M'y2q&2B஻CC,ABCi☢PCD,CDui☢PCD,ᡠcAB஻i☢PCD,_i☢P4Bni☢PC஺=1,4Bui☢PCO,ᡠc1஻4B,ᑣ/¢i☢PACᡠᡂhḄST#I4B¢i☢PACᡠᡂhḄST#c4&ᙶ᪗¥¦§᝞©ᡠªḄ«¬hᙶ᪗®r-xyz,ᑣ4(0,0,0),P(0,V2,V2).6(2,0,0),஺(2,2¯,0)ᡠc°=(0,±,&)AC=(2,2V2.0).AB=(2,0,0).³i☢P4CḄ´µᔣ·&ᐗ=(x,y,z),¹,º0,(n-AC=0ᓽ»y+»z=0,ᓽ¼:-ᦔ(2x+2yj2y=01z=-y¾¿=1,3À=(Á,1,1)³/¢i☢24cᡠᡂh&஺ᑣsin஺=|cos\=1n•Ã=Ä=Å|7l||/1JD|NXNZ

17_q&஺G[0,Èᡠc1¢i☢PACᡠᡂh&ூM᪆௃(1)ACIB஺:ᵫk☢ᚖḄឋÍUᳮ3ACIP஺3k☢ᚖᯠÏ13☢☢ᚖ((2)ᵫk☢iÐḄᑨUUᳮ¢ឋÍUᳮ஻qÒ?3k¢i☢PACᡠᡂhᓽ1§᝞©ᡠªḄ«¬hᙶ᪗®ᵨ«¬ᔣ·µ?k☢h.<⚪ὃCD☢☢ᚖḄk☢hḄ.HI7᫏⚪.19.ூ$ᫀ௃M'(1)n=l,aa=4a-1,M3a2=3,r2tᵫ⚪|a/n+l=4S—1জ,<^n+lan+2=4S—1ঝnn+1ᵫঝজ3Ù+1(ᓽ+2-Ú)=4¢+1q&ᓽ>0ᡠca+2-a"=4,nIf'ᦪᑡÝa"Ḅ᜻ᦪ⚗fcᵫ=1&✌⚗c4&áḄáᦪᑡᏔᦪ⚗fc=3&✌⚗c4&áḄáᦪᑡᡠcÝᓽãḄä⚗>a-2n-18n(2)ᵫ(1)13å=(-l)n(2n-l)(2n+1),T=஺ä2+a2a3-a3a4+a4a5...+(-l)naai=a(-a+a)+a(-a+nnn+21343as)……+(-l)nan«n+i>æ&ᏔᦪT=4(a+<24+....+a)=4ஹ'"'"௃)=2n(n+1)-n2nn&᜻ᦪT=4(a+a+....+ᓽ-rMan+i=43êë(2n-l)(2n+n241)=-2n2—2n+1,2n2+2n,n&Ꮤᦪì,ᦪᑡÝ4ãḄín⚗஻=-2n2-2n+l,n&᜻ᦪூM᪆௃(1)î#ᓽ1?ïa?,ᑭᵨᓽ¢ñḄᐵ®1?3ÝᓽãḄ⌴ôᐵ®?ïa(n(2)+nᑖ᜻Ꮤõön&Ꮤᦪ,÷ᵨø⚗µ?n&᜻ᦪ,T=*Eaa.nnn+1<⚪ὃCDᦪᑡḄ⌴ô>ø⚗µ?HI7᫏⚪.20.ூ$ᫀ௃M'(1)XḄᡠᨵ1#&3,2,1,P(X=3)=(,P(X=2)=(+q=',P(X=l)=%ᐸᑖûᑡ&18⚓ᐳ21⚓

18X321111P424y2z³ü3ᑖ&y,ᑣyḄ#&5,4,3,2,r=5z=ixi=±Pyy=4z=ixiixi=^m+cஹ12,115cஹ121Pyy=3z=-x-+-x-=—,P)Y=2z=-x-=-'7234312v,436y543211151p2030126ᡠcEy=5x/+4x2+3x^+2x=3.3.ZUSU1ZOூ᪆௃(1)ᢥ᯿ᦣ!"#ᑖ%ᑡḄ()ᓽ+;(2)-.ᢥ᯿ᦣ!"#ᑖ%ᑡḄ()ᓽ+./⚪ὃ23ᦣ!"#Ḅᑖ%ᑡ456789:᫏⚪.'C_1a~221.ூ<ᫀ௃(1)>⚪?:2»+2=17Ba2=4,b2=3,a24b2a2-b2=c2ᦑFᙊCḄIJKL+M=1.43(2)NB(P7K)C(%27y2)7.••SBKFᙊEUV9WஹY⚔SḄ[S,ᑣ-]BC^4_`aᔠ7ᑣ+NK_=zny+34FᙊIJὶfB(362+4)y2+6mty+3t2-12=0,ᑣ4=367n2t2_12(362+4)(m—4)>0,+Bm<3m2+4,ᵫstuᳮ+Bw+%=xy7w%=᫔{

19-]B4ḄIJKy=|(}x2),~x=tBSMᙶ᪗VM=4W7ᳮ+B7SNᙶ᪗2.X2-20ஹwஹMஹNSᐳᙊ7ᵫᒘ]uᳮ+B|P*|P0|=|PM|-|PN|,ᓽ-2)=lywy/vb_%4-2)2_%K(52)2___________K2(♦2)2___________''M'N-(-2)(X-2)-(my+t-2Xmy+t-2')-m2yy+7n(t-2)(y+y)+(t-2)2-X12121212___________3(-4)(t-2________________3(t+2)(t-2)2________3m2(t2-4)-67n2t(t-2)+(37n2+4)(t-2)2-37n2(t+2)-67n2t+(3?n2+4)(t-2)=__,3()(1)2__=3(t+&)(&2)232)(t_2).=3m2t+6m2-6m2t+3m2t-6m2+4t-84(t-2)4''ᵫt>2,ᦑt(t—2)=[(t+2)(t—2),Bt=6.ூ᪆௃(1)᪷ᩩᑡᐵ9aஹbஹcḄIJᓽ+(2)N஺0272)7N8(7K}=^1+᝞4FᙊIJὶf᪷stuᳮB᪷4ᦪḄᐵ.⊤BAஹC4ḄIJ7~x=tᑖ¡MஹNḄᙶ᪗.0ஹAஹMஹNSᐳᙊ7ᵫᒘ]uᳮ+B|P*•|P0|=|PM|-|PN|,¢ᐭᔜSᙶ᪗ᓄ¦ᓽ+BtḄ§./⚪¨⌕ὃ2FᙊIJḄ7-]4ᙊ┵«]Ḅ¬ᐵ7stuᳮ®ᐸ°ᵨ²³,89:²⚪.22.ூ<ᫀ௃(l)/(x)=x2—X+2sinx,f(x)=2x—1+2cosx.~g(x)—f'(x)=2x-1+2cosx,ᑣg'(x)=2—2sinx>0,ᦑg(x)=2x-1+2cosx´RUḄµ¶ᦪ7·K((0)=1,ᡠ¹IJ=1ᨵ»xx=0.¼20⚓7ᐳ21⚓

20¾/(0)=0,ᑣᑗSK(0,0),ᓽÀ᳛K1Ḅᑗ]IJ´y=X;(2)/'(%)=2%—a+2cosx,~g(x)=/'(%)=2x—a+2cosx,ᑣg'(%)=2—2sinx>0,ᦑg(%)=2%—a+2cos%´RUḄµ¶ᦪ,¾g(^^)=2•—a+2cos^-=—3+2cos^-^-<0,2,222,a+3ஹ஺a+3஺x+30na+3n5(—)=2---------a+2cos—=3+2cos—>0,222ᦑÂᙠ»xÄᦪ4e(²,²)7Åg(4)=0.ᙠÆÇ(-8,X஺)U7g(x)=f'(x)<0,f(x)ᓫÉ⌴Ë,ᙠÆÇQo,+8)U,g(x)=f'(x)>0,/(x)ᓫÉ⌴µ,ᦑ&´f(x)Ḅ»x᩽§S7ÎK᩽ϧS.¾஻0)=0,ᦑf(x)ᙠÆÇ(0,2ᐔ)Uᨵ»xÒS²Ó9Ô>0Î/(2ᐔ)>0,ᦑ>dB20ᡠ¹ÄᦪaḄÕ§×K(2,2ᐔ).ூ᪆௃(1)ضᦪḄÙ¶ᦪ7~g(%)=(஺)=2%-1+2cosx,ᐸÙ¶ᦪ7Úᔠ®¶ᦪḄᓫÉឋ+BᑗSK(0,0),BᑮÀ᳛K1Ḅᑗ]IJ´y=%(2)/'(%)=2%-a+2cosx,~g(%)=f'(x)=2%-a+2cosx,ᑭᵨÙᦪẆ߶ᦪḄᓫÉឋ7ᑭᵨ¶ᦪÒSᑨáuᳮ+BÂᙠ»xÄᦪX஺e(â,²)7Åg(&)=0,Bᑮ&´/Xx)Ḅ»x᩽§S7ÎK᩽ϧS.ã⚪äᓄKᐵ9aḄ^²L./⚪ὃ2ᑭᵨÙᦪẆßå«]UæSᜐḄᑗ]IJ7ὃ2¶ᦪÒSḄᑨu®°ᵨ7ὃ2ᓄè4äᓄéê7ὃ2ëìíw7´:᫏⚪.