2021-2022学年甘肃省金昌市永昌某高级中学高一（下）第一次月考数学试卷（附答案详解）

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2021-2022
ᵞᾓḕᨴὃᦪᔁ1.a=(x,3),b=(3,1),"#஻b,ᑣx=()A.9B.-9C.1D.-12./482Ḅᑁ54,8,஺Ḅ78ᑖ:#஻"<"4=8=4,£1=2⍩ᑣ5஺=A.?B.WC.ABD.CD3.sina=I"a#┦5ᑣcosa+?=A.-⌹B.C.OD.101010104./ABCḄᑁ5A,B,CḄ78ᑖ:#a,b,c,P4=60஺a=V3,ᑣ..T஺sinF+sinCVWA.-B.V3C.-D.2225.PᔣZ[B\]|_|=1,`|=2,ala+K,ᑣcBdḄᜳ5#A.-B.-C.—D.—63366.ga#┦5᜛#ij▲5"cosa=||,sinS=—|,ᑣcosa-/?Ḅn#A.--B.--C.-D.-656565657.gtana=4,ᑣ…rMaḄn#sin2aA.18B.-C.16D.-448.ᙠ/ABCPtu+v2=0,ᑣ/ABCḄwxyzA.V8i5wB.{5i5wC.VῪi5wD.VῪ{5i5w9.}☢ᔣZ=1,0,_=1,2ᑣᑡuḄzA.|a+6|=16B.a+b-a=2C.ᔣZc+BB_Ḅᜳ5#30°D.ᔣZc+dᙠḄᢗᔣZ#2c10.ᑡᨵᐵᔣZ⚪uḄzA.P|=\b\,ᑣ=B.⊤ᜧ6,"c•=b•ᑣc=bC.Pc—b,b=c,ij8—cD.P:=>ᑣ=\b\^a//b11.ᑡᔜn#Ḅz

1A.B.2tanl50•cos215°l-tan222.5<,c^cos2—--sin2—D---+ߟ-31231216sm50°16cos50012.ᙠ/ABCᔜ5ᡠ78ᑖ:#q,b,c,ᑡ¢£uḄᨵ()aA.¤ᑣ/4BC#V8i5wcosAB.g(a+b+c)(a+b—c)=3ab,Ij/C=60°C.a=7,b=4V3,c=V13,ᑣᨬ¦ᑁ5Ḅ§ᦪ#30஺D.ᙠa=5,A=60°,b=4,¨i5wᨵ©¨13.ᔣZa=(1,=(3,—1),P_>Lb,ᑣm=.14.V3tan87°tan33°—tan87°—tan33°=.15.ᙠ/ABCA=60",AC=4,BC=273,ᑣ/4BCḄ☢¬VW.16.ᙠ/ABC=®¯?zBNḄ±P²=+³ᑣ´ᦪµḄn#——.17.¶ᔣZc=(1,2),b=(1,-1).(·)¸|+252(·)P¹=c+3,BC=a-2b,CD=4a-26,¸º®A,C,஺i±ᐳ¼.18.tan(a—£)=5tan/?="aஹ/?6(0,TT).(1)¸tanaḄn2(2)¸2a—/?Ḅn.19.᝞Àg/4BC஺#BCḄ±AE=^EC,AD,BEÂW±F,¶Ã=ÄAD=b.পᵨÄdᑖ:⊤ÇᔣZEB-.(2)P_=t¸´ᦪfḄn.20.ÉজbsinC=V3ccosB,ঝ᮱+ac=a?+c?Í©ÎᩩÐÑ⌱Î⊡ᐙᑮ☢ᩩÐÖרØ.2⚓ᐳ12⚓

2/ABC5A,B,CḄ78ᑖ:#a,b,c,".(ÚᑏজᡈঝÝÞ<⌱ßÎ᪗áâãäᩩÐÖרØ.)প¸8(2)Pb=2,/4BCḄ☢¬#å,¸a.21.᝞ÀæçÉéêAëìíu_ᔣî×ïᛞBᙠéêAᓅòó75஺Ḅ_ᔣ"BéêAôõïᛞCᙠéêAᓅòö30஺Ḅ_ᔣ"BéêAôõ8yf3nmile,÷çî×ᑮ஺ᜐùúᑮïᛞBᙠᓅòó135஺Ḅ_ᔣ.(1)¸஺BéêAḄõû:(2)¸஺BïᛞCḄõû.22.ᔣZü=(sinx,—J,n—(V3cosx,cos2x),ýᦪ/(þ)=ÿᐗ(1)
ᦪ/(x)Ḅᨬᜧᨬᕜ(2)ᦪy=f(x)Ḅᔣgᓫᑮᦪy=g(x)Ḅ
g(x)ᙠ6[0Ḅ.

3!ᫀ#$᪆1.ூ!ᫀ௃Aூ$᪆௃$*•••ᔣ,-஻/,9—%=0,$%=9.ᦑ⌱*A.ᑭᵨᔣ,ᐳ;<ᳮᓽ?@.A⚪ὃDEᔣ,ᐳ;<ᳮFGHẠ⚪.2.ூ!ᫀ௃Bூ$᪆௃$*᪷L⚪MN48CRc=4,a=2®ᑣA>C,ᑣᨵC<ᡈᵫ[<ᳮ?*\=\]ᵫ/=$c=4,a=2A/6,ᑣsinC=dᔞᲝ=g,a2yf6~2]ᵫC

4BsinCsin2••b=2sinB,c=2sinC,ᑣ——=2.sinF+sinC4⚓ᐳ12⚓

5ᦑ⌱*D.ᵫsᔠ[<ᳮᓽ?¡
$.A⚪⌕ὃDE[<ᳮḄ}ᵨFGHẠ¢⚪.5.ூ!ᫀ௃Cூ$᪆௃$*¤1(4+¦)a-(a+Z7)=a24-a-Z)=l+a-&=0,Aa-h=-1,,¬ஹ¤¯T1cos=—=r=—=—,|a||d|1X22]Va,b>G[0,7l],¤z/Ḅᜳk-´ᦑ⌱*c.᪷Lᩩ¶?@/=—1ᯠ¸ᓽ?
@cosீ¤º»Ḅ¼½@!ᫀ.A⚪ὃDEᔣ,ᜳkḄ[ᔣ,ᚖḄᐙ⌕ᩩ¶ὃDEuvÀFGHẠ⚪.6.ூ!ᫀ௃Aூ$᪆௃$*Á-a-┦k/?-j▲kcosa=||,sinS=-|,ᡠÆsina=jcos᜛=-ᑣcos(a-£)=cosacosp+sinasinp=—x—x(-f)=-—,13513565ᦑ⌱*A.
@sina=ᔁcos£=sᔠkÊḄ[ᦪËᐭ
ᓽ?.A⚪ὃDk#Ḅjkᦪ
FGHẠ⚪.7.ூ!ᫀ௃Dூ$᪆௃$*?tana=4,.τÎÏ_2cos2a+8sin2a_l+4tan2a_1+64_65**ஹ2sinacosatana44'ᦑ⌱*D.ÒᑖÓᑭᵨÔÕkḄ[ᦪᓄ×ØᑭᵨkjkᦪÙHAᐵoÚlÛËᐭuvᓽ?
@.Ü⚪ὃDEkjkᦪHAᐵoḄÝᵨÞßàáHAᐵoâ$A⚪Ḅᐵã.8.ூ!ᫀ௃Bூ$᪆௃$:ᙠZiABCRå•æ+ç2=AB.(AB+BC)=AB-AC=0,.-.ABLAC,.•.è4=éᑣN4BC-kjkl

6ᦑ⌱*B.ᵫᩩ¶
å•ê=0?ë1¯ᦑ44=],ᵫÜ?NABCḄlí.A⚪⌕ὃDᔣ,ᚖḄᩩ¶jkllíḄᑨ<FGHẠ⚪.9.ூ!ᫀ௃BDூ$᪆௃ூᑖ᪆௃A⚪ὃDE☢ᔣ,ᦪ,ðḄÝvᔣ,Ḅᜳkᢗòᔣ,ÛóFGHẠ⚪.᪷Lᔣ,ᙶ᪗;ឋÝv#÷Ḅᙶ᪗⊤ùᓽ?ᑨúA᪷Lᔣ,ᦪ,ðḄᙶ᪗⊤ùᓽ?ᑨú᪷Lcosீ1+ûa>=üýᓽ?ᑨúC᪷Lᢗòᔣ,Ḅ<þᓽ?ᑨúD.\a+b\\a\ூ௃2+3=(2,2ᑣ|+9|=74+12=4,ᦑA┯(a4-Z))-2=2,ᦑ8cosீ஺+&a)'0°<7=27ᦑ஺.|a|ᦑᫀ⌱BD.10.ூᫀ௃ABூ᪆௃ᔣ.ᵫCD⌕F0ᔣGHIJKA┯L7஻NO1HᚖRSTᡂVWX8YZ[\]B┯L_=3,b=c,ᵫᔣ.ḄZ`ab7=cCd[ᔣ.ed[0ᔣdf஺⌱⚗.ᦑ⌱AB.᪷iᔣ.Ḅᭆk1ᔣ.ḄeḄᭆk⌲Yᑖ᪆ᔜD⌱⚗ᓽab.p⚪r⌕ὃtuᔣ.Ḅᭆk1ᔣ.ḄeḄᭆkḄvᵨx\yẠ⚪.11.ூᫀ௃ABCூ᪆௃ூᑖ᪆௃p⚪ὃt{4ឤ[}ᣚḄᔠvᵨx\᫏⚪.Rᑭᵨ{4ᦪᐵḄឤ[}ᣚḄdᐵᑨAஹBஹCஹ஺Ḅ1ᔲ.ூ௃6⚓ᐳ12⚓

7A-7J-ri-~T~4tan22.5°12tan22.5°1._1_LL.-r--஽,\oA------=-x-------=-tan45°=-,ᦑA\32tanl5°-cos215°=2sinl5°cosl5°=sin300=ᦑ5\C—cos2———sin2—=-cos-=ᦑC312312362\D_^+/_=Lx(a⌭¤)=Lx¥g§=¨ᦑ஺┯16sin50°16cos50°16IsinSO^cosSO0J16%inl00042ᦑ⌱ABC.12.ூᫀ௃ABCூ᪆௃\4L®—°®{cos?lcosBcosCrn.isin?!sinFsinC´!J--7=---------7cosAcosBcosCᓽtanA=tanB=tanC,µA,B,஺{4¶ᑁ4ᡠ,4=B=C,ᓽ¸4BC[¹{4¶ᦑA\8,ᵫ(a+b+c)(a+b—c)=3Q5abQ24-b2—c2=ab,r-|ca2+b2-c21ᡠr,HcosC=-------=2ab2µ0

8ூ᪆௃•■a-b=3—m=0,:.m=3.ᦑᫀ3.᪷i01BabË7-b=0,ÒÓᔣ.ᙶ᪗Ḅᦪ.Ö×ÐᓽaØËmḄÙ.p⚪ὃtuᔣ.ᚖRḄᐙ⌕ᩩÜᔣ.ᙶ᪗Ḅᦪ.Ö×ÐὃtuÏÐÝÃx\Þ᧕⚪.14.ூᫀ௃V3ூ᪆௃tanl20°=tan(87°+33°)=taM+
an33
=Yà..tan87°+taan33°=1—tan87tan33—V3+V3tan870tan33°,V3tan87°tan33°—tan87°—tan33°=V3tan87°tan33°—(—V3+V3tan87°tan33°)=V3,ᦑᫀV3.ᵫ⚪áᑭᵨC4GḄᑗÏÐØb⌕ØãḄÙ.p⚪r⌕ὃtC4GḄᑗḄvᵨx\yẠ⚪.15.ூᫀ௃2V3ூ᪆௃ூᑖ᪆௃ᑭᵨ{4¶ḄÄZᳮØË4B,äᑭᵨ{4¶Ḅ☢ÖØËA4BCḄ☢Ö.p⚪ḼçὃtuèË{4¶ḄC¹GᐸY¹Ḅ4ØêḄ☢Ö.ÄZᳮஹR4{4¶ஹ{4¶Ḅ☢Ö[ëìx\yẠ⚪.ூ௃•••¸ABCA=60°,AC=4,BC=2®ᵫÄZᳮbÝ=1sinXsinfi2V34sin60°sinB'bsinB=1,v00/3x4xsin30°=2A/3.ᦑᫀ2®16.ூᫀ௃211ூ᪆௃:Pñ8N:ḄYò,óô=4ôᵫô=õ᪃8⚓ᐳ12⚓

9ᑣ÷=AB+BP=AB+A^N=AB+A(AN-AB)=(1-A)AB+AANYA—.=(1-^AB+-AC4m=1—A,-=—411b4=*m=-^ᦑᫀøóô=4ùᡃû᧕üQ⊤ÿ(1-
)ᓭ+(?Ḅ᪷☢ᔣḄᳮᡃ᧕᪀⌼ᐵ%Ḅ!"!#ᓽ%&ᑮ(Ḅ).⚪,⌕ὃ/0Ḅ1234☢ᔣḄᳮ5ᐸ78"9⚪Ḅᐵ:4᪷☢ᔣḄᳮ᪀⌼ᐵ;”Ḅ!=>᫏⚪.17.ூ9ᫀ௃"C(I)a+2b=(1,0),|a+2b|=V1+0=1D(஻)GHC••J=L+J=M+N+OP29=2QPᩈ.■.CD=2(2a-b)=2AC,.■■A,C,஺T3ᐳV.ூ"᪆௃(/)ᑭᵨᔣᙶ᪗]^ឋ`ஹbḄc^dᓽ%&efgD(஻)ᑭᵨᔣᐳVᳮᓽ%GHfg.⚪ὃ/0ᔣᙶ᪗]^ឋ`ஹbḄc^dஹᔣᐳVᳮὃ/0hᳮijkc^ij=Ạ⚪.18.ூ9ᫀ௃"C(1),tan(a—᜛)=Jtan/?=nQஹ᜛€(0,ᐔ)11tan(a-/?)+tan/??~71•tana=tan[(a-£)+£]=---------D-----———-=------——=-"8q1Ptan(a—S)tan£.,113v1+27ফtan(2a-2£)=s^4tan(2a-20)+tan/?tan(2aP£)=tan(2a-2£+£)=------------------------=1kq1Ptan(2a-20)tan/?va>£6(0,ᐔ)tan£=—vw°66ᐔ)tana—tan(a—0+0)=C,a6(0,—ct-PG(—7i,0)9tan(a—/?)=[,a—/?G(—7r,—1)2a—0=a-/7+a€(—TI,0)

10ytan(2a—/?)=12.ct—°<——.ூ"᪆௃পᵫtana=tan[(a-£)+£],ᑭᵨᑗᳮief.(2)ᐜᑭᵨwdetan(2a-20)=£ᵫᑭᵨtan(2a—£)=tan(2a-2/?+/?)etan(2a-4)=1,ᵫie2a-⚪ὃ/Tᦪ)Ḅ4>᫏⚪"⚪⌕⚪7ᑗᦪᳮḄᔠᳮ]ᵨ.19.ூ9ᫀ௃"C(1)ᵫ⚪7஺BCḄ>3n=C■■■AB+AC=2AD,AB=2b—a,=AB-AE=2b-a--a=--a+2b=33(2)vAF=t=tb,:.FB=AB—AF=—¡+(2—t)b,-EB=-Ja+2b,FF,JᐳV•_-1—_2-_t••ߟ2’3t=-.2ூ"᪆௃⚪ὃ/ᔣḄVឋ]^ὃ/ᔣᐳVᩩ¨Ḅ]ᵨ=>᫏⚪.(1)ᑭᵨᔣḄVឋ]^ᓽ%ᵨ©ᑖ«⊤ᔣEB;(2)®¯=tᑭᵨ°°ᐳV±ᦪ1Ḅ).20.ூ9ᫀ௃"C(1)⌱³ᩩ¨জCᵫµᳮ5bsinC=V3ccosB,1sinBsinC=V5sinCcosB,¼sinC*0,ᡠ¿sinB=V3cosB,ᡠ¿tanB==V3,¼Be(0,7T),ᡠ¿B=g.⌱³ᩩ¨ঝCᵫŵᳮ1cosB=a+c—ZacZac2ᡠ¿8=ᐰ(2)¼ÈjBCḄ☢ÉS=[acsinB=gacXÌ=Íᡠ¿ac=4,ᵫ(1)1j2+஺஺=஺2+,ᡠ¿4+4=a2+c2,ᓽ8=a2+(92,Î10⚓ᐳ12⚓

11ᓄÑ&a"-8a2+16=0,"&a?=4.ya>0,ᡠ¿a=2.ூ"᪆௃(1)⌱³ᩩ¨জCᑭᵨµᳮᓄÕÖfᔠ×TᦪḄᖪᦪᐵÙ&"D⌱³ᩩ¨ঝCᑭᵨŵᳮecosBḄ)ᓽ%D(2)ᵫS=[acsinB,%&ac=4,Öfᔠŵᳮ&".⚪ὃ/"TÚÛÜݵᳮஹŵᳮ4"⚪Ḅᐵ:ὃ/Þßhᳮijà]^ij=Ạ⚪.21.ூ9ᫀ௃"C(1)ᵫ⚪7&ᙠÈ4BD>AB=4V6,4DAB=75°,4ADB=45°,ᵫµᳮ&AD_ABs

12z.ABDsinZ-ADB1AB-sinZ-ABDAD=-------------sinCADB4V6xsin(180°-75°-45஺)sin45°=Unmile.(2)ᵫপ&4D=12,ᙠÈ4CD>AC=8V3,/.DAC=30°,ᵫŵᳮ&CD2=AC2+AD2-2AC-ADcos30°=48,CD=4>/3nm.ile.ூ"᪆௃⚪Ḅὃ34"TḄ±▭åᵨ,⌕ὃ/0µᳮஹŵᳮᙠ"T>Ḅåᵨ"æ±▭ç⚪Ḅᐵ:4⌕è±▭ç⚪éᓄᦪêç⚪ᯠ#ᑭᵨᦪê12ìí".(1)ᙠAABD>ᵫµᳮ%AOD(2)ᙠÈACD>ᵫŵᳮ&CD.22.ூ9ᫀ௃"C(l)f(x)=m-n=V3sinxcosx—|cos2x=fsin2x-^cos2x=sin(2x—ó2ᡠ¿/(%)Ḅᨬᜧ)1,ᨬöᕜø(2)ᵫ(1)&f(%)=sin(2xPg).ùᦪy=/(%)Ḅûüᔣýþgÿᓫ
ᑮy=66sin[2(x+7)-7]=sin(2x+g)Ḅ.666

13g(%)=sin(2xH—),ExE[0,—],ᡠ#2xH—6[—,$],sin(2x-\—)G[—,1].6266662ᦑg(x)ᙠ[0,*+Ḅ,-.ூ0᪆௃(1)ᑭᵨᔣ6Ḅᦪ68#9:;<=>Ḅ?;@ᦪᓄB@ᦪḄ0᪆CDᯠF0@ᦪḄᕜH#9ᨬ,.(2)ᑭᵨ@ᦪḄJᣚFL@ᦪḄ0᪆CDᯠF0@ᦪḄ,-.M⚪ὃPᔣ6=?;@ᦪQRᔠD:;<=>Ḅ?;@ᦪDὃP?;@ᦪḄ=ឋU#9VWXY.Z12⚓Dᐳ12⚓