福建省龙岩市第一中学2023届高三上学期第二次月考数学答案

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龙岩一中2023届高三上学期第二次月考数学答案1-8:DDCCBABD9.ABD10.AB11.AC12.BC13.4314.()0,−和()1,015.2−16.1317.(1)因为22258bcabc+−=,由余弦定理可得222-5cos216bc

aAbc+==.故5cos16A=·························································································································

···············4(2)因为sin2sinCB=,所以2cb=.由余弦定理得2222152cos4abcbcAb=+−=,则152ab=.····························································

············································································6因为ABC的周长为615+,所以1536152bb+=+,解得2b=.··························8所以A

BC的面积为215231212164bb−=·········································································1018.(1)因为函数()2ππ2sinsin

3sincoscos44fxxxxxx=+−++2222232cossincossinsin2cos22222xxxxxx=+−++()2231cos2

cossinsin222xxxx+=−++π13sin232x=++·············································································································

··············3令π2π,3xkk+=Z,解得ππ62kx=−+,即对称中心π1π,622kk−+Z,··································5当π0,2x时,则ππ4π2,333x+,再

结合三角函数图像可得()11,32fx−+所以,函数对称中心:π1π,622k−+,kZ,值域:11,32−+.···········································7(2)因

为函数()gx的图像与函数()fx的图像关于y轴对称,则()()π13sin232gxfxx=−=−++,························································································

·9令ππ3π2π22π232kxk+−++,kZ,解得7ππππ,1212kxkk−+−+Z·······························11当1k=时,即为5π11π,1212

所以当()0,πx时,()gx的单调递增区间:5π11π,1212.·················1219.(1)由题表知,随着时间x的增大,y的值随x的增大,先减小后增大,而所

给的函数(0)yaxba=+,()log0,0,1byaxabb=和(0)aybax=+在(0,)+上显然都是单调函数,不满足题意,故选择()20yaxbxca=++.·········································

··············································3(2)把()2,102,()6,78,()20,120分别代入2yaxbxc=++,得42102,36678,40020120,abcabcabc++=++

=++=解得12a=,10b=−,120c=∴()221110120107022yxxx=−+=−+,,()0x+.···································································5∴当10x=时,y有最小值,且

min70y=.故当该纪念章上市10天时,市场价最低,最低市场价为每枚70元.·······································7(3)令()()()1701010210fxgxxxx==−+−−(10,)x+,················

················································8因为存在()10,x+,使得不等式()0gxk−成立,则()minkgx.··················································

·················································································9又()()()17017010210235210210gxxxxx=−

+−=−−∴当10235x=+时,()gx取得最小值,且最小值为235,∴235k.···································································

···································································1220.解:(1)由21()2ln(21)(0)2fxxaxaxa=−+−,可得2()21fxaxax=

−+−.因为(1)2211faaa=−+−=+,13(1)21122faaa=−+−=−,所以切点坐标为3(1,1)2a−,切线方程为:()311(1)2ayax−−=+−,因为切线经过(0,0),所以3112aa−=+,解得4a=.··················

·················································4(2)解:由题可知()fx的定义域为(0,)+,21()[(21)2]fxaxaxx=−−−−,令()0fx=

,则2(21)20axax−−−=,解得1xa=−或2x=,························································6因为0,a所以10a−,所以12a−,令()0fx,即2(21)20axax−−−,

解得:12xa−,令()0fx,即2(21)20axax−−−,解得:1xa−或2x,···················································8又()fx的定义域为(0,)+,所以,()f

x增区间为(0,2),减区间为(2,)+.因为()22()211gtttt=−=−−,所以函数()gt在区间(0,2]的最大值为0,··································9函数()fs在(0,2)上单调递增,故在区间(0,2]上max()(2)2ln

222fsfa==+−,·······················10所以2ln2220a+−,即ln210a+−,故1ln2a−,所以a的取值范围是(0,1ln2)−.·····1221.(1)取BC中点O,连接AO,1AO,1AC,因为ABAC=,所以AOBC⊥,·

··································································2因为11AABAAC=,11,ABACAAAA==,所以11AABAAC,所以1

1ABAC=,所以1AOBC⊥,····························································4因为1AOAOO=,1,AOAO平面1AAO,所以BC⊥平面1AAO,因为1AA平面1AAO,所以1AABC⊥;········

·················································································6(2)连接OD,则平面1AAO即为平面1AADO,由(1

)知BC⊥平面1AADO,因为BC平面ABC,且BC平面11BCCB,故平面1AADO⊥平面ABC,平面1AADO⊥平面11BCCB,过O作1OMAD⊥于M,则OM⊥平面ABC,过1A作1AHOD

⊥于H,则1AH⊥平面11BCCB,因为11DOBBAA∥∥知DOBC⊥,在ABC中:2,22ABACBC===,所以11122BDBSDBDO==△,所以11111111214336BABDABDBBDBAAVVShh−−====△

,所以1172AAHh==,························································································································8法

一:设MOD=,则1DAH=,在1RtAHD△中117142cos42AHAD===,所以214sin,cos22DMDOOMOD====,又12AD=,所以点M为线段1AD的中点,以O为原点,分别以,,OAOBOM分别为x,y,z轴正方向建立空间

直角坐标系,1214(2,0,0),(0,2,0),(0,2,0),,0,22ABCA−,1214214,2,,,0,2222BD−−,设面1ABD的法向量为()1111,,xnyz=,则有11111111121

420222142022nBAxyznBDxyz=−+==−−+=,两式相减得:10x=,所以1114202yz−+=,令12z=,可得:17y=,所以1(0,7,2)n=,设面11CBBC的法向量为()2222,,nxyz=,则有2211222

20214022nCBynCBxz===−+=,解得:20y=,令21z=,解得:27x=所以2(7,0,1)n=,设锐二面角为,则有121222cos2122477nnnn===++.·······································

·········12法二:过H做HEBD⊥,连接1AE,1AH⊥面11BCCB,1AHDB⊥,则DB⊥面1AHE,1AEBD⊥,则1AEH即为所求二面角.在1RtADH△中,117,22AHAD==,则12DH=,在RtDOB中,2,2,6DOOBDB===,由RtRtDEHDOB可得

:HEDHOBDB=,36HE=,则1116AE=,11122cos2222HEAEHAE===.·························································

····································1222.解:(1)()ecosxfxax=−,···········································································

·······························1①当01a时,因为()0,x,所以cos1ax,1eex,()0fx,()fx在()0,上单调递增,没有极值点,不合题意,舍去;②当1a时,令()=()gxfx,则()esinxgxax=+,因为(

)0,x,所以()0gx,所以()fx在()0,上递增,又因为(0)10fa=−,2e02f=,所以()fx在()0,上有唯一零点1x,且10,2x,所以()10,

xx,()0fx;1,2xx,()0fx,所以()fx在()0,上有唯一极值点,符合题意.综上,(1,)+a.···································································

··························································4(2)由(1)知1a,所以,2x时,()ecos0xfxax=−,所以()10,xx,()0fx,()fx单

调递减;()1,xx,()0fx,()fx单调递增,所以()10,xx时,()(0)0fxf=,则()10fx,又因为()e10f=−,所以()fx在()1,x上有唯一零点2x,即()fx在(0,)上有唯一零点2x.··

·····································6因为()112211112esin21e2sincos1xxfxaxaxx=−−=−−,由(1)知()10fx=,所以11ecosxax=,·····················

·····································································································

····7则()112112e2esin1xxfxx=−−,构造2()e2esin1,0,2ttpttt=−−,·······································8所以()2()2e2e(sincos)2eesincostttt

pttttt=−+=−−,记()esincos,0,2ttttt=−−,则()ecossintttt=−+,显然()t在0,2上单调递增,所以()(0)0t=,········

········································································································

················9所以()t在0,2上单调递增,所以()(0)0t=,所以()0pt,所以()pt在0,2上单调递增,所以()(0)0ptp=,·········································

········································································10所以()()1220fxfx=,由前面讨论可知:112xx,12xx,且()fx在()1,xx单调递增,所以122xx.·

············································································································

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