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2023~2024学年度第一学期期中学业水平诊断高三数学参考答案一、选择题:1.B2.A3.D4.A5.D6.A7.C8.C二、选择题9.ACD10.AB11.BC12.BCD三、填空题13.2614.815.7416.(5,4)−−四、解答题17.解:(1)由题
知,π22T=,所以,2ππT==,所以,2=.·················2分所以,π()2sin(2)4fxx=+.···············································
············3分所以,πππ2π22242kxk−++,即3ππππ88kxk−+,·················4分故()fx的单调递增区间为3ππ[π,π]()88kkk−+
Z.································5分(2)将函数()fx图像上所有点横坐标伸长到原来的2倍(纵坐标不变),得π2sin()4yx=+,再向右平移π4个单位长度,得()2singxx=.···············6分所以()2sin(sincos)hxx
xx=+2π22sin2sincossin(2)42xxxx=+=−+,···································8分因为π02x,ππ3π2444x−−,所以ππ242x−=,3π8x=时,()hx取得最大值为212+.················
·········································································10分18.解:(1)当1n=时,2112aa=,则10a=或12a=,因为11a,所以12a=
;································································2分当2n时,22112122nnnnSanSan−−=+−=+−,两式相减得,22121nnnaaa−=−+,即221
(1)nnaa−=−,因为1na,所以11nnaa−=−,即11nnaa−−=,·····4分故数列{}na是以2为首项,1为公差的等差数列.···································5分(2)由
(1)知,2(1)11nann=+−=+,所以12,1,(2)nnnbnnn+=+为奇数为偶数,·····················································
······7分21232nnTbbbb=++++1321242()()nnbbbbbb−=+++++++242111(222)()24462(22)nnn=++++++++4(14)1111111[
()()()]1422446222nnn−=+−+−++−−+·······················10分所以,1244344nnnTn+−=++.····························
··························12分19.解:(1)由题知,每年的追加投入是以80为首项,14155−=为公比的等比数列,所以,41()4580400400()4515nnna−==−−;··································
············3分同理,每年牧草收入是以60为首项,15144+=为公比的等比数列,所以,51()5460240()2405414nnnb−==−−.················································6分(2)设至少经
过n年,牧草总收入超过追加总投入,即0nnba−,即5454240()240(400400())240()400()64004545nnnn−−−=+−,············8分令4()(01)5nt
t=,则上式化为2404006400tt+−,即25830tt−+,··············································································
·9分解得305t,即43()55n,所以,43lglg55n,即3lglg3lg5lg3lg2152.24lg4lg53lg21lg5n−+−==−−,所以3n.·························11分所以,至少经过3年,牧草
总收入超过追加总投入.····································12分20.解:若选①:(1)由正弦定理得,3sinsin3sincosBCAC=+,···············1分因为sinsin()BAC=+,所以3sin()sin3
sincosACCAC+=+,即3cossinsinACC=,又因为(0,π)C,sin0C,·······················3分所以1cos3A=.·······························
···············································4分(2)在ABC中,1cos3A=,则22sin3A=,sinsin()sincoscossin221sinsinsin3tan3bBACACACcCCCC++===
=+.···········6分因为ABC是锐角三角形,所以π02π02BC,即π2π02ACC+,即ππ022CA−,所以πsin()πcos22tantan()π2sin4cos()2AAC
AAA−−===−,所以1022tanC,···································································7分所以1(,3)3bc.·································
·············································8分设btc=,则221122222bcbctbccbt+=+=+,令122tyt=+,1(,3)3t,则222111222tytt−=−=,令0y=,则1t=,则y在1(,1)3上单调
递减,在(1,3)上单调递增,··································10分所以1151223tt+,即222bcbc+的取值范围为5[1,)3.··························12分若选②:(1)因
为2222()Sabc=−−,所以22()220bcaS−−+=,所以22222sin0bcabcbcA+−−+=,···········································1分所以2cos22sin0bcA
bcbcA−+=,所以sin22cosAA=−.·····························································3分又22sincos1AA+=,解得1cos3A=或cos1A=(舍),所以1cos3A=
.··············································································4分(2)在ABC中,1cos3A=,则22s
in3A=,sinsin()sincoscossin221sinsinsin3tan3bBACACACcCCCC++====+,··········6分因为ABC是锐角三角形,所以π02π02BC,即π2π02ACC+,即ππ022CA−,所以πs
in()πcos22tantan()π2sin4cos()2AACAAA−−===−,所以1022tanC,···································································7分所以1
(,3)3bc.··············································································8分设btc=,则221122222bcbctbccbt+=+=+
,令122tyt=+,1(,3)3t,则222111222tytt−=−=,令0y=,则1t=,则y在1(,1)3上单调递减,在(1,3)上单调递增,··································10分所以1151223tt+,即222bcb
c+的取值范围为5[1,)3.··························12分若选③:(1)由正弦定理得,coscos()42cossinaAaBCbAC+−=,····1分因为coscos()ABC=−+,所以cos()cos()42cossinaBCaBCbAC
−++−=,所以2sinsin42cossinaBCbAC=,所以2sinsinsin42sincossinABCBAC=.··································3分又因为,(0,π)BC,sin0,sin0BC,所
以sin22cos0AA=,又22sincos1AA+=,解得1cos3A=.··············································4分(2)在ABC中,1cos3A=,则22sin3A
=,sinsin()sincoscossin221sinsinsin3tan3bBACACACcCCCC++====+,··········6分因为ABC是锐角三角形,所以π02π02BC,即π2π02ACC+,即ππ022CA−,所以πsin()
πcos22tantan()π2sin4cos()2AACAAA−−===−,所以1022tanC,···············································
····················7分所以1(,3)3bc.··············································································8分设btc=,则221122222bcbctbccbt+
=+=+,令122tyt=+,1(,3)3t,则222111222tytt−=−=,令0y=,则1t=,则y在1(,1)3上单调递减,在(1,3)上单调递增,························
··········10分所以1151223tt+,即222bcbc+的取值范围为5[1,)3.··························12分21.解:(1)由题知,()(1)(e)xfxxa=+−,···
·········································1分所以,当0a时,e0xa−恒成立,所以,令()0fx=,解得1x=−.所以,当(,1)x−−时,()0fx,()fx在(,1)−−上单调递减;当(1,)x−
+时,()0fx,()fx在(1,)−+上单调递增;·················3分当0a时,令()0fx=,解得1x=−或lnxa=,所以,当ln1a−,即1ea时,(1,ln)xa−时,()0fx,()fx在(1,ln)a−上单调递减,当(,
1)(ln,)xa−−+时,()0fx,()fx在(,1)−−和(ln,)a+上单调递增;·························································
·······························4分当ln1a−,即10ea时,(ln,1)xa−时,()0fx,()fx在(ln,1)a−上单调递减,当(,ln)(1,)xa−−+时
,()0fx,()fx在(,ln)a−和(1,)−+上单调递增;···························································································5分当ln1a=−时,()0f
x在(,)−+上恒成立,所以,()fx在(,)−+上单调递增.·····················································6分(2)由(1)知,当1a时,()fx在(1,ln)a−上单调递减,在(,1)−−和(ln,)a+
上单调递增,且当x→−时,()fx→−,当x→+时,()fx→+,所以,若方程()fxb=始终有三个不相等的实根,则(ln)(1)fabf−,即21(ln)22eaaab−−在(1,)a+上恒成立.··············
··························8分当1a时,显然1112e2ea−−.···························································9分令2()(ln)2a
gaa=−,则21()(ln)ln2gaaa=−−,因为1a,所以,ln0a,所以,21()(ln)ln02gaaa=−−恒成立,所以,()ga在(1,)+上单调递减,所以,()(1)0gag=
.··············································································11分综上,若方程()fxb=始终有三个不相等的实根,b的取值范围为1102eb−.·················
···········································12分22.解:(1)由题得,2ln(),(0)xxafxxx−−=,································
···1分令()ln,(0)hxxxax=−−,则函数()fx有两个极值点,即方程()0hx=有两个正实数根.············································································
···················2分因为11()1xhxxx−=−=,所以当(0,1)x时,()0hx,()hx单调递减,当(1,)x+时,()0hx,()hx单调递增,所以,min()(1)1hxha==−,且当0x→时,()hx→+,x→+
时,()hx→+.······································································4分所以,方程()0hx=有两个正实数根,只需(1)10ha=−,解得1
a,···················································································5分即函数()fx有两个极值点时,a的范围为(1,)+.·
·······························6分(2)若12xx且123xx,则令21(1,3]xtx=,由(1)知,12()()0hxhx==,即1122lnlnaxxxx=−=−,则2211lnlnxxxtx−==,即11lntxxt−
=,解得,1ln1txt=−,所以,21ln1ttxtxt==−.···················8分所以,1212(1)lnlnln21lnln11tttttttxxaxxt=++++=−=−+−,··
···············9分令(1)ln,(1,1(3)]ttttt+−=,则()2211(ln)(1)(1)ln2ln(1)()1tttttttttttt++−−+−+−=−−=,········
·················10分令12ln()tPttt−+−=,则22221(1)10,(1,)3(]tttttPt−=−++=所以函数()Pt在(1,3]上单调递增,且(1)0P=,所以,()0
Pt,··········11分所以,当(1,3]t时,()0t,所以,()t在(1,3]上单调递增,所以,当3t=时,max()(3)2ln3t==.获得更多资源请扫码加入享学资源网微信公众号www.xiangxue100.
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