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高一数学答案第1页共6页2019~2020学年度第二学期质量检测高一数学答案及评分标准2020.7一、单项选择题:本大题共8小题,每小题5分,共40分.1~4ADCB5~8DCBD二、多项选择题:本大题共4小题,每小题5分,共20分.全部选对的得5分,部分选对的得3分,有选错的得0分.
9.ACD10.ACD11.BC12.BCD三、填空题:本题共4小题,每小题5分,共20分.13.{|(21)π,}xxkk≠+∈Z14.①②③15.0或316.0.65四、解答题:本题共6小题,共70分.解答应写出文字说明、证明过程或演算
步骤.17.(本题满分10分)解:(1)圆锥SO的母线长22345l=+=.······························································1分圆锥SO的侧面积π3515πS=××=侧.·
·······················································3分圆锥SO的体积21(π3)412π3V=×××=.···············································
·····5分(2)设圆锥SO的内接圆柱OO′的底面半径为r,则hRrHR−=,即343hr−=.解得334hr=−.········································7分内接圆柱OO′的侧面积32π2π
(3)4hSrhh==−·············································8分侧面积2233(4)[4(2)]22Shhhππ=−+=−−.当2h=时,S有最大值6π,即圆柱侧面积的最大值为6π.···············
···10分18.(1)2ππ()3sincos(sincos)(sincoscossin)244fxxxxxxx=++−················1分31sin2cos222xx=−·······················
·····················································3分πsin(2)6x=−················································
·········································4分所以()fx的最小正周期2ππ2T==.··············································
················5分BOASO'C高一数学答案第2页共6页令πππ2π22π()262kxkk−−+∈Z��,解得ππππ()63kxkk−+∈Z��.所以()fx的单调递增区间是ππ[π,π]()63kkk−+∈Z.··
·······························6分(2)由(1)及题设条件知π1()sin()263AfA=−=.因为π02A<<,所以πππ663A−<−<.····················
·········································7分从而22ππ122cos()1sin()1()6633AA−=−−=−=.·······················
··············8分所以ππcoscos[()]66AA=−+··············································································10分ππππcos()cossin()sin6666AA=−−−·
·················································11分2231126132326−=×−×=.···············································
··········12分19.(本小题满分12分)解:(1)1111()3333BDBCACABACAB==−=−uuuruuuruuuruuuruuuruuur.············1分所以2133ADABBDABAC=+=
+uuuruuuruuuruuuruuur.··········2分所以111236AEADABAC==+uuuruuuruuuruuur.··············3分所以2136EBABAEABAC=−=−uu
uruuuruuuruuuruuur;·······························································4分(2)1536ECACAEABAC=−=−+uuuruuuruuuruuuruuur.·····················
··················································5分解法1:如图,以ACuuur,ABuuur所在方向分别为x轴,y轴的正方向,建立平面直角坐标系xAy,则(0,3)B,(6,0)C.·······························
·························6分则2136EBABAC=−uuuruuuruuur(1,2)=−;·······························7分1536ECABAC=−+uuuruuuruuur15
(0,3)(6,0)(5,1)36=−+=−.·······8分yxEDBCAEDBAC高一数学答案第3页共6页所以·(-1)52(-1)-7EBEC=×+×=uuuruuur,····················································
················9分22(1)25EB=−+=uuur,225(1)26EC=+−=uuur.······································11分设ECuuur与EBuuur的夹角为θ,则77130cos130526EB
ECEBECθ⋅−===−×⋅uuuruuuruuuruuur.·····················································12分解法2:因为2136EBABAC=−uuuruuuruuur,1536ECABAC=−+uu
uruuuruuur.所以2115()()3636EBECABACABAC⋅=−⋅−+uuuruuuruuuruuuruuuruuur22251193618ABACABAC=−−+⋅uuuruuuruuuruuur222511360793618
=−×−×+×=−.···················································7分22221412||()369369EBABACABACABAC=−=+−⋅uuuruuuruuuruu
uruuuruuuruuur2241236059369=×+×−×=;···································9分222151255||()369369ECABACABACABAC=−+=+−
⋅uuuruuuruuuruuuruuuruuuruuur221255360269369=×+×−×=.····························11分设ECuuur与EBuuur的夹角为θ,则77130cos130526EBECEBECθ⋅−
===−×⋅uuuruuuruuuruuur.·········12分20.(1)所有的样本点为:1234AAAA,1234AAAA,1234AAAA,1234AAAA;1234AAAA,1234AAAA,1234AAAA,1234AAAA;1234AAAA,1234AAAA,1234A
AAA,1234AAAA;1234AAAA,1234AAAA,1234AAAA,1234AAAA.·······································4分注:少一个样本点不得分.EDBAC高一数学答案第4页共6页(2)用B表示事件“李明通过面
试”,则李明恰好答对3道题或答对4道题.所以12341234123412341234()()()()()BAAAAAAAAAAAAAAAAAAAA=UUUU.·················································
··············································································6分由互斥事件的概率加法公式与互为独立事件的概率乘法公式,得12341234123412341234()()()()()()PBPA
AAAPAAAAPAAAAPAAAAPAAAA=++++··········································································································
·····················8分1114113112111111111123452345234523452345=×××+×××+×××+×××+×××·······························································
····················································10分111111(43211)2345120=××××++++=.················································12分注
:(2)严格按步骤评分.21.(本小题满分12分)证明:(1)如图,设G为PA的中点,连接GE,GF.又F为PB的中点,所以GF为PAB△的中位线.所以GFABP,且12GFAB=.·················1分矩形ABCD中,点E是CD的中点
,故ECABP,且12ECAB=.··········································································2分所以GFECP,且GFEC
=.所以四边形GFCE为平行四边形.···································································3分所以CFEGP.····························
·····································································4分又CF⊄平面PAE,EG⊂平面PAE,所以CFP平面PAE.··········
···········6分GFPECAB高一数学答案第5页共6页(2)设H为AE的中点,连接PH,BH.在矩形ABCD中,24ABAD==,点E是CD的中点,故2PAPE==,90APE∠=°,故PHAE⊥,且2PH=.····················8分在ABH△中,2AH
=,4AB=,45HAB∠=°,由余弦定理,得2222cos45BHABAHABAH=+−⋅⋅°2162242102=+−×××=.······························································9分在
PHB△中,23PB=,2PH=,10BH=,故222PBPHBH=+,故PHBH⊥.··························································10分又PHAE⊥,BHAEH=I,BH⊂平面ABCE,AE⊂平面ABCE,故PH⊥
平面ABCE.······················································································11分又PH⊂平面PAE,故平面PAE⊥平面ABCE.·····
···································12分22.(本小题满分12分)解:(1)根据题意,[70k∈,100),按组距为5可分成6个小区间,分别是[70,75),[75,80),[80,85),[85,90),[90,95),[95,100)
.因为70100k<�,由55(1)nkn<+�,*n∈N,·················································1分所以14n=,15,16,17,18,19.····································
·································2分每个小区间对应的频率值分别是20339,14,15,16,56052,17,18,19.nnnYan−−�=�=��⋅=�所以3(14151639)5(8+4+2)160a×++
−+=.························································3分解得1100a=.·········································································
·····························4分FPHECBA高一数学答案第6页共6页(2)由(1)中的数据,得[85k∈,90)的频率为20171250.4100−××=;[90k∈,95)的频率为20181250.2100−××=;[95k∈,100]的频
率为20191250.1100−××=.利用按比例分配分层随机抽样抽取的7件产品中,[85k∈,90)的有4件,分别记作1A,2A,3A,4A;[90k∈,100)的有3件,分别记作1B,2B,3B.
················6分从抽取的7件产品中任取2件产品,则样本空间{Ω=12AA,13AA,14AA,11AB,12AB,13AB,23AA,24AA,21AB,22AB,23AB,34AA,31AB,3
2AB,33AB,41AB,42AB,43AB,12BB,13BB,23}BB,所以()21nΩ=.·························································7分事件A
=“随机抽取的2件产品中至少有一件A级品”,则{A=11AB,12AB,13AB,21AB,22AB,23AB,31AB,32AB,33AB,41AB,42AB,43AB,12BB,13BB,23}BB,所以()15nA=.由古典概型公式,得()155()()217nPAnAΩ===.······
··························8分(3)[70k∈,75)的频率为3143916020×−=;[75k∈,80)的频率为3153916010×−=;[80k∈,85)的频率为3163936020×−=;[85k∈,90)的频率为0.4;[90
k∈,95)的频率为0.2;[95k∈,100]的频率为0.1.·········································································9分11372.577.582.587.50.492.50.
297.50.1201020k=×+×+×+×+×+×·························································································
····································10分87=.·································································································
············12分