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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)4h
Srhh==−·············································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2
ππ2T==.······························································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==−=−uuuruuuruuur
uuuruuuruuur.············1分所以2133ADABBDABAC=+=+uuuruuuruuuruuuruuur.··········2分所以111236AEADABAC==+uuuruuuruuuruuur.··············3分所以2136EBABAEABA
C=−=−uuuruuuruuuruuuruuur;·······························································4分(2)1536ECACAEABA
C=−=−+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=−uuuruuur
uuur(1,2)=−;·······························7分1536ECABAC=−+uuuruuuruuur15(0,3)(6,0)(5,1)36=−+=−.·······8分y
xEDBCAEDBAC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77130cos130526EBECEBECθ
⋅−===−×⋅uuuruuuruuuruuur.·····················································12分解法2:因为2136EBABAC=−uuuruuuru
uur,1536ECABAC=−+uuuruuuruuur.所以2115()()3636EBECABACABAC⋅=−⋅−+uuuruuuruuuruuuruuuruuur22251193618ABACABAC=−−+⋅uuuruuuruuuruuur
222511360793618=−×−×+×=−.···················································7分22221412||()369369EBABACABACABAC=−=+−⋅uuuruuuruuur
uuuruuuruuuruuur2241236059369=×+×−×=;···································9分222151255||()369369ECABACABACABAC=−+=+−⋅uuuruuu
ruuuruuuruuuruuuruuur221255360269369=×+×−×=.····························11分设ECuuur与EBuuur的夹角为θ,则77130cos130
526EBECEBECθ⋅−===−×⋅uuuruuuruuuruuur.·········12分20.(1)所有的样本点为:1234AAAA,1234AAAA,1234AAAA,1234AAAA;1234AAAA,1234AAAA,123
4AAAA,1234AAAA;1234AAAA,1234AAAA,1234AAAA,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()()()()()()P
BPAAAAPAAAAPAAAAPAAAAPAAAA=++++·································································································
······························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12G
FAB=.·················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P
AE,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P
HAE⊥,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,32AB,33AB,41AB,42AB,43AB,12BB,13
BB,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)[70
k∈,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;[90k∈,95)的频率为
0.2;[95k∈,100]的频率为0.1.·········································································9分11372.577.582.587.50.49
2.50.297.50.1201020k=×+×+×+×+×+×··································································································
···························10分87=.·············································································································12分