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高一数学试卷参考答案及评分标准第1页(共4页)2020-2021学年度下学期期末质量监测高一数学试卷参考答案及评分标准一.单选题:本大题共8小题,每小题5分,共40分.在每小题给出的四个选项中,只有一项是符合题目要求的
.题号12345678答案ADCCBCAD二.多选题:本大题共4小题,每小题5分,共20分.在每小题给出的选项中,有多项符合题目要求,全部选对的得5分,有选错的得0分,部分选对得2分.题号9101112答案ACDCDBCAC三.填空题:本大题共4小题
,每小题5分,共20分.把正确答案写在答题卡相应题的横线上.13、114、22815、9445(注:第一空2分;第二空3分)16、25四.解答题:共70分,解答应写出文字说明、解答过程或演算步骤.17.
(10分)解:选①BabtansinA2,由正弦定理可得BBAABcossinsinsinsin2······················································2分0sins
in0sin,sin0BABABA,,21cosB·······················································3分3B······
·························································································································
················································5分选②BbcasincsinCsinA)(,由正弦定理可得acbca222······················
···························2分212cos222acbcaB·························································································
··················································3分30BB························································
·······························································································5分(2)acBacSABC43sin21
························································································································
···········6分acacacaccabB221cos222···············································································
···················8分243bS2max43bSca时当且仅当····························································
·····································9分23432bb····················································
······························································································10分高一数学试卷参考答案及评分标准第2页(共4页)18.(1
2分)解:(1)设事件A为甲第一轮活动中猜对1个谜语,事件B为乙第一轮活动中猜对1个谜语事件C为“星队”在第一轮活动中猜对1个谜语··················································
··················································1分则PBPAP)(43)(,PBPAP1)(41)(,····························
····························································2分则BABAC,因为:BABA与互斥,且A与B,A与B分别相互独立·······································3分所以321254
1)1(43)()()()()()()(PPPBPAPBPAPBAPBAPCPP的值为32··································································
···································································································6分(2)设事件21AA,分别为甲两轮
猜对1个,2个谜语,事件21BB,分别为乙两轮猜对1个,2个谜语.事件D为两轮活动“星队”猜对3个谜语·················································································
·······························7分则,,169)43()(8341432)(221APAP.94)32()(9431322)(221BPBP,··································
·········································································9分则,1221BABAD且21BA与12BA互斥,1A与2B,2A与1B分别相互独立,··················
···················10分所以)()()()()()()(12211221BPAPBPAPBAPBAPDP.125941699483······················11分因此,“星队”在两
轮活动中猜对3个谜语的概率是.125·················································································12分19.(12分)解:(1)证明:取BP中点F,连结
FE,FA,∵E是CP中点CBEF21//··················································································
··1分∵AD//BC,∠ABC=900,∴∠DAB=900∵AD=AB=1,∴BD=2,∠ADB=450∴∠DBC=450又∵CD=2,∴∠CDB=90°∴CB=��2+��2=2CBAD21//············································
·······························································································································3分
ADEF//∴四边形ADEF为平行四边形∴DE//AF·································································································5分又∵DE平面ABP
,AF⊂平面ABP∴DE//平面ABP·····························································································6分(
2)取BC中点O,连DO∴DO⊥CB∵平面PBC⊥平面ABCD,平面PBC∩平面ABCD=CB,∴DO⊥平面PBC∴DO为三棱锥D-EBP的高········································
····································································9分又∵BC=BP=2∴BE⊥CP∴BE=12CP=2·················································
···································································10分∴31)2221(131)21(3131EPBEDOS
DOVVEBPEBPDDBPE三棱锥三棱锥三棱锥E-DBP的体积为31···································································
········································································12分高一数学试卷参考答案及评分标准第3页(共4页)20.(
12分)解:(1)连EC,由余弦定理可得EC2=ED2+DC2-2ED·DC·cos120°=3,∴EC=3·····························2分由DC=ED,∠CDE=120°∴∠ECD=30°∵∠DCF=90°∴∠ECF=60°·····················
···········································3分在ECF中,CFECEFCFECECF2cos222∴0632CFCF∴32),(3或舍CF∴道路CF的长度为km32·
··················································································································
·······················6分(2)设∠FCA=α,在△CFA中,由正弦定理可得42332)60sin(sin0ACFACCF)60sin(40AC···········································
········································································································8分cos)60sin(4cos,//9000
ACABCABCFABABCDCAB3)602sin(232sincos23cossin2cos3202AB·······························10分0000024060260900∴当009
0602,即015时,AB取最大值为32故AB两地的最大距离为km)32(············································································
············································12分21.(12分)解:(1)估计该组数据的众数为8628884··············································
·····································································2分该组数据的频率统计为因为0.08+0.12+0.15=0.
35<0.5,0.08+0.12+0.15+0.3=0.65>0.5,所以中位数落在区间[84,88)内,设中位数为,x由,5.0)84(075.035.0x得到86x,因此中位数为86···························
··············································································································6分(2)因为[72,76)与[76,80)的党
员人数的比值为2∶3,采用分层随机抽样法抽取5人,则在[72,76)中抽2人,[76,80)中抽3人.···················································
·········7分设[72,76)抽取的2人编号为a1,a2;[76,80)抽取的3人编号b1,b2,b3;则5人中任选2人进行问卷调查对应的样本空间Ω={(a1,a2),(a1,b1),(a1,b2),(a1,b3),(a2,b1),(a2,b2),(a2,
b3),(b1,b2),(b1,b3),(b2,b3)},共有10个样本点····················································································
··············································································9分至少有1人成绩低于76分的有:(a1,a2),(a1,b1),(a1,b2),
(a1,b3),(a2,b1),(a2,b2),(a2,b3)共有7个样本点所以这2人中至少有1人成绩低于76分的概率为7.0107···································································
···············12分区间[72,76)[76,80)[80,84)[84,88)[88,92)[92,96)[96,100]频率0.080.120.150.30.160.10.09高一数学试卷参考答案及评分标准第4页(共4页)22.(12分)解:(1)取PA中点G,AC中点O,连
接GO,GB,OB在PACRt中,APOGCPOGAPCP//,,在等边PAB中,APBGOGB为二面角BPAC的平面角····················································
······················································3分AB32321CPOG23BG,26OB,90222GOBGBOBOG33cosGBO
GOGB二面角BPAC的余弦值为33··················································································································6分(2)过点N作B
MNQ//交CP于点NQANBMANQ,3CMCPCM在MCB中,33360cos22222MCBCMCBCBM)1(32BM设yPBPNMBQ
NyBPBN1,)1(3)1(,32yQNyBN···········································8分在NBA中,yyBNABBNABAN33360cos22222)1
(32yyAN)1)(1(3)1(1yPMyPQyPMPQ····························································
·······················10分在PAQRt中,22222)1()1(33yPQAPAQ①在ANQRt中,)1()1(3)1(3222222yyyNQ
ANAQ②由①=②,得yy1]5241[]3231[,,yBPBN的取值范围为]5241[,·············································································
························································12分(其他解法,请酌情给分)