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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BBAABcossinsin
sinsin2······················································2分0sinsin0sin,sin0BABABA,,21cos
B·······················································3分3B·····································································
··········································································································5分选②Bbc
asincsinCsinA)(,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.(12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3212541)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,∴∠DA
B=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(3131EPBEDOSDOVVEBPEB
PDDBPE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∴0632
CFCF∴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,//9000ACABCABCFABABCDCAB3)602sin(232sincos23cossin2cos3202
AB·······························10分0000024060260900∴当0090602,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8
6x,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),(a
2,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,
90222GOBGBOBOG33cosGBOGOGB二面角BPAC的余弦值为33······················································
····························································6分(2)过点N作BMNQ//交CP于点NQANBMANQ,3CMCPCM在MCB
中,33360cos22222MCBCMCBCBM)1(32BM设yPBPNMBQNyBPBN1,)1(3)1(,32yQNyBN···········································8分在N
BA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(3222222yyyNQANAQ②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)