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高一数学参考答案第1页(共5页)ABCDEFGH2020-2021学年度第二学期期末学业水平诊断高一数学参考答案一、选择题CACDBCDA二、选择题9.AC10.BCD11.ABD12.ACD三、填空题13.0.941
4.366615.2222ABCACDADBBCDSSSS∆∆∆∆++=16.283π四、解答题17.证明:(1)连接BD,因为,EF分别为棱,ABAD的中点,所以11//,22EFBDEFBD=.·································2分同理11//
,22HGBDHGBD=.································4分所以//EFGH且EFGH=.··································5分所以四边形EFGH是平行四边
形.····························6分(2)当ACBD⊥且ACBD=时,四边形EFGH为正方形.·····················10分18.解:(1)设事件,,,ABCD分别表示“被评为等级,,,ABCD”.由
题意,事件,,,ABCD两两互斥,所以3131()1483232PD=−−−=.·························································2分又CD=“延迟送达且
被罚款”,所以1()()()8PCDPCPD=+=.······················································4分因此“延迟送达且被罚款”的概率为18.······
···········································5分高一数学参考答案第2页(共5页)(2)设事件,,,iiiiABCD表示“第i单被评为等级,,,ABCD”,1,2i=.则“两单共获得的奖励为0元”即事件221221()()()AB
ACAC,·············7分且事件221221,,ABACAC互斥,又22111()8864PAB=×=.····································································8分又
1221339()()432128PACPAC==×=.··················································9分所以221221[()()()]PPABA
CAC=221221()()()PABPACPAC=++··············································10分1133528843232=×+××=.··················
·······································12分19.解:(1)由题意,435761179869440y×+×+×+×+×+×=··································
·········2分6.5=.··························································································3分22221[3(
46.5)7(56.5)11(66.5)40ys=×−+×−+×−2229(76.5)6(86.5)4(96.5)]+×−+×−+×−·······························5分1.95=.····································
····················································6分(2)由(1)可得,1(5040)90zxy=+·······························
·······················8分1(507.4406.5)790=×+×=.·····································9分2225040[2.6(7.47)][1.95(6.5
7)]9090zs=+−+×+−·································11分11345=.·············································
·····································12分高一数学参考答案第3页(共5页)20.解(1)证明:因为1ABAA⊥,ABAC⊥,1ACAAA=所以AB⊥平面11ACCA,·············
····················································2分1AC⊂平面11ACCA,所以1ABAC⊥.···············································3分又因为直三
棱柱111ABCABC−中,1AAAC=,所以四边形11ACCA为正方形,所以11ACAC⊥.···································4分因为1ACABA=,所以1AC⊥平面1ABC,···············
························5分1BC⊂平面1ABC,所以11ACBC⊥.···················································6分(2)过1A作111ADBC⊥,垂足为D,连CD,则1AD⊥平面11BCCB,1ACD∠为1AC与
平面11BCCB所成的角.···················8分因为11AAAC==,则12AC=,所以11112sin42ADADACDACa∠===,所以112AD=.··················
········9分在11RtACD∆中,111111sin2ADACDAC∠==,所以1130ACD∠=.在111RtABC∆中,11113tan303ABAC==.······································10分所以1111331132318AA
BCBAACVV−−==××××=.·································12分21.解:(1)由已知()0.0040.0060.020.030.024101m+++++×=,···
··········2分解得0.016m=.·············································································3分(2)
测试成绩的平均数450.04550.06650.2750.3850.24950.16x=×+×+×+×+×+×·········4分76.2=.··································
···················································5分测试成绩落在区间[40,70)的频率为()0.0040.0060.02100.3++×=,落在在区间[4
0,80)的频率为()0.0040.0060.020.03100.6+++×=,所以设第57百分位数为a,有()0.3700.030.57a+−×=,··················6分解得79a=.·
·················································································7分DCBAC1B1A1高一数学参考答案第4页(共5页)FGMEDCB
ANNABCDEM(3)由题知,测试分数位于区间[80,90)、[90,100)的人数之比为0.2430.162=,所以采用分层随机抽样确定的5人,在区间[80,90)中3人,用123,,AAA表示,在区间[90,100)中2人,用12,BB表示.········
············································8分从这5人中抽取2人的所有可能情况有:12(,)AA,13(,)AA,11(,)AB,12(,)AB,23(,)AA,21(,)AB,22(,)AB,31(,)AB,32(
,)AB,12(,)BB,共10种.······10分其中“落在区间[80,90)和[90,100)”有6种.···········································11分所以3()5PA=.··
·················································································12分22.解:(1)证明:取AE中点N,连MN,则//MNDE,12
MNDE=.······1分又因为//BCDE,12BCDE=,所以//BCMN,BCMN=.···································2分所以四边形BCMN为平行四边形,所
以//BNCM.····3分又因为BN⊂平面ABE,CM⊄平面ABE,所以//CM平面ABE.···········································5分(2)延长,EBDC交于点F,则AF为平面ABE与平面ACD的交线.·············6分因为/
/BCDE,12BCDE=,所以2BFBE==.三角形ADE中,因为2ADDEAE===,N为AE的中点,所以DNAE⊥,又因为DEBE⊥,AEBE⊥,DEAEE=所以BE⊥平面ADE,DN⊂平面ADE,所以DNBE⊥.又因为BEAEE=,所以D
N⊥平面ABE.AF⊂平面ABE,所以DNAF⊥.························································8分在三角形AEF中,过N作NGAF⊥,垂足为G,连接DG,因为DNN
GN=.所以AF⊥平面DNG,DG⊂平面DNG,所以AFDG⊥.所以DGN∠为二面角DAFE−−的平面角.············································9分高一数学参考答案第5
页(共5页)在RTAEF∆中,2,4AEEF==,41625AF=+=,由AEFAGH∆∆,所以AFANAEGN=,255NG=.···································10分在RTDNG∆中,253,5
DNNG==,22955DGDNNG=+=,所以219cos19NGDGNDG∠==.···························································11分即
面ABE与面ACD所成二面角(锐角)的余弦值为21919.······················12分