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12021年学业水平考试数学答案一、选择题ACCDDCADBADA二、填空题13.14.15.16.三、解答题17.(1)过线段中点.....................................
............................................................1分由坐标可得.........................................
..................................................2分.................................................................
.....................................................3分则整理成一般式为:.........................................4分(2)设①.....
.................................................................................................5分②..............................
...........................................................7分由①②消可得..................................................................................
....9分则点坐标为或.......................................................................................10分18(1)第一组的频率为0.
15,第二组的频率为0.25所以第30百分位数·······················································6分(2)日销售量都不低于100个的概率为0.6,日销售量低于50的概率为0.152设
A=“在未来连续3天里,有连续2天的日销售量都不低于100个且另1天的日销售量低于50”······························································
·················································8分P(A)=2×0.6×0.6×0.15=0.10812分19.(1)设圆心,则半径为.....................................
...........................................1分圆心到直线的距离为..................................................
.....2分.......................................................................................3分则......................................
...............................4分由,解得............................................................................
................5分圆方程:或.........................................6分(2)当圆心在第一象限时,圆心,....................................
.....7分①斜率不存在时,直线方程:,圆心到直线的距离是2,是圆的切线..........9分②斜率存在时,设过点的圆的切线方程为,即..10分圆心到直线距离,解得,切线方程为.综上,过点的圆的切线方程为和............................
..........12分20.(1)设A=“从袋中一次摸出2个小球恰为异色球”.............................................................1分.........................
.................................................................4分(2)设B=“3个球中,黑球与白球的个数都没有超过红球的个数”.................
..........................5分.......................................................8分3(3)设C=“得分为8分”..........................
....................................................................................9分........................................
......................12分21.(1)·····················································································4分(2)设直线,,,,代入得:
,·····································································6分·······················································
·····································8分点到直线的距离·································································10分·····················
··················································12分22.(1)············································
·········································2分(2)由(1)得,,又直线的斜率不为零,故可设的方程为,由,代入得,由于恒成立,故可设交点,,又直线,所以,则有,,·············
··································4分4相除可得,又直线的方程为,又由,所以,···································6分所以直线的方程为,故直线恒过定点.······················
·············7分(3)设交点,,设①直线斜率存在时,代入得,,·······················································9分,得:所以,·
·················································11分②直线斜率不存在时,或5综上,···································································12分