2020届数学高考安徽省江南十校2020届高三冲刺联考(二模)数学理 doc
发布时间:2019-07-21 08:21:14
发布时间:2019-07-21 08:21:14
2020年“江南十校”高三学生冲刺联考(二模)
理科数学
第Ⅰ卷(选择题 共60分)
一、选择题(本大题共12个小题,每小题5分,共60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)
1.已知复数满足word/media/image2.gif,则word/media/image3.gif的模为( )
A.word/media/image4.gif B.word/media/image5.gif C.word/media/image6.gif D.word/media/image7.gif
2. word/media/image8.gif为第三象限角,word/media/image9.gif,则word/media/image10.gif( )
A.word/media/image11.gif B.word/media/image12.gif C.word/media/image13.gif D.word/media/image14.gif
3.已知全集为word/media/image15.gif,集合word/media/image16.gif,word/media/image17.gif,则word/media/image18.gif( )
A.word/media/image19.gif B.word/media/image20.gif C.word/media/image21.gif D.word/media/image22.gif
4.不等式word/media/image23.gif所表示的区域为word/media/image24.gif,函数word/media/image25.gif的图象与word/media/image26.gif轴所围成的区域为word/media/image27.gif.向word/media/image24.gif内随机投一个点,则该点落到word/media/image27.gif内概率为( )
A.word/media/image28.gif B.word/media/image29.gif C.word/media/image30.gif D.word/media/image31.gif
5.直线word/media/image32.gif过抛物线word/media/image33.gif:word/media/image34.gif的焦点且与word/media/image26.gif轴垂直,则直线word/media/image32.gif与word/media/image33.gif所围成的面积等于( )
A.word/media/image36.gif B.word/media/image37.gif C.word/media/image38.gif D.word/media/image39.gif
6.一个几何体的三视图如图所示,则该几何体表面积为( )
A.word/media/image40.gif B.word/media/image41.gif C.word/media/image42.gif D.word/media/image43.gif
7.阅读如图所示程序框图,运行相应的程序.当输入的word/media/image44.gif时,则输出word/media/image45.gif的范围是( )
A.word/media/image46.gif B.word/media/image47.gif C.word/media/image48.gif D.word/media/image49.gif
8.函数word/media/image50.gif的图象沿word/media/image26.gif轴向右平移word/media/image51.gif个单位后,得到word/media/image52.gif为偶函数,则word/media/image53.gif的最小值为( )
A.word/media/image54.gif B.word/media/image55.gif C.word/media/image56.gif D.word/media/image57.gif
9.平面word/media/image58.gif内有word/media/image59.gif个点(无三点共线)到平面word/media/image60.gif的距离相等,能够推出word/media/image61.gif,三个平面将空间分成word/media/image53.gif个平面,则word/media/image62.gif的最小值为( )
A.word/media/image63.gif B.word/media/image64.gif C.word/media/image65.gif D.word/media/image66.gif
10.已知word/media/image26.gif,word/media/image45.gif满足word/media/image67.gif,word/media/image68.gif的最小值、最大值分别为word/media/image69.gif,word/media/image70.gif,且word/media/image71.gif对word/media/image72.gif上恒成立,则word/media/image73.gif的取值范围为( )
A.word/media/image74.gif B.word/media/image75.gif C.word/media/image76.gif D.word/media/image77.gif
11.向量word/media/image78.gif,word/media/image79.gif,word/media/image80.gif满足:word/media/image81.gif,word/media/image82.gif,word/media/image83.gif,则word/media/image84.gif最大值为( )
A.word/media/image85.gif B.word/media/image86.gif C.word/media/image87.gif D.word/media/image88.gif
12. word/media/image89.gif的导函数满足:当word/media/image90.gif时,word/media/image91.gif,则( )
A.word/media/image92.gif B.word/media/image93.gif
C.word/media/image94.gif D.word/media/image95.gif
第Ⅱ卷(非选择题 共90分)
二、填空题(本大题共4小题,每小题5分,共20分.把答案填在答题卡的相应位置)
13.二项式word/media/image96.gif展开式中,只有第word/media/image97.gif项的二次项系数最大,则展开式中常数项是 .
14.已知两个圆word/media/image98.gif,word/media/image99.gif与两坐标系都相切,且都过点word/media/image100.gif,则word/media/image101.gif .
15.在《九章算术》方田章圆田术(刘徽注)中指出:“割之弥细,所失弥之,割之又割,以至于不可割,则与圆周合体而无所失矣”注述中所用的割圆术是一种无限与有限转化思想.比如在word/media/image102.gif中“...”即代表无限次重复,但原数中有个定数word/media/image26.gif,这可以通过word/media/image103.gif确定出来word/media/image104.gif,类似地可得到:word/media/image105.gif .
16. word/media/image106.gif中,角word/media/image107.gif,word/media/image108.gif,word/media/image109.gif所对边分别为word/media/image69.gif,word/media/image70.gif,word/media/image110.gif.word/media/image111.gif是word/media/image112.gif边的中点,且word/media/image113.gif,word/media/image114.gif,word/media/image115.gif,则word/media/image106.gif面积为 .
三、解答题(本大题共6小题,共70分.解答应写出文字说明、证明过程或演算步骤.解答写在答题卡上的指定区域内)
17.数列word/media/image116.gif满足word/media/image117.gif.
(1)求数列word/media/image116.gif的通项公式;
(2)设word/media/image118.gif,求word/media/image119.gif的前word/media/image59.gif项和word/media/image120.gif.
18.甲乙两个班进行物理测试,其中女生word/media/image121.gif人,男生word/media/image122.gif人,从全部word/media/image123.gif人任取一人及格的概率为word/media/image124.gif,并且男生和女生不及格人数相等.
(1)完成如下word/media/image125.gif列联表
(2)根据表中数据,能否在犯错误的概率不超过word/media/image126.gif的前提下认为物理成绩及格与学生性别有关?
(3)从两个班有放回的任取word/media/image127.gif人,记抽取的word/media/image127.gif人中不及格人数为word/media/image128.gif,求word/media/image128.gif的数学期望和方差.
附:word/media/image129.gif.
19.平行六面体word/media/image140.gif中,底面word/media/image141.gif为菱形,word/media/image142.gif,word/media/image143.gif,word/media/image144.gif.
(1)证明:平面word/media/image146.gif平面word/media/image147.gif;
(2)设word/media/image148.gif与word/media/image149.gif交于word/media/image150.gif点,求二面角word/media/image151.gif平面角正弦值.
20.已知椭圆word/media/image33.gif:word/media/image152.gif,点word/media/image107.gif、word/media/image108.gif、word/media/image109.gif都在椭圆word/media/image33.gif上,word/media/image150.gif为坐标原点,word/media/image111.gif为word/media/image153.gif中点,且word/media/image154.gif.
(1)若点word/media/image109.gif的坐标为word/media/image155.gif,求直线word/media/image153.gif的方程;
(2)求证:word/media/image106.gif面积为定值.
21.设word/media/image156.gif.
(1)word/media/image157.gif在word/media/image158.gif上单调,求word/media/image69.gif的取值范围;
(2)已知word/media/image159.gif在word/media/image160.gif处取得极小值,求word/media/image69.gif的取值范围.
请考生在22、23题中任选一题作答,如果多做,则按所做的第一题记分,作答时请写清题号.
22.选修4-4:坐标系与参数方程
已知曲线word/media/image24.gif的参数方程为:word/media/image161.gif(word/media/image58.gif为参数),曲线word/media/image27.gif的极坐标方程为word/media/image162.gif.
(1)求曲线word/media/image24.gif的普通方程与曲线word/media/image27.gif的直角坐标方程;
(2)曲线word/media/image24.gif与曲线word/media/image27.gif有两个公共点,求word/media/image53.gif的取值范围.
23.选修4-5:不等式选讲
已知word/media/image163.gif.
(1)解不等式:word/media/image164.gif;
(2)不等式word/media/image165.gif对任意word/media/image166.gif恒成立,求word/media/image26.gif的范围.
2020年“江南十校”高三学生冲刺联考(二模)
理科数学参考答案
一、选择题
1-5: CBCAC 6-10: CDDCB 11、12:DC
二、填空题
13. word/media/image167.gif 14. word/media/image168.gif 15. word/media/image169.gif 16. word/media/image170.gif
三、解答题
17.解:(1)当word/media/image171.gif时,word/media/image172.gif;
当word/media/image173.gif,word/media/image174.gif word/media/image175.gif,可得word/media/image176.gif,
又∵当word/media/image171.gif时也成立,∴word/media/image176.gif;
(2)word/media/image177.gifword/media/image178.gif,
∴word/media/image179.gifword/media/image180.gif
word/media/image181.gif.
18.解:(1)
(2)由word/media/image185.gifword/media/image186.gif,犯错误概率不超过word/media/image126.gif的前提下,没有足够的证据说明物理成绩及格与性别有关;
(3)由题意可知word/media/image187.gif,∴word/media/image188.gif,∴word/media/image189.gif.
19.(1)证明:设word/media/image149.gif,word/media/image148.gif交于点word/media/image150.gif,∵底面word/media/image141.gif为菱形,∴word/media/image190.gif,又∵word/media/image144.gif,word/media/image150.gif是word/media/image148.gif的中点,∴word/media/image191.gif,word/media/image192.gif,∴word/media/image193.gif平面word/media/image194.gif,又∵word/media/image195.gif平面word/media/image147.gif,∴平面word/media/image146.gif平面word/media/image147.gif;
(2)解:∵word/media/image196.gif,word/media/image150.gif是word/media/image149.gif的中点,∴word/media/image197.gif,word/media/image198.gif,word/media/image199.gif,word/media/image200.gif两两垂直,以word/media/image199.gif,word/media/image200.gif,word/media/image198.gif分别为word/media/image26.gif,word/media/image45.gif,word/media/image3.gif轴建立空间直角坐标系如图所示,
设word/media/image201.gif,由题得word/media/image202.gif,word/media/image203.gif,word/media/image204.gif,则
word/media/image205.gif,word/media/image206.gif,word/media/image207.gif,word/media/image208.gif,
设word/media/image209.gif是平面word/media/image210.gif的一个法向量,
word/media/image211.gif,word/media/image212.gif,
word/media/image213.gif,可得word/media/image214.gif,
设word/media/image215.gif是平面word/media/image216.gif的一个法向量,
word/media/image217.gif,word/media/image218.gif word/media/image219.gif,
word/media/image220.gif,可得word/media/image221.gif,
word/media/image222.gifword/media/image223.gif,
∴二面角word/media/image151.gif平面角正弦值为word/media/image224.gif.
20.解:(1)设word/media/image226.gif,word/media/image227.gif,word/media/image228.gif,∵word/media/image154.gif,∴word/media/image229.gif,将word/media/image107.gif,word/media/image108.gif带入椭圆方程中,可得word/media/image230.gif化简可得
word/media/image231.gif,
∴word/media/image232.gifword/media/image233.gif,
∴直线word/media/image234.gif的方程为word/media/image235.gif;
(2)证明:设word/media/image236.gif,∴word/media/image237.gif,
①当直线word/media/image153.gif的斜率不存在时,word/media/image238.gif,由题意可得word/media/image239.gif,word/media/image240.gif,word/media/image241.gif或word/media/image242.gif,word/media/image243.gif,word/media/image244.gif,此时word/media/image245.gif;
②当直线word/media/image153.gif的斜率存在时,word/media/image246.gif,由(1)word/media/image247.gif,
∴word/media/image153.gif:word/media/image248.gif,即直线word/media/image153.gif:word/media/image249.gif word/media/image250.gif,
即word/media/image251.gif,word/media/image252.gif word/media/image253.gif,
∴word/media/image254.gif,word/media/image255.gif,∵word/media/image154.gif,
word/media/image256.gif
word/media/image257.gif
word/media/image258.gif,
word/media/image150.gif到word/media/image153.gif的距离word/media/image259.gif,
word/media/image260.gifword/media/image261.gif.∴word/media/image262.gif为定值.
21.解:(1)由word/media/image263.gif,
即word/media/image264.gif,word/media/image265.gif,
word/media/image266.gif,
①word/media/image267.gif在word/media/image158.gif上单调递增,∴word/media/image268.gif对word/media/image269.gif恒成立,即word/media/image270.gif对word/media/image269.gif恒成立,得word/media/image271.gif;
②word/media/image267.gif在word/media/image158.gif上单调递减,∴word/media/image272.gif对word/media/image269.gif恒成立,即word/media/image273.gif对word/media/image269.gif恒成立,得word/media/image274.gif,
由①②可得word/media/image69.gif的取值范围为word/media/image275.gif;
(2)由(1)知,
①word/media/image276.gif,word/media/image277.gif在word/media/image278.gif上单调递增,∴word/media/image279.gif时,word/media/image280.gif,word/media/image159.gif单调递减,word/media/image281.gif时,word/media/image282.gif,word/media/image159.gif单调递增,∴word/media/image159.gif在word/media/image160.gif处取得极小值,符合题意;
②word/media/image283.gif时,word/media/image284.gif,又word/media/image277.gif在word/media/image285.gif上单调递增,∴word/media/image279.gif时,word/media/image280.gif,∴word/media/image286.gif时,word/media/image282.gif,∴word/media/image159.gif在word/media/image287.gif上单调递减,word/media/image288.gif上单调递增,word/media/image159.gif在word/media/image160.gif处取得极小值,符合题意;
③word/media/image289.gif时,word/media/image290.gif,word/media/image277.gif在word/media/image287.gif上单调递增,∴word/media/image281.gif上单调递减,
∴word/media/image265.gif时,word/media/image291.gif,word/media/image159.gif单调递减,不合题意;
④word/media/image292.gif时,word/media/image293.gif,当word/media/image294.gif时,word/media/image282.gif,word/media/image159.gif单调递增,当word/media/image281.gif时,word/media/image280.gif,word/media/image159.gif单调递减,∴word/media/image159.gif在word/media/image160.gif处取得极大值,不符合题意;
综上所述,可得word/media/image295.gif.
22.解:(1)在曲线word/media/image24.gif中word/media/image296.gifword/media/image297.gifword/media/image298.gif,
∴曲线word/media/image24.gif的普通方程为word/media/image299.gif,word/media/image300.gif.
在曲线中:由可得,∴曲线的直角坐标方程为;
(2)联立word/media/image301.gifword/media/image302.gif,word/media/image300.gif有两解,
令word/media/image303.gif,在word/media/image304.gif上有两解,
∴word/media/image305.gif,
∴word/media/image306.gif.
23.解:(1)①word/media/image307.gif,
②word/media/image308.gifword/media/image309.gif,
③word/media/image310.gifword/media/image311.gif,
由①②③可得word/media/image312.gif;
(2)①当word/media/image313.gif时,word/media/image314.gif,∴word/media/image315.gif;
②当word/media/image316.gif时,即word/media/image317.gif对word/media/image53.gif恒成立,
word/media/image318.gif,当且仅当word/media/image319.gif,即word/media/image320.gif时取等号,
∴word/media/image321.gif,解得word/media/image322.gif.
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