河北省石家庄实验中学2019-2020学年高二上学期第一次月考数学试题
发布时间:2020-11-18 17:27:48
发布时间:2020-11-18 17:27:48
2019-2020学年高二上学期第一次月考
数学试卷
(说明:本试卷满分150分,考试时间120分钟,学生答题时不可使用计算器.)
一.选择题:(本大题共12小题,每小题5分,满分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.数列word/media/image2_1.png的一个通项公式是 ( )
A. word/media/image3_1.png B. word/media/image4_1.png C. word/media/image5_1.png D. word/media/image6_1.png
2.下列双曲线中,渐近线方程为word/media/image7_1.png的是( )
A.word/media/image8_1.png B.word/media/image9_1.png
C.word/media/image10_1.png D.word/media/image11_1.png
3.已知双曲线word/media/image12_1.png的一条渐近线方程是y=word/media/image13_1.png,它的一个焦点是,则双曲线的方程为( )
A.word/media/image15_1.png B. word/media/image16_1.png word/media/image17_1.pngC.word/media/image18_1.png D.word/media/image19_1.png
4.已知椭圆E的中心为坐标原点,离心率为word/media/image20_1.png,E的右焦点与抛物线word/media/image21_1.png的焦点重合,word/media/image22_1.png是C的准线与E的两个交点,则word/media/image23_1.png ( )
A. word/media/image24_1.png B.word/media/image25_1.png C. word/media/image26_1.png D.word/media/image27_1.png
5.已知word/media/image28_1.png,则word/media/image29_1.png的最小值为( )
A. 4 B. 16 C. 8 D. 10
6、已知直线word/media/image30_1.png和平面word/media/image31_1.png,则下列四个命题正确的是( )
A. 若word/media/image32_1.png, word/media/image33_1.png,则word/media/image34_1.png B. 若word/media/image35_1.png, word/media/image36_1.png,则word/media/image37_1.png
C. 若word/media/image38_1.png, word/media/image39_1.png,则word/media/image40_1.png D. 若word/media/image41_1.png, word/media/image42_1.png,则word/media/image43_1.png
7、直三棱柱ABC﹣A1B1C1中,若∠BAC=90°,AB=AC=AA1,则异面直线 BA1与AC1所成的角为( )
A.60° B.90° C.120° D.150°
8、直线word/media/image45_1.png恒经过定点( )
A. word/media/image46_1.png B. word/media/image47_1.png C. word/media/image48_1.png D. word/media/image49_1.png
9.设word/media/image50_1.png是等差数列word/media/image51_1.png的前word/media/image52_1.png项和,若word/media/image53_1.png,则word/media/image54_1.png ( )
A. word/media/image55_1.png B. word/media/image56_1.png1 C. word/media/image57_1.png D. word/media/image58_1.png
10、若点到直线的距离为,则的值为( )
A. B. C.或 D.或
11、在三棱锥中,,,.的中点为, 的余弦值为,若都在同一球面上,则该球的表面积为( )
A. B. C. D.
12、若直线与圆有两个不同交点,则点与圆的位置关系是( ) A.点在圆上 B.点在圆内 C.点在圆外 D.不能确定
二.填空题:(本大题共4小题,每小题5分,共20分).
13.若数列word/media/image82_1.png的通项满足word/media/image83_1.png,那么word/media/image84_1.png是这个数列的第__________项.
14、一个圆柱和一个圆锥的底面直径和它们的高都与某一个球的直径相等,这时圆柱、圆锥、球的体积之比为 .
15、某几何体的三视图如图所示,则该几何体的体积是________.
16.数列word/media/image87_1.png满足 word/media/image88_1.png,写出数列word/media/image89_1.png的通项公式__________.
三、解答题:(第17题10分,其余各题12分,解答应写出文字、符号 说明,证明过程或演算步骤.)
17.设word/media/image90_1.png是等差数列,word/media/image91_1.png,且word/media/image92_1.png成等比数列.
(1)求word/media/image90_1.png的通项公式
(2)求数列word/media/image90_1.png的前word/media/image93_1.png项和word/media/image94_1.png
18、(本小题满分12分)如图,在四棱锥中,底面是矩形,底面,是的中点.已知,,.
求: (1).三角形的面积;(2).异面直线与所成的角的大小.
19.某工厂生产的某种产品,当年产量在word/media/image107_1.png吨至word/media/image108_1.png吨之间时,年生产总成本word/media/image109_1.png(万元)与年产量word/media/image110_1.png(吨)之间的关系可近似地表示成word/media/image111_1.png,问年产量为多少时,每吨的平均成本最低?并求出该最低成本.
20.已知公差不为零的等差数列word/media/image112_1.png中,word/media/image113_1.png,且word/media/image114_1.png成等比数列.
(1).求数列word/media/image112_1.png的通项公式;
(2).令word/media/image115_1.png,求数列word/media/image116_1.png的前n项和word/media/image117_1.png.
21.(本小题满分12分)
word/media/image118_1.png如图,在四棱锥word/media/image119_1.png中,底面word/media/image120_1.png是平行四边形,word/media/image121_1.png平面word/media/image122_1.png,word/media/image123_1.png,word/media/image124_1.png,word/media/image125_1.png是棱word/media/image126_1.png上的一点.
(1)证明:word/media/image127_1.png平面word/media/image128_1.png;
(2)若word/media/image129_1.png∥平面word/media/image130_1.png,求word/media/image131_1.png的值;
(3)在(2)的条件下,三棱锥word/media/image132_1.png的体积是word/media/image133_1.png,
求word/media/image134_1.png点到平面word/media/image135_1.png的距离.
22.(本小题满分12分)
在直角坐标系word/media/image136_1.png中,曲线word/media/image137_1.png与word/media/image138_1.png轴交于word/media/image139_1.png,word/media/image140_1.png两点,点word/media/image141_1.png的坐标为word/media/image142_1.png.当word/media/image143_1.png变化时,解答下列问题:
(1)能否出现word/media/image144_1.png的情况?说明理由;
(2)证明过word/media/image145_1.png,word/media/image146_1.png,word/media/image147_1.png三点的圆在word/media/image148_1.png轴上截得的弦长为定值.
参考答案
1-5 BADCC 6-10 CACAD 11-12 AC
13.答案:5解析:由word/media/image149_1.png可知, word/media/image150_1.png,令word/media/image151_1.png,得word/media/image152_1.png.
14、3:1:2; 15 word/media/image153_1.png-16 16.答案:word/media/image154_1.png解析:因为word/media/image155_1.png,所以word/media/image156_1.png,两式相减得word/media/image157_1.png,即word/media/image158_1.png,又word/media/image159_1.png,所以word/media/image160_1.png,因此word/media/image161_1.png
17. 解:(1)因为word/media/image162_1.png,且word/media/image163_1.png成等比例,
所以word/media/image164_1.png,解得word/media/image165_1.png.
所以word/media/image166_1.png.
(2)因为word/media/image167_1.png,所以word/media/image168_1.png.
18. (1).因为底面,所以,又,所以平面,从而.因为,,所以三角形的面积为.(2.)方法一:取的中点,连接,,则,从而(或其补角)是异面直线与所成的角.在中,由,,知是等腰直角三角形,,所以.因此,异面直线与所成的角的大小是.
19..答案:年产量为200吨时,每吨的平均成本最低,最低为10万元.
设每吨的平均成本word/media/image191_1.png(万元/t),则word/media/image192_1.png,
当且仅当word/media/image193_1.png,word/media/image194_1.png(t)的每吨平均成本最低,且最低成本为10万元.
20.答案:
(1).由题意:word/media/image195_1.png 化简得word/media/image196_1.png,因为数列word/media/image112_1.png的公差不为零,
word/media/image197_1.png,故数列word/media/image112_1.png的通项公式为word/media/image198_1.png.
(2).由1知word/media/image199_1.png,
故数列word/media/image116_1.png的前n项和word/media/image200_1.png.
21.【解析】:(1)由已知条件可知word/media/image201_1.png,所以word/media/image202_1.png.
因为word/media/image203_1.png平面word/media/image204_1.png,所以word/media/image205_1.png.
又因为word/media/image206_1.png,所以word/media/image207_1.png平面word/media/image208_1.png.
因为word/media/image209_1.png是平行四边形,所以word/media/image210_1.png∥word/media/image211_1.png,
所以word/media/image212_1.png平面word/media/image213_1.png. …………………4分
(2)连接word/media/image214_1.png交word/media/image215_1.png于word/media/image216_1.png,连接word/media/image217_1.png,
则word/media/image218_1.png是平面word/media/image219_1.png与平面word/media/image220_1.png的交线.
因为word/media/image221_1.png∥平面word/media/image222_1.png,所以word/media/image223_1.png∥word/media/image224_1.png.
又因为word/media/image225_1.png是word/media/image226_1.png中点,所以word/media/image227_1.png是word/media/image228_1.png的中点,
所以word/media/image229_1.png. ……………………………………8分
(3)由(1)(2)知点word/media/image230_1.png到平面word/media/image231_1.png的距离等于word/media/image232_1.png,所以word/media/image233_1.png,
所以word/media/image234_1.png,即word/media/image235_1.png. ……………………10分
因为word/media/image236_1.png平面word/media/image237_1.png,所以word/media/image238_1.png,word/media/image239_1.png,又因为word/media/image240_1.png,word/media/image241_1.png,
所以word/media/image242_1.png,word/media/image243_1.png,所以word/media/image244_1.png是等边三角形,则word/media/image245_1.png.
设word/media/image246_1.png点到平面word/media/image247_1.png的距离为word/media/image248_1.png,因为word/media/image249_1.png,
所以word/media/image250_1.png,解得word/media/image251_1.png.
所以word/media/image252_1.png点到平面word/media/image253_1.png的距离为word/media/image254_1.png. ……………12分
22.【解析】:(1)不能出现word/media/image144_1.png的情况,理由如下:
设word/media/image255_1.png,word/media/image256_1.png,则,word/media/image258_1.png满足word/media/image259_1.png,所以word/media/image260_1.png.
又word/media/image141_1.png的坐标为word/media/image261_1.png,故word/media/image262_1.png的斜率与word/media/image263_1.png的斜率之积为word/media/image264_1.png,
所以不能出现word/media/image144_1.png的情况.
(2)word/media/image263_1.png的中点坐标为word/media/image265_1.png,可得word/media/image263_1.png的中垂线方程为word/media/image266_1.png.
由(1)可得word/media/image267_1.png,所以word/media/image268_1.png的中垂线方程为word/media/image269_1.png.
联立word/media/image270_1.png,又word/media/image271_1.png,可得word/media/image272_1.png,
所以过word/media/image139_1.png、word/media/image140_1.png、word/media/image141_1.png三点的圆的圆心坐标为word/media/image273_1.png,半径word/media/image274_1.png.
故圆在word/media/image275_1.png轴上截得的弦长为word/media/image276_1.png,即过word/media/image139_1.png、word/media/image140_1.png、word/media/image141_1.png三点的圆在word/media/image275_1.png轴上的截得的弦长为定值.