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1、昌平区20212021学年第一学期高三年级期末质量抽测数 学 试 卷文 科 总分值150分,考试时间120分钟考生须知:本试卷共6页,分第一卷选择题和第二卷非选择题两局部。答题前考生务必将答题卡上的学校、班级、姓名、考试编号用黑色字迹的签字笔填写。答题卡上第I卷(选择题)必须用2B铅笔作答,第II卷(非选择题)必须用黑色字迹的签字笔作答,作图时可以使用2B铅笔。请按照题号顺序在各题目的答题区内作答,未在对应的答题区域内作答或超出答题区域作答的均不得分。修改时,选择题局部用塑料橡皮擦涂干净,不得使用涂改液。保持答题卡整洁,不要折叠、折皱、破损。不得在答题卡上做任何标记。考试结束后,考生务必将答题
2、卡交监考老师收回,试卷自己妥善保存。第一卷选择题 共40分一、选择题共8小题,每题5分,共40分。在每题列出的四个选项中,选出符合题目要求的一项.(1) 全集,集合,那么集合(A) (B) (C) (D)开始, 输出结束 否是(2) 在复平面内表示复数的点位于 (A)第一象限 (B)第二象限 (C)第三象限 (D)第四象限(3) 假设某程序框图如右图所示,那么该程序运行后输出的等于 (A)63 (B)31 (C)127 (D)15(4) “是“直线与平行的A充分而不必要条件 B必要而不充分条件 C充分必要条件 D既不充分也不必要条件(5) 设是两条不同直线, 是两个不同的平面,以下命题正确的选
3、项是(A) 且,那么 (B) 且,那么(C) ,那么 (D) ,那么(6)将函数的图象向右平移个单位长度,再将所得图象上的所有点的横坐标缩短到原来的纵坐标不变,得到的函数解析式为(A) (B) (C) (D) (7)函数的值域为,那么实数的取值范围是(A) (B) (C) (D) (8) 函数的图象上相邻的一个最大值点与一个最小值点恰好都在圆上,那么函数的图象的一条对称轴可以是(A)直线 (B) 直线 (C)直线 (D)直线 第二卷非选择题 共110分二、填空题本大题共6小题,每题5分,共30分.(9) 向量,假设,那么_ .(10) 假设实数满足,那么的最大值是_ .(11) 抛物线的准线方
4、程是,那么实数的值为_ . (12) 设,当时,从小到大的顺序是_ .(13) 假设是2和8的等比中项,那么_ ,圆锥曲线的离心率是_ . (14) 函数的定义域为,假设对于任意,当时,都有,那么称函数在在上为非减函数,且满足以下三个条件:;. 那么_ ;_ .三、解答题本大题共6小题,共80分.解容许写出文字说明,证明过程或演算步骤.(15)(本小题总分值13分)的内角的对边分别为,.()求;()求的面积. (16)(本小题总分值13分) 为了参加某项环保活动,用分层抽样的方法从高中三个年级的学生中,抽取假设干人组成环保志愿者小组,有关数据见下表:年级相关人数抽取人数高一36高二72高三54
5、3分别求出样本中高一、高二年级志愿者的人数,;用表示样本中高一年级的志愿者,表示样本中高二年级的志愿者,现从样本中高一、高二年级的所有志愿者中随机抽取2人.1按照以上志愿者的表示方法,用列举法列出上述所有可能情况;2求二人在同一年级的概率.(17)(本小题总分值14分)如图,在直三棱柱中,,为的中点.求证:平面;() 求证:平面;求三棱锥的体积. (18)(本小题总分值13分)设函数.假设曲线在点处的切线方程为,求实数的值;II假设,求函数的最大值. (19)(本小题总分值13分)椭圆的焦距为,过焦点且垂直于长轴的直线被椭圆截得的弦长为,过点的直线与椭圆交于两点.求椭圆的方程;设为椭圆上一点,
6、且满足为坐标原点,求实数的取值范围.(20)(本小题总分值14分)无穷数列中,是首项为,公差为的等差数列,是首项为,公比为的等比数列(其中),并对任意的,均有成立.()当时,求;()假设,试求的值;()判断是否存在,使得成立?假设存在,试求出的值;假设不存在,请说明理由.昌平区20212021学年第一学期高三年级期末质量抽测一、选择题共10小题,每题5分,共50分。在每题列出的四个选项中,选出符合题目要求的一项。题 号12345678答 案CBBABDCD二、填空题共6小题,每题5分,共30分。 9 10 11 12(方向写反扣2分)13答出一个给1分,两个给2分; 或答出一个给1分,两个给3
7、分 14 ;第一空2分,第二空3分三、解答题本大题共6小题,共80分.解容许写出文字说明,证明过程或演算步骤.(15)(本小题总分值13分)解:() 因为为的内角,且,所以.所以 . 7分() 在中,由正弦定理,解得.所以的面积为. 13分(16)(本小题总分值13分) 解:依题意,分层抽样的抽样比为. 所以在高一年级抽取的人数为人 , 在高二年级抽取的人数为人. 4分1用表示样本中高一年级的2名志愿者,用表示样本中 共15种. 9分2设为事件“抽取的二人在同一年级.因为抽取的二人在同一年级的情况是共7种. 所以抽取的二人是同一年级的概率为. 13分(17)(本小题总分值14分)解:在直三棱柱
8、中,连结交于,连结. 因为,所以四边形、为正方形. 所以为中点.在中,因为为的中点,所以. 因为平面,平面,所以平面. 5分()因为三棱柱为直三棱柱,所以平面.因为平面,所以.又,, 所以平面.因为,所以. 因为是正方形,所以.又,所以平面. 10分因为为等腰直角三角形,所以.因为平面,所以. 14分(18)(本小题总分值13分)解:() 函数的定义域为.,因为曲线在处与直线相切,所以解得 6分() 当时,.因为,1当时,. 因为时, 所以在上单调递减,无最大值.2当时, 所以时, 所以在上单调递减,无最大值.3当时,. 因为时, 时,所以在上单调递增,在上单调递减. 所以. 13分(19)(
9、本小题总分值13分)解:I因为所求椭圆的方程为,焦距为,所以.设过焦点且垂直于长轴的直线为.因为过焦点且垂直于长轴的直线被椭圆截得的弦长为,代入椭圆方程解得:,即. 由解得 所以所求椭圆的方程为: . 6分设过点的直线的斜率为,显然存在.1当时,所以.2当时,设直线的方程为.由消并整理得.当时,可得.设,那么,. 因为, 所以. 所以 , .由点在椭圆上得. 解得.因为,所以.所以.所以.所以.所以.所以.所以. 所以. 综合(1) (2)可知 13分(20)(本小题总分值14分)解:() 时,数列的周期为. 因为,而是等比数列中的项, 所以. 4分()设是第一个周期中等比数列中的第项,那么.
10、 因为,所以等比数列中至少有项,即,那么一个周期中至少有项. 所以最多是第三个周期中的项. 假设是第一个周期中的项,那么,所以;假设是第二个周期中的项,那么,所以,;假设是第三个周期中的项,那么所以, .综上,. 9分()因为是此数列的周期, 所以表示64个周期及等差数列的前3项之和. 所以最大时, 最大. 因为 ,当时,;当时,;当时, . 当时,取得最大值,那么取得最大值为.由此可知,不存在,使得成立. 14分【各题假设有其它解法,请酌情给分】9JWKffwvG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRW
11、XmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK
12、8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2
13、zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFye
14、Q!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxG89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK
15、!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8vG#tYM*Jg
16、&6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$
17、UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmUE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7
18、JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*
19、3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#
20、KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#
21、KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWR
22、rWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89A
23、mYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89A
24、mYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWF
25、A5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum
26、&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpa
27、zadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqvadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&
28、qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9Cpb
29、K!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9
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38、pP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#Q
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47、$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcU
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59、Dx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#
60、QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwc
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