河南省名校联盟2020-2021学年高三上学期11月联考卷

河南省名校联盟2020-2021学年高三11月联考试卷

数学（文科）

2020.11

一、选择题（每题5分，共12道小题，共60分）

1. 已知集合{}{}Z x x x x B A ∈≤-=-=,02,2,0,12，则=B A （ ） A. {}2 B. {}2,0 C . {}2,0,1- D.{}21,0,1，-

2.若a ＞b ，则下列不等式成立的是（ ）

A. 22b a ＞

B. 33b a ＞ C . 12＜b a - D.()1lg ＜b a - 3.△ABC 中，角A ，B ，C ，的对边分别为c b a ,,，若,6

,3,3π===B c a 则 b=（ ）

A. 32

B. 22 C . 3 D.2

4.已知向量,23,2==b a 若向量a 与b 的夹角为4

π，则=-b a 2（ ） A. 10 B. 10 C . 13 D.132

5.已知幂函数()()1275++-=m x m m x f 为奇函数，则实数m 的值为（ ） A. 4或3 B. 2或3 C . 3 D. 2

6.已知5

3sin =α，α为第二象限角，则=α2sin （ ） A. 2524-

B. 257 C 2512- D.25

24

提升至λ,则C 大约增加了20%,则λ的值约为（ ）（ 参考数据：

9120

10,3.02lg 96.3≈≈）

A. 7596

B. 9119 C . 11584 D. 14469

8.已知函数()??

???-≥-=,0,4

,0,22＜x x x x x x x f ，则函数()3-=x f y 的零点个数为（ ） A. 1 B. 2 C . 3 D. 4 9.已知数列{}n a 满足()

*111

2

,3N n a a a a n n n ∈--==+，则=2020a （ ） A.

21 B. 3 C . 1 D. 3

1 10.()()??

?

?

?+=2,0,0sin π＜＞＞?ω?ωA x A x f 的部分图象如图所示，若将函数

()x f 的图象向右平移2

π

个单位长度，得到函数()x g 的图象，则（ ）

A. ()???

??-

=122

1sin 2πx x g B. ()??

?

??+

=1221

sin 2πx x g C . ()??

?

?

?-

=122sin 2πx x g C. ()??

?

??+

=122sin 2πx x g

11.如图，在离地面h 的热气球M 上，观察到山顶C 处的仰角为θ，在山脚A 处观察到山顶C 处的仰角为60°，若A 到热气球的距离

2400=AM ，山的高度，?=∠=45,600ACM BC ，则θ=（ ）

A. ?30

B. ?25 C . ?20 D. ?15 12.已知函数()x x e

e x x x

f 2

243-

+-=，其中e 是自然对数的底数，若()

()0122≤--+a f a f ，则实数a 的取值范围为（

）

A. ??

????-1,21 B. ??

???

?-21,1 C . []1,2- D. []2,1-

二、填空题（每小题5分，共4小题，共20分）

13，已知变量x,y 满足约束条件??

?

??≥≥≤+≤+,0,0,1832,

8y x y x y x ，则目标函数y x z 3+=的

最大值为 。

14.已知{}n a 为等比数列，n S 为其前n 项和，若10,74236=+-=a a S S ，则

=1a 。

15.曲线x x ax y ln 22++=在点()()1,1f 处的切线与x 轴交点的横坐标为2，则实数a 的值为 。

16.已知一个圆锥内接于球O （圆锥的底面圆周及顶点均在球面上），

若球的半径R=5，圆锥的高是底面半径的2倍，则圆锥的侧面积为

三、解答题（共6道小题，共70分） 17.（10分）设等差数列{}n a 的前n 项和为n S ，3313==a a

（1）求n S ，

（2）求数列?

??

???n S 1的前n 项和n T 。

18.（12分）

在△ABC 中，角A ，B ，C ，的对边分别为c b a ,,A

B

a A c C sin 3sin cos cos =

+ （1）求c 的值。

（2）若3

π

=C ，求△ABC 面积的最大值。

19. 已知()()??

? ?

?=+=2,0sin 3π＜＞?ω?ωa x x f 的图象过点??

?

??a ,12

π，且图象

的相邻两条对称轴的距离为2

π

。 （1）求函数()x f 的单调区间； （2）若()x f 在区间??

?

???-212π，π上的最大值与最小值之和为3，求实数a 的值。

20. （12分）已知向量??

? ?

?

-=??

? ?

?=x x b x x a 43sin ,4

3cos ,45sin ,4

5cos ，且??

????∈2,0πx

（1）求b a ?及b a +；

（2）求函数()b a b a x f +-?=的最值以及对应x 的值。

21.（12分）

如图，已知平面四边形ABCP 中，D 为PA 的中点，且PA ⊥AB ，且 CD ∥AB ，且PA=CD=2AB=2.将四边形ABCP 沿CD 折起，使平面PCD ⊥ABCD ，连接PA 、PB. （1）求证：BC ⊥平面PBD.

（2）设Q 为PC 的中点，求点A 到平面QBD 的距离

22.（12分）已知函数()()()m x e x x f x g x

x x f x

-+??

???

?-=-+=1,11ln

（1）求()x f 的单调区间；

（2）当??

?

???∈e e x ,1

时，判断函数()x g 的零点个数.

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