广西柳州市2022届高三上学期第二次模拟考试数学（文）试题答案

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柳州市 2022 届高三第二次模拟考试
文科数学参考答案
一、选择题（本题共 12 小题，每小题 5 分，共 60 分．）
二、填空题（本题共 4 小题，每小题 5 分，共 20 分．）
13．
5
； 14．
3
28
； 15．
 
,1
； 16．
3
33
．
三、解答题（共 70 分，解答应写出文字说明、证明过程或演算步骤．）
17．解：（1）完成列联表（单位：人）：
成绩合格
成绩不合格
合计
男性
40
10
50
女性
30
20
50
合计
70
30
100
···················································································································· 4 分
由列联表，从参与的市民中随机抽取 100 人，有 70 人竞答成绩合格，所以估计该市参与此次冬奥知
识竞答的市民的成绩合格率为
7.0
．···································································· 6 分
（2）由列联表，
的观测值
841.3762.4
21
100
30705050
)30102040(100 2
k
，
∴有 95%的把握认为该市参与此次冬奥知识竞答的市民的成绩与性别有关．···············12 分
18．（1）∵点
 
1
,
n n
S S
在直线
)(1
1*
Nnnx
n
n
y
上，
∴
)1(
1
1
nS
n
n
Snn
，·················································································1 分
同除以
1n
，则有
1
1
1
n
S
n
Snn
，·································································· 2 分
∴
1
1
1
n
S
n
Snn
，·························································································· 3 分
∴数列
n
S
n
 
 
 
是以
2
11
1a
S
为首项，
1
为公差的等差数列．·································· 4 分
（2）由（1）可知
3)1()1(2 nn
n
Sn
，·················································· 5 分
∴
nnSn3
2
，·························································································· 6 分
∴当
1n
时，
231
11 Sa
；····································································· 7 分
当
2n
时，
42)1(3)1(3 22
1nnnnnSSa nnn
，··················· 8 分
题号
1
2
3
4
5
6
7
8
9
10
11
12
答案
C
D
D
B
B
C
A
C
A
D
C
A
2
经检验，当
1n
时也成立．∴
)(42 *
Nnnan
．········································9 分
∵
1 2 1n n n
T b b b b
 
 
n
nnT 22424222022 4321
··············································· 10 分
 
15432 224242220222
n
nnT
··········································· 11 分
 
14321 2242222222222
nn
nnT
 
1223 2n
nnT
·················································································· 12 分
19．（1）证明：设
AC
的中点为
O
，连接
BO
，
PO
.
由题意，得
2PA PB PC 
，
1PO
，
1AO BO CO
.
∵在
PAC
中，
PA PC
，
O
为
AC
的中点，
∴
PO AC
，·································································································1 分
∵在
POB
中，
1PO
，
1OB
，
2PB
，
2 2 2
PO OB PB 
，
∴
PO OB
.···································································································2 分
又∵
AC OB O
，························································································· 3 分
,AC OB
平面
ABC
，∴
PO
平面
ABC
，··························································· 4 分
∵
PO
平面
PAC
，∴平面
PAC
平面
ABC
.······················································· 5 分
（2）解：由（1）知，
BO PO
，
BO AC
，
BO
平面
PAC
，
∴
BMO
是直线
BM
与平面
PAC
所成的角，························································· 6 分
且
1
tan BO
BMO OM OM
 
，∴当
OM
最短时，
BMO
最大，
即当
M
是
PA
的中点时，直线
BM
与平面
PAC
所成的角最大，································· 7 分
设点
M
到平面
PBC
的距离为 h，
∵
CPMBPBCM VV
，·······················································································8 分
∴
1
2
2
2
2
1
3
1
2
6
2
2
1
3
1h
·······················································10 分
∴
3
3
h
，·······························································································11 分
即点
M
到平面
PBC
的距离为
3
3
．···································································· 12 分
20.（1）由题意知
2
3
a
c
.·············································································· 1 分
1
4
13
22 ba
··································································································2 分
1,2 ba
································································································· 3 分
椭圆
1
4
:2
2
y
x
C
······················································································ 4 分
（2）由已知得
 
2,0A
、
 
2,0B
，设椭圆 C上动点
 
,P x y
，
则利用两点连线的斜率公式可知
0
2
PA
y
kx
，
0
2
PB
y
kx
，
∴
  
2
2 2
2 2
1
0 0 1
4
2 2 2 2 4 4 4
PA PB
x
y y y y
k k x x x x x x
 
  
 
，································· 5 分
设直线
PA
方程为：
 
2y k x 
，则直线
PB
方程为：
 
12
4
y x
k
 
，······················ 6 分
广西柳州市2022届高三上学期第二次模拟考试数学（文）试题答案.pdf

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