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重庆市万州第一中学高三数学10月月考试题 理


万一中高 2017 级高三 10 月月考理科数学试题
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.已知函数 y ? 1 的定义域为集合 A,集合 B ? x 0 ? x ? 1 ,则 A? B ? ( ) x

A. ?0, ???

B. ?0,1?

C. ?0,1?

D. ?0,1?

2.设 x ? 30.5 , y ? log3 2 , z ? cos 2 ,则(



A. z ? y ? x B. z ? x ? y C. y ? z ? x D. x ? z ? y

3.已知向量 a =(1,2),向量 b =(x,-2),且 a ? (a ? b) ,则实数 x 为( )

A.9

B.4

C.0

D.-4

4.在 ?ABC 中,a ,b ,c 为角 A ,B ,C 的对边,若 A ? ? ,cosB ? 3 ,b ? 8 ,则 a ? ( )

6

5

A. 40 3

B.10

C. 20 3

D. 5

5.下列命题中 假命题 的个数是( )
???

(1)“若 a ??0,1? ,则关于 x 的不等式 ax2 ? 2ax ?1 ? 0 解集为 R ”的逆命题为真.

(2) C 表示复数集,则有 ?x ? C, x2 ?1 ? 1.
(3)已知向量 a, b ,若| a |? 2 10 , b ? (?1, 3) ,且 a ?b ? 10 ,则 a与b 的夹角为 ? 。 3
A.0 B.1 C.2 D.3

6.函数 f (x) ? 2x log 1 x ?1的零点个数为 (



2

A. 1

B. 2

C. 3

D. 4

7. 已 知 函 数 f (x) 是 定 义 在 R 上 的 偶 函 数 , 且 在 区 间 [0, ??) 单 调 递 增 . 若 实 数 a 满 足

f (log2 a) ? f (log1 a) ? 2 f (1) , 则 a 的最小值是(



2

A. 3

B.1

2

C. 1 2

D.2

8.函数 f (x) ?sin x ?ln( x 2?1) 的部分图象可能是( )

y

y

y

y

O

.

x

O.

x

O

x

O

x

A

B

C

D

9.已知 f (x) 是定义在 R 上的偶函数,对任 x ? R,都有f (x ? 6) ? f (x) ? 2 f (3),且f (0) ? 3,

则f (2016) ? ( )

A. 1

B. 2

C. 3

D. 4

10.函数 f (x) ? sin(?x ??)(? ? 0) 的图象如图所示,为了得到函数 y ? cos(2x ? ? ) 的图象,只需 6

将 y ? f (x) 的图象( )

A.向左平移 ? 个单位长度 B.向右平移 ? 个单位长度

6

6

C.向左平移 ? 个单位长度 D.向右平移 ? 个单位长度

3

3

11.已知函数 f ?x? ? ax3? 1 x2 在 x ? ?1 处取得极大值,
2

记 g?x? ?

f

1
'?x?

.程序框图如图所示,若输出的结果

s

?

2014 , 2015

则判断框中可以填入的关于 n 的判断条件是( ) A.n ? 2014? B.n ? 2015? C.n ? 2014? D.n ? 2015?

12.若 f ??x? 是 f ?x? 的导函数, f ??x? ? 2 f ?x? ?x ? R?, f ?? 1 ?? ? e ,
?2?

则 f ?ln x? ? x 2 的解集为( )

A.?? 0, e ?? ? 2?

B.?? e , e ?? ?2 ?

C.?? 1 , e ?? ?e 2?

?D. 0, e ?

二、填空题(本大题共 4 小题,每小题 5 分,共 20 分。将答案填在答题卷相应位置上.
13.已知复数 z 满足 (1? i)z ? i2015 (其中 i 为虚数单位),则 z 的虚部为

14.已知? , ? ? ?? 0,? ?? , tan? ? 2, sin?? ? ? ? ? 3 .则 cos? 的值=

? 2?

5

15.已知函数 f ? x? ? x2 ? x ? a ln ? x ?1? ,其中 a ? 0. 若 f ? x? 在定义域内既有极大值又有极小值,

则实数 a 的取值范围为

16.已知函数

f

?x?

?

????

1 2

x

? ?

2x2

?

1 4

?? x ? 2

x ?[0, 1]

2 ,g ? x? ? a sin(? x ? 3? ) ? 2a ? 2(a ? 0) ,给出下列

x

?

? ??

1 2

,1???

32

结论,其中所有正确的结论的序号是



①函数 f ? x? 的值域为[0, 2] ;
3
②函数 g ? x? 在?0,1?上是增函数;

③若存在

x1, x2

??0,1? ,使得

f

( x1 )

?

g(x2 ) ,则实数 a

的取值范围是[ 4 9

,

4]. 5

三、解答题:本大题共 6 小题,满分 70 分.解答须写出文字说明、证明过程和演算步骤

17.(本小题满分 12 分)三角形 ABC 中,已知 sin2 A ? sin2 B+sin Asin B ? sin2 C ,其中,角 A、B、C 所对的边分别为 a、b、c . (Ⅰ)求角 C 的大小; (Ⅱ)求 a ? b 的取值范围.
c

18.(本小题满分 12 分)已知集合U ? R ,集合 A ? {x | (x ? 2)(x ? 3) ? 0},函数 y ? lg x ? (a2 ? 2) a?x

的定义域为集合 B.

(1)若

a

?

1 2

,求集合

A?

(CU

B) ;

(2)命题 p : x ? A ,命题 q : x ? B ,若 q 是 p 的必要条件,求实数 a 的取值范围.

19.(本小题满分 12 分)已知函数 f(x)=(x-k)ex. (1)求 f(x)的单调区间; (2)求 f(x)在区间[0,1]上的最小值.

20.(本小题满分 12 分)已知 f ? x? ? 3 cos 2x ? 2sin(3? ? x)sin(? ? x), x ? R
2 (1)求 f (x) 的最小正周期及单增区间;
(2)已知锐角 ?ABC 的内角 A, B,C 的对边分别为 a,b, c ,且 f ? A? ? ? 3 , a ? 3,求 S???C 的最
大值.

21.(本小题满分 12 分)

已知函数

f (x) ?

x ex

? x2

? x (其中 e ? 2.71828L

).

(Ⅰ)求 f (x) 在 (1, f (1))处的切线方程;

(Ⅱ) 若 函 数 g(x) ? ln[ f (x) ? x2 ? x] ? b 的 两 个 零 点 为 x1, x2 , 证 明 :

g?(x1)

+

g?(x2 )

?

g?(

x1

? 2

x2

)

.

请考生在第 22、23 题中任选一题作答,如果多做,则按所做的第一个计分.

22.选修 4-4:坐标系与参数方程(本小题 10 分)

在直角坐标系中,以原点为极点, x 轴的正半轴为极轴 建立极坐标系,曲线

?

2

C

:

?

sin2 ?

?

2a cos? , (a

?

0)

,过点

P(?2, ?4)

的直线

I

的参数方程为

?? ?

x

?

?2

?

2

t, ( t 为参

? ??

y

?

?4

?

2 t, 2

数), I 与 C 分别交于 M , N .

(Ⅰ)写出 C 的平面直角坐标系方程和 I 的普通方程; (Ⅱ)若| PM |,| MN |,| PN |成等比数列,求 a 的值.

23.选修 4-5:不等式选讲(本小题 10 分)
设函数 f (x) ?| x ?1| ? | x ? 2 | . (Ⅰ)求 f (x) 的最小值,并求出 f (x) 取最小值时 x 的取值 范围; (Ⅱ)若不等式 f (x) ? a(x ?1) 的解集为空集,求实数 a 的取值范围.

万州一中高 2019 级高三 10 月月考试题答案 一、选择题 DAADC BCBCA BD

二、填空题

13. 1 14. 2 5

2

5

15. 0 ? a ? 1 8

16.①②③ .

三、解答题:本大题共 6 小题,满分 70 分.解答须写出文字说明、证明过程和演算步骤

17. 解析:(Ⅰ)由正弦定理得: a2 ? b2 ? c2 ? ?ab ………………2 分

?由余弦定理得: cos C ? a2 ? b2 ? c ? ? 1 ,?C ? 2? .……………………5 分

2ab

2

3

(Ⅱ)由正弦定理得: a ? b ? sin A ? sin B ? 2 3(sin A ? sin B) …………………………7 分

c

sin C

3

又 A ? B ? ? ,? B ? ? ? A,…………………………8 分

3

3

?sin A ? sin B ? sin A ? sin(? ? A) ? sin( A ? ? ) ,…………………………10 分

3

3

而 0 ? A ? ? ,?? ? A ? ? ? 2? ,

33

33

?sin A ? sin B ? ( 3 ,1],? a ? b ? (1, 2 3 ] .………………12 分

2

c

3

18.(本小题满分 12 分)

解析:(1)因为集合 A ? {x | 2 ? x ? 3},因为 a ? 1 2

函数

y

?

lg

x

?

(a2

?

2)

= lg

x

?

9 4

,由

x

?

9 4

>

0



a?x

1?x

1?x

2

2

可得集合 B={x | 1 ? x ? 9}…………2 分 24

CU

B

?

{x

|

x

?

1 2

或x

?

9} , 4

… ……… ………………………………4 分



A

?

(CU

B)

?

{x

|

9 4

?

x

?

3}

.

……………………………6 分

(2)因为 q 是 p 的必要条件等价于 p 是 q 的充分条件,即 A ? B

由 A ? {x | 2 ? x ? 3},而集合 B 应满足 x ? (a2 ? 2) ? 0 , a?x

因为 a2 ? 2 ? a ? (a ? 1)2 ? 7 ? 0 24

故 B ? {x | a ? x ? a2 ? 2} ,

……………………8 分

依题意就有:

?a ? 2 ??a2 ? 2 ? 3 ,

………………………………………10 分

即 a ? ?1 或1? a ? 2

所以实数 a 的取值范围是(- ?,- 1] [1,2].

…………………12 分

19.(本小题满分 12 分) 解 (1)由题意知 f′(x)=(x-k+1)ex. 令 f′(x)=0,得 x=k-1. f(x)与 f′(x)随 x 的变化情况如下表:

x

(-∞,k-1)

k-1

(k-1,+∞)

f′(x) f(x)



0



-ek-1

所以,f(x)的单调递减区间是(-∞,k-1);单调 递增区间是(k-1,+∞). (2)当 k-1≤0,即 k≤1 时,f(x)在[0,1]上单调递增, 所以 f(x)在区间[0,1]上的最小值为 f(0)=-k; 当 0<k-1<1,即 1<k<2 时, f(x)在[0,k-1]上单调递减,在[k-1,1]上单调递增, 所以 f(x)在区间[0,1]上的最小值为 f(k-1)=-ek-1; 当 k-1≥1,即 k ≥2 时,f(x)在[0,1]上单调递减, 所以 f(x)在区间[0,1]上的最小值为 f(1)=(1-k)e. 综上,当 k≤1 时,f(x)在[0,1]上的最小值为 f(0)=-k; 当 1<k<2 时,f(x)在[0,1]上的最小值为 f(k-1)=-ek-1; 当 k≥2 时,f(x)在[0,1]上的最小值为 f(1)=(1-k)e.

20. 解析:(1) f ? x? ?

3

cos

2x

?

sin

2x

?

?2 sin

? ??

2

x

?

? 3

? ??

? f ? x?的最小正周期为? ,

由 2k? ? ? ? 2x ? ? ? 2k? ? 3? (k ? Z ) ? k? ? 5? ? x ? k? ? 11? (k ? Z )

2

3

2

12

12

?

f

(x)的单增区间为??x | ?

k?

?

5? 12

?

x

?

k?

? 11? 12

(k

? Z )?? ?

(2)由 f ? A? ? ?

3



sin

? ??

2A

?

? 3

? ??

?

3 2

,

又A

?

? ??

0,? 2

???,?

A=

? 3

由余弦定理得 a2 ? b2 ? c2 ? 2bc cos A得9=b2 ? c2 ? bc ? bc

即bc ? ( 9 当且仅当b=c时取等号)



S???C

=

1 2

bc

sin?



9

3 4

21、【解析】(Ⅰ)由题意得

f

?(x)

?

1? x ex

+2x

?1,

f

(1)

?

1 e



∴ f (x) 在 (1, f (1))处的切线斜率为 f ?(1) ? 1,

∴ f (x) 在 (1, f (1))处的切线方程为 y ? 1 ? x ?1,即 ex ? ey ? e ?1 ? 0 . ……………4 分 e

令?(t) ? t ? 1 ? 2(t ?1) ? ln t , t t ?1

??(t)

?1?

1 t2

?1? t

4 (t ?1)2

?

t4

? t3 ? 4t2 ? t t 2 (1? t)2

?1 ,

………10 分

令 m(t) ? t4 ? t3 ? 4t 2 ? t ?1 ,

m?(t) ? 4t3 ? 3t 2 ? 8t ?1,设 h(x) = 4t3 ? 3t2 ? 8t ?1,

h?(t) ? 12t2 ? 6t ? 8 ? 0 对 t ?1恒成立,即?m?(t) 在 ?1, ??? 上单调递增, ?m?(t) ? m?(1) ? 0 ,?m(t) 在 ?1, ??? 上单调递增,?m(t) ? m(1) ? 0 , 即??(t) ? 0 , ?? (t) 在 ?1, ??? 上单调递增, ??(t) ? ?(1) ? 0 ,即原不等式成立.………12
分 22.选修 4-4:坐标系与参数方程
解析:(Ⅰ)曲线 C 的直角坐标方程为 y2 ? 2ax(a ? 0) ; 直线 l 的普通方程为 x ? y ? 2 ? 0 .………………5 分 (Ⅱ)将直线 l 的参数方程与 C 的直角坐标方程联立, 得 t2 ? 2(4 ? a) 2t ? 8(4 ? a) ? 0 (*) ? ? 8a(4 ? a) ? 0 . 设点 M , N 分别对应 参数 t1, t2 ,恰为上述方程的根. 则| PM |?| t1 | ,| PN |?| t2 | ,| MN |?| t1 ? t2 | . 由题设得 (t1 ? t2 )2 ?| t1t2 | ,即 (t1 ? t2 )2 ? 4t1t2 ?| t1t2 | . 由(*)得 t1 ? t2 ? 2(4 ? a) 2 , t1t2 ? 8(4 ? a) ? 0 ,则有 (4 ? a)2 ? 5(4 ? a) ? 0 ,得 a ?1,或 a ? ?4 . 因为 a ? 0 ,所 以 a ?1.………………10 分

23.选修 4-5:不等式选讲
解析:(Ⅰ) f (x) ?| x ?1| ? | x ? 2 |?| (x ?1) ? (x ? 2) |? 3

当且仅当 (x ?1)(x ? 2) ? 0 时取等号

? f (x)min ? 3 ,此时 x ?[?1, 2]

(Ⅱ)

??2x ?1(x ? ?1)

f (x) ? ??3

(?1 ? x ? 2)

??2x ?1 (x ? 2)

由于 y ? a(x ?1) 的图像是过点 P(?1, 0) ,斜率为 a 的直线,由图可得不等式 f (x) ? a(x ?1) 的解集

非空时 a 的取值范围是 KAC ? a ? KPB ,即 a ?[?2,1) ………………10 分


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