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广东省广州市2014届高三调研测试数学文试题-Word版包含答案)


广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

试卷类型:A

广州市 2014 届高三年级调研测试 数 学(文 科)
本试卷共 4 页,21 小题,满分 150 分.考试用时 120 分钟.

2014.1

注意事项: 1.答卷前,考生务必用黑色字迹钢笔或签字笔将自己的姓名和考生号、试室号、座位号填写在答
题卡上.用 2B 铅笔将试卷类型(A)填涂在答题卡相应位置上. 2.选择题每小题选出答案后,用 2B 铅笔把答题卡上对应题目选项的答案信息点涂黑,如需改动,
用橡皮擦干净后,再选涂其他答案,答案不能答在试卷上. 3.非选择题必须用黑色字迹的钢笔或签字笔作答,答案必须写在答题卡各题目指定区域内的相应
位置上;如需改动,先划掉原来的答案,然后再写上新的答案;不准使用铅笔和涂改液.不按以上要求 作答的答案无效.
4.作答选做题时,请先用 2B 铅笔填涂选做题的题号对应的信息点,再作答.漏涂、错涂、多涂 的,答案无效.
5.考生必须保持答题卡的整洁.考试结束后,将试卷和答题卡一并交回.

参考公式:
锥体体积公式V ? 1 Sh ,其中 S 为锥体的底面积, h 为锥体的高. 3

一.选择题:本大题共 10 小题,每小题 5 分,满分 50 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.函数 y ? 4 ? x 的定义域是

A. ???, 4?

B. ???, 4?

C. ?4, ???

D.?4, ???

2.命题“若 x2 ? 1,则 ?1 ? x ? 1”的逆否命题是

A.若 x2 ? 1,则 x ? 1,或x ? ?1

B.若 ?1 ? x ? 1,则 x2 ? 1

C.若 x ? 1,或x ? ?1,则 x2 ? 1

D.若 x ? 1,或x ? ?1,则 x2 ? 1

3.如图 1 是 2013 年某大学自主招生面试环节中,七位评委为某考生打出的分数的茎叶统计图,去掉

一个最高分和一个最低分后,所剩数据的平均数和众数依次为

A. 85,84

B. 84,85

C. 86,84

D. 84,86

4.设 z ?1?i ( i 是虚数单位),则复数 2 ? i2 的虚部是 z

A. ?i

B. ?1

C. i

1 / 10

79 8 44 64 7 93
图1
D.1

广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

5.若集合 A, B 满足 A ? ?x?Z | x ? 3? , B ? N ,则 A B 不.可.能.是

A.{0,1, 2}

B. {1, 2}

C. {?1}

D. ?

?x ? 2 y ? 2 ? 0,

6.若实数

x



y

满足不等式组

? ?

x

?

y

?

1

?

0,

则 x ? y 的最大值为 ks5u

??x ? 2 y ? 2 ? 0,

A. 4

B. 5

C. 6

7.执行如图 2 的程序框图,如果输入的 N 的值是 6,那么输出的 p 的值是

D. 7

A.15

B.105

C.120

D.720

开始

输入 N

k ? 1, p ? 1 p ? p ? k

k ? N ? 否 输出 p

结束



k ?k?2

图2

8.某几何体的三视图(如图 3 所示)均为边长为 2 的等腰直角三角 形,则该几何体的表面积是

A. 4 ? 4 2

B. 4 2

C. 4 ? 2

D.8 ? 4 2

正视图

侧视图

俯视图

图3

9.若点 A(1,0) 和点 B(4, 0) 到直线 l 的距离依次为 1 和 2,则这样的直线有

A.1 条

B.2 条

C.3 条

D.4 条

10.函数 f (x) ? sin x ? x 在区间?0, ??? 内
A.没有零点 C.有且仅有 2 个零点

B.有且仅有 1 个零点 D.有且仅有 3 个零点

二.填空题: 本大题共 5 小题,考生作答 4 小题,每小题 5 分,满分 20 分. (一)必做题(11~13 题)
11.若向量 m ? ?1,2? , n ? ? x,1? 满足 m ? n ,则| n |? __________.

12.在等比数列{an}中,若 a2 ? a3 ? 3a1 ,则 a4 ?



13.在边长为 2 的正方形 ABCD 内部任取一点 M ,则满足 ?AMB ? 90? 的概率为_______.

(二)选做题(14~15 题,考生只能从中选做一题) 2 / 10

广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

14.(几何证明选讲选做题)
如图 4, AC 为⊙ O 的直径, OB ? AC ,弦 BN 交 AC 于点 M .

若 OC ? 3 , OM ?1,则 MN 的长为



C

15.(坐标系与参数方程选讲选做题)

若点

P(x,

y)

在曲线

?x

? ?

y

? ?

?2 ? cos? sin?

(?

为参数,?

?R

)上,则

y x

的取值范围是

B

M

O

A

N

图4



三.解答题: 本大题共 6 小题,满分 80 分.解答应写出文字说明、证明过程或演算步骤.
16.(本小题满分 12 分)ks5u 在△ ABC 中,角 A , B , C 所对的边分别为 a , b , c ,且 cos A ? C ? 3 . 23 (1)求 cos B 的值; (2)若 a ? 3, b ? 2 2 ,求 c 的值.

17.(本小题满分 12 分)
某单位 N 名员工参加“社区低碳你我他”活动.他们
的年龄在 25 岁至 50 岁之间.按年龄分组:第 1 组
[25,30) ,第 2 组[30,35) ,第 3 组[35, 40) ,第 4 组

[40, 45) ,第 5 组[45,50],得到的频率分布直方图

如图 5 所示.下表是年龄的频率分布表.

图5

区间 [25,30) [30,35) [35, 40) [40, 45) [45,50]

人数

25

a

b

(1)求正整数 a , b , N 的值;

(2)现要从年龄较小的第 1,2,3 组中用分层抽样的方法抽取 6 人,则年龄在第 1,2,3 组的人数分

别是多少?

(3)在(2)的条件下,从这 6 人中随机抽取 2 人参加社区宣传交流活动,求恰有 1 人在第 3 组的概率.

18.(本小题满分 14 分)

P

如图 6,在三棱锥 P ? ABC 中, PA ? AC , PC ? BC ,

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M

A

C

广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)
M 为 PB 的中点, D 为 AB 的中点,且△ AMB 为正三角形. (1)求证: BC ? 平面 PAC ; (2)若 BC ? 4 , PB ?10 ,求点 B 到平面 DCM 的距离.

19.(本小题共 14 分)

设数列?an ?满足 a1

?

a2 2

?

a3 22

?

?

an 2n?1

?

2n , n ? N*



(1)求数列?an ?的通项公式;

(2)设 bn

?

? an

an
?1? ? an?1

?1?

,求数列?bn? 的前 n

项和

Sn



ks5u

20.(本小题满分 14 分)
在圆 x 2 ? y 2 ? 4 上任取一点 P ,设点 P 在 x 轴上的正投影为点 D .当点 P 在圆上运动时,动点
M 满足 PD ? 2MD ,动点 M 形成的轨迹为曲线 C . (1)求曲线 C 的方程;
(2)已知点 E?1,0? ,若 A, B 是曲线 C 上的两个动点,且满足 EA ? EB,求 EA ? BA 的取值范围.

21.(本小题满分 14 分)
已知函数 f ? x? ? ln x ? ax2 ? ?a ? 2? x .
(1)若 f (x) 在 x ?1 处取得极值,求实数 a 的值;
(2)求函数 f (x) 在区间[a2 , a] 上的最大值.

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广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

广州市 2014 届高三年级调研测试

数学(文科)试题参考答案及评分标准

说明:1.参考答案与评分标准指出了每道题要考查的主要知识和能力,并给出了一种或几种解法供参 考,如果考生的解法与参考答案不同,可根据试题主要考查的知识点和能力比照评分标准给 以相应的分数.
2.对解答题中的计算题,当考生的解答在某一步出现错误时,如果后继部分的解答未改变该题 的内容和难度,可视影响的程度决定后继部分的得分,但所给分数不得超过该部分正确解答 应得分数的一半;如果后继部分的解答有较严重的错误,就不再给分.
3.解答右端所注分数,表示考生正确做到这一步应得的累加分数. 4.只给整数分数,选择题和填空题不给中间分. 一.选择题:本大题主要考查基本知识和基本运算.共 10 小题,每小题 5 分,满分 50 分.

题号 1 2 3 4 5 6 7 8 9 10 答案 B D A D C D B A C B

二.填空题:本大题主要考查基本知识和基本运算.本大题共 5 小题,考生作答 4 小题,每小题 5 分, 满分 20 分.其中 14~15 题是选做题,考生只能选做一题.

11. 5

12.3

13. ? 8

14.1

? 15. ??
?

3, 3

3?

3

? ?

三.解答题: 本大题共 6 小题,满分 80 分.解答应写出文字说明、证明过程或演算步骤.

16.(本小题满分 12 分)

解:(1)在△ ABC 中, A ? B ? C ? ?.……………ks5u…………………………………1 分

所以 cos A ? C ? cos ? ? B …………………………………………………………………………2 分

2

2

? sin B ? 3 .………………………………………………………………………3 分 23
所以 cos B ? 1? 2sin2 B ……………………………………………………………………………5 分 2
? 1 .………………………………………………………………………………………7 分 3
(2)因为 a ? 3, b ? 2 2 , cos B ? 1 , 3

由余弦定理 b2 ? a2 ? c2 ? 2ac cos B ,………………………………………………………………9 分

得 c2 ? 2c ?1 ? 0 .……………………………………………………………………………………11 分 解得 c ?1.……………………………………………………………………………………………12 分

17.(本小题满分 12 分)

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广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

解:(1)由频率分布直方图可知,[25,30) 与[30,35) 两组的人数相同,
所以 a ? 25 人.………………………………………………………………………………………1 分 且 b ? 25? 0.08 ? 100 人.……………………………………………………………………………2 分
0.02 总人数 N ? 25 ? 250 人.………………………………………………………………………3 分
0.02 ? 5 (2)因为第 1,2,3 组共有 25+25+100=150 人,利用分层抽样在 150 名员工中抽取 6 人,每组抽取的
人数分别为:
第 1 组的人数为 6 ? 25 ? 1,…………………………………………………………………………4 分 150
第 2 组的人数为 6 ? 25 ? 1,…………………………………………………………………………5 分 150
第 3 组的人数为 6? 100 ? 4 ,………………………………………………………………………6 分 150
所以第 1,2,3 组分别抽取 1 人,1 人,4 人.……………………………………………………7 分
(3)由(2)可设第 1 组的 1 人为 A ,第 2 组的 1 人为 B ,第 3 组的 4 人分别为 C1, C2 , C3, C4 ,则从
6 人中抽取 2 人的所有可能结果为:
( A ,B ),( A, C1) ,( A, C2 ) ,( A, C3 ) ,( A, C4 ) ,(B, C1) ,(B, C2 ) ,(B, C3 ) ,(B, C4 ) ,(C1, C2 ) ,
(C1, C3 ) , (C1, C4 ) , (C2 , C3 ) , (C2 , C4 ) , (C3, C4 ) ,共有15 种.……………………………9 分
其中恰有 1 人年龄在第 3 组的所有结果为:
( A, C1) , ( A, C2 ) , ( A, C3 ) , ( A, C4 ) , (B, C1) , (B, C2 ) , (B, C3 ) , (B, C4 ) ,共有 8 种.
………………………………………………………11 分
所以恰有 1 人年龄在第 3 组的概率为 8 .……………ks5u…………………………12 分 15

18.(本小题满分 14 分)
(1)证明:在正 ?AMB 中, D 是 AB 的中点,所以 MD ? AB .……………………………………1 分

因为 M 是 PB 的中点, D 是 AB 的中点,所以 MD // PA ,故 PA ? AB .……………………2 分

又 PA ? AC , AB AC ? A, AB, AC ? 平面 ABC ,

P

所以 PA ?平面 ABC .…………………………………4 分

因为 BC ? 平面 ABC ,所以 PA ? BC .……………5 分

M

又 PC ? BC, PA PC ? P, PA, PC ? 平面 PAC ,

A

C

所以 BC ? 平面 PAC .………………………………7 分 (2)解法 1:设点 B 到平面 DCM 的距离为 h ,………8 分

D B

因为 PB ?10 , M 是 PB 的中点,所以 MB ? 5 .

因为 ?AMB 为正三角形,所以 AB ? MB ? 5.……………………………………………………9 分

因为 BC ? 4, BC ? AC ,所以 AC ? 3.

所以 S?BCD

?

1 2

S?ABC

?

1? 2

1 ? BC ? 2

AC

?

1? 2

1 ? 4? 3 ? 3 .…………………………………10 2



6 / 10

广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

因为 MD ? 52 ? ?? 5 ??2 ? 5 3 , ?2? 2

由(1)知 MD // PA,所以 MD ? DC . 在 ?ABC 中, CD ? 1 AB ? 5 ,
22

所以 S ?MCD

?

1 2

? MD ? CD

?

1 2

?

53 2

?

5 2

?

25 8

3

.…………………………………………11



因为VM ?BCD ? VB?MCD ,……………………………………………………………………………12 分

所以

1 3

S ?BCD

?

MD

?

1 3

S ?MCD

?h



即 1 ? 3? 5 3 ? 1 ? 25 3 ? h .………………………ks5u…………………………13 分

3

238

所以 h ? 12 . 5

故点 B 到平面 DCM 的距离为 12 .………………………………………………………………14 分 5

解法 2:过点 B 作直线 CD 的垂线,交 CD 的延长线于点 H ,…………………………………………8 分

由(1)知, PA ?平面 ABC , MD // PA,

P

所以 MD ? 平面 ABC .

因为 BH ? 平面 ABC ,所以 MD ? BH .

因为 CD MD ? D ,所以 BH ?平面 DCM .

所以 BH 为点 B 到平面 DCM 的距离.………………9 分

M

因为 PB ?10 , M 是 PB 的中点,所以 MB ? 5 . 因为 ?AMB 为正三角形,所以 AB ? MB ? 5.……10 分 因为 D 为 AB 的中点,所以 CD ? BD ? 5 .
2

A D
H

C E B

以下给出两种求 BH 的方法:

方法 1:在△ BCD 中,过点 D 作 BC 的垂线,垂足为点 E ,

则 DE ? 1 AC ? 3 .…………………………………………………………………………………11 分

2

2

因为 1 ? CD ? BH ? 1 ? BC ? DE ,………………………………………………………………12 分

2

2

所以 BH

?

BC ? DE

?

4?

3 2

?

12 .

CD

55

2

方法 2:在 Rt △ BHD 中, BH 2 ? DH 2 ? BD2 ? 25 . 4
在 Rt △ BHC 中,因为 BC ? 4 ,

①…………………………11 分

7 / 10

广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

所以 BH 2 ? CH 2 ? BC2 ,



BH

2

?

? ??

DH

?

5 2

?2 ??

? 16



②…………………………………12 分

由①,②解得 BH ? 12 . 5

故点 B 到平面 DCM 的距离为 12 .………………………………………………………………14 分 5

19.(本小题满分 14 分)

解:(1)因为 a1

?

a2 2

?

a3 22

?

? an ? 2n , n ? N* , 2n?1



所以当 n ? 1时, a1 ? 2 .……………………………………………………………………………1 分

当n

?

2时, a1

?

a2 2

?

a3 22

?

?

an?1 2n?2

?

2?n ?1? ,

② …………………………………2 分

①-②得, an 2n?1

?

2

.…………………………………………………………………………………4



所以 an ? 2n .…………………………………………………………………………………………5 分

因为 a1 ? 2 ,适合上式,
? ? 所以 an ? 2n n ? N* .………………………………………………………………………………6 分

(2)由(1)得 an ? 2n .

? ?? ? 所以bn

?

? an

an
?1? ? an?1

?1?

?

2n

…………………………………………………8 分

2n ?1 2n?1 ?1

?

1 2n ?1

?

1 2n?1 ?1

.……………………………………ks5u………………10



所以 Sn ? b1 ? b2 ? ? bn

?

???1 ?

1 3

? ??

?

? ??

1 3

?

1 7

? ??

?

? ??

1 7

?

1 15

? ??

?

?

? ??

1 2n ?

1

?

1 2n?1

?

1

? ??

………………………………12



?

1

?

1 2n?1

?1

.………………………………………………………………………………14



20.(本小题满分 14 分)
(1)解法 1:由 PD ? 2MD 知点 M 为线段 PD 的中点.……………………………………………1 分
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广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

设点 M 的坐标是 (x, y) ,则点 P 的坐标是 ? x, 2y? .……………………………………………2 分

因为点 P 在圆 x 2 ? y 2 ? 4 上,
所以 x2 ? ?2y?2 ? 4 .…………………………………………………………………………………3 分

所以曲线 C 的方程为 x 2 ? y 2 ? 1.…………………………………………………………………4 分 4
解法 2:设点 M 的坐标是 (x, y) ,点 P 的坐标是 ?x0 , y0 ? ,

由 PD ? 2MD 得, x0 ? x , y0 ? 2 y .……………………………………………………………1 分

因为点 P ?x0 , y0 ? 在圆 x 2 ? y 2 ? 4 上, 所以 x02 ? y02 ? 4 .

①………………………2 分

把 x0 ? x , y0 ? 2 y 代入方程①,得 x 2 ? 4 y 2 ? 4 .……………………………………………3 分

所以曲线 C 的方程为 x 2 ? y 2 ? 1.…………………………………………………………………4 分 4

(2)解:因为 EA ? EB,所以 EA ? EB ? 0 .…………………………………………………………5 分
? ?2
所以 EA? BA ? EA? EA? EB ? EA .……………………………………………………………7 分

? ? 设点 A

x1, y1

,则

x12 4

?

y12

? 1,即

y12

?1?

x12 4

.………………………………………………8



所以

EA? BA

?

2
EA

?

? x1

?1?2

?

y12

?

x12

?

2x1

?1?1?

x12 4

?

3 4

x12

?

2x1

?

2

?

3 4

? ??

x1

?

4 3

?2 ??

?

2 3

.……………………………………………………………10



因为点 A? x1, y1 ? 在曲线 C 上,所以 ?2 ? x1 ? 2 .………………………………………………11 分

所以

2 3

?

3 4

? ??

x1

?

4 3

?2 ??

?

2 3

?

9 .……………………………………………………………………13



所以

EA

?

BA

的取值范围为

? ??

2 3

,9???

.………………………………………………………………14



21.(本小题满分 14 分)
解:(1)因为 f (x) ? ln x ? ax2 ? (a ? 2)x ,
所以函数 f ? x? 的定义域为 (0, ??) .……………………ks5u……………………………1 分
9 / 10

广东省广州市 2014 届高三调研测试数学文试题-Word 版包含答案)

且 f ?(x) ? 1 ? 2ax ? (a ? 2) .………………………………………………………………………2 分 x
因为 f (x) 在 x ?1处取得极值,

所以 f ??1? ?1? 2a ? ?a ? 2? ? 0 .

解得 a ? ?1.…………………………………………………………………………………………3 分

当 a ? ?1时, f ?(x) ? 1 ? 2x ? 3 ? (2x ?1)(x ?1) ,

x

x

当 0 ? x ? 1 时, f ?(x) ? 0 ;当 1 ? x ? 1时, f ?(x) ? 0 ;当 x ?1时, f ?(x) ? 0 .

2

2

所以 x ?1是函数 y ? f (x) 的极小值点.

故 a ? ?1.……………………………………………………………………………………………4 分

(2)因为 a2 ? a ,

所以 0 ? a ? 1.…………………………………………………………………………………………5 分

由(1)知 f ?(x) ? ? (2x ?1)(ax ?1) . x

因为 x ?(0, ??) ,所以 ax ?1 ? 0 .

当 0 ? x ? 1 时, f ?(x) ? 0 ;当 x ? 1 时, f ?(x) ? 0 .

2

2

所以函数

f

(x)



? ??

0,

1 2

? ??

上单调递增;在

? ??

1 2

,

??

? ??

上单调递减.………………………………7



①当 0 ? a ? 1 时, f (x) 在[a2, a] 上单调递增, 2

所以? f (x)? ? f (a) ? ln a ? a3 ? a2 ? 2a .………………………………………………………9 分 max

②当

? ?? ? ?

a a2

? ?

1, 2 1.



1 2

?

a

?

2 2

时,

f

(x)



? ??

a2,

1 2

? ??

上单调递增,在

? ??

1 2

,

a

? ??

上单调递减,

?? 2

所以? f (x)? max

?

f

? ??

1 2

? ??

?

?

ln

2

?

a? 4

a?2 2

?

a 4

?1? ln 2 .……………………………………11 分

③当 1 ? a2 ,即 2 ? a ? 1时, f (x) 在[a2 , a] 上单调递减,

2

2

所以? f (x)? ? f (a2) ? 2ln a ? a5 ? a3 ? 2a2 .…………………………………………………13 分 max
综上所述:

当 0 ? a ? 1 时,函数 f (x) 在[a2, a] 上的最大值是 ln a ? a3 ? a2 ? 2a ; 2

当 1 ? a ? 2 时,函数 f (x) 在[a2, a] 上的最大值是 a ?1? ln 2 ;

2

2

4

当 2 ? a ? 1时,函数 f (x) 在[a2 , a] 上的最大值是 2ln a ? a5 ? a3 ? 2a2 .……ks5u……14 分 2

10 / 10


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