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www.x740.com:2016年成都三诊数学文科试题及答案,四川省成都市2013届高三第三次诊断考试数学(文)试题及答案.doc_图文

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三、解答题:(本大题共 6 小题,共 75 分)

16. 解:(I) f (x) ? 3 sin 2x ? sin(2x ? ? ) ? 3 2

? 3 s i nx2? c oxs?2 ? 3 2xs?i?n ( 2? , ) …3…………………3 分 6

2k? ? ? ≤ 2x? ?≤ 2?k ? ? (?k Z,)

2

6

2

得 k? ? ? ≤x≤k? ? ? (k ? Z )

3

6

∴函数

f

(x)

的单调递增区间为

???k?

?

? 3

, k?

?

? 6

? ??

(k

?

Z)

.

……………………6 分

(II)由(I), f ( A) 1? 3?

,即

sin

? ??

2

A

?

? 6

? ??

?

1 2

.

∵V

1 ? 3 S梯形ABHG

FC, S梯形ABHG

1 ? S?ABC ? S?GHC , BC ? CF ? 2 AB ? 1,

∴ S梯形ABHG =S?ABC

? S?GHC

?

1 ?1? 2

3?1?1? 3 ? 3 3 22 2 8

………………10 分

∴V

?

1 3 S梯形ABHG

FC= 1 ? 3 3 ?1= 38

3 8

…………………12 分

18. 解:(I)用 A 表示“从这 20 名参加测试的学生中随机抽取一名,抽到语言表达能力

优秀或逻辑思维能力优秀的学生”.

∵语言表达能力优秀或逻辑思维能力优秀的学生共有(6+n)名.

∴P (A) ? 6 ? n ? 2 ,解得 n=2. 20 5

∴m=4.

……………6 分

(II)∴用 B 表示“从参加测试的语言表达能力良好的学生中任意抽取 2 名,其中至少

有一名逻辑思维能力优秀的学生”.

∵从参加测试的语言表达能力良好的学生中任意抽取 2 名有 36 种情况,没有逻辑

思维能力优秀的学生的情况有 15 种,

………………10 分

(II)易知直线 l1 , l2 的斜率存在且不为 0. 设直线 l1 的斜率为 k , A(x1, y1), B(x2 , y2 ) .

则直线 l1

的方程为

y

?

k(x

?1) ,

P(

x1

? 2

x2

,

y1

? 2

y2

)

? y2?4x
由 y?k ( x?1)

消去 y , 得 k 2 x2 ? (2k 2 ? 4)x ? k 2 ? 0 .

……………7 分

? ? (2k 2 ? 4)2 ? 4k 4 ? 16k 2 ?16 ? 0



x1

?

x2

?

2

?

4 k2

,

y1

?

y2

?

k ( x1

?

x2

?

2)

?

4 k

…………8 分

由 f '(x) ? 0, 解得1 ? x ? ?a ? 3. ∴ f (x)在(1,? a ? 3)上单调递减.. 综上所述,当 a ? ?4 时, f (x) 在 R 上单调递增;

当 a ? ?4 时, f (x) 在 ???, ?a ?3?,(1, ??)上单调递增;

??a ?3,1?上单调递减;

当 a ? ?4时, f (x) 在 ???,1?,(?a ?3, ??)上单调递增;

?1, ?a ?3?上单调递减;

(Ⅱ) ∵ x ?[0,1] , 由(I)可知

①当 a≤? 4 时, f (x)在x ?[0,1]上单调递增.

∵函数 f (x) 的图象恒在直线 y ? e 的上方,



f

( x)min

?

f

(0) ?

?2a ? 3 ? e, 解得a ? ? 3 ? 2

e 2

.

而 ? 3 ? e ? ?4 ,∴ a≤? 4 . 22

②当 ?4 ? a ? ?3时, 0 ? ?a ?3 ?1.

∴ f (x) 在 (0, ?a ? 3) 上单调递增,在 (?a ? 3,1) 上单调递减.

∵函数 f (x) 的图象恒在直线 y ? e 的上方,


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