福建省福州市2014届高三上学期期末质量检测数学文试题

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福州市2013—2014学年第一学期高三期末质量检测

数学(文科)试卷 参考答案与评分标准

第Ⅰ卷 (选择题 共60分)

一、选择题(本大题共12小题,每小题5分,共60分.在每小题所给的四个答案中有且只有一个答案是正确的.把正确选项涂在答题卡的相应位置上.)

1. D 2.D 3. B 4.A 5. D 6. D 7. D 8. B 9. C. 10.C 11. C

12. A

第Ⅱ卷 (非选择题 共90分)

二.填空题(本大题共4小题,每小题4分,共16分.把答案填在答题卡的相应位置上.

13.16 14.9 15.42 16..②④

三、解答题(本大题共6小题,共74分,解答应写出文字说明、证明过程或演算过程.

17.(本小题满分12分)

解: (Ⅰ)xbxg2sin1)(22 ······················································· 2分

由0)(xg得Zkkxx202sin即 Zkkx2 ······················· 5分

故方程)(xg=0的解集为Zkkxx2 ······································· 6分

(Ⅱ)12sin3cos21)2sin,1()3,cos2(1)(22xxxxbaxf ···· 7分

)62sin(22sin32cosxxx ········································· 9分

∴函数)(xf的最小周期22T ···················································· 10分

由Zkkxk226222得Zkkxk63

故函数)(xf的单调增区间为Zkkk6,3. ( 开区间也可以)

···································································································· 12分

18. (本小题满分12分) 解:(Ⅰ)1111,033nnnnaaaanQ ks5u

1111==n13n13nnaaagQ,又 ······················································ 2分

nna11为首项为,公比为的等比数列33 ····································· 4分

n1n11n==n333nnaa, ····························································· 6分

(Ⅱ) 1231233333nnnSL……① ················································· 7分

231112133333nnnnnSL……② ········································ 8分

①-② 得:123121111333333nnnnSL ·························· 9分

1111331313nnn ······································· 10分

3114323nnnnS

133243nnnnS ··························································· 12分

19. (本小题满分12分)

. 解:(Ⅰ)设“从该批电器中任选1件,其为”B”型”为事件1A, ············· 1分

则15059()5010PA ································································· 3分

所以从该批电器中任选1件,求其为”B”型的概率为910. ·················· 4分

(Ⅱ)设“从重量在[80,85)的5件电器中,任选2件电器,求其中恰有1件为”A”型”为事件2A,记这5件电器分别为a,b,c,d,e,其中”A”型为a,b.从中任选2件,所有可能的情况为ab,ac,ad,ae,bc,bd,be,cd,ce,de,共10种. ······································································································ 8分

其中恰有1件为”A”型的情况有ac,ad,ae,bc,bd,be,共6种. ········ 10分

所以263()105PA.

所以从重量在[80,85)的5件电器中,任选2件电器,其中恰有1件为”A”型的概率为35. ······································································································· 12分

20.(本小题满分12分)

解:依题意得g(x)3x,设利润函数为f(x),则f(x)(x)g(x)r,

所以20.5613.5(0x7)f(x),10.5(x7)xxx ·································· 2分

(I)要使工厂有盈利,则有f(x)>0,因为

f(x)>0⇔20x770.5613.5010.50xxxx或, ····························· 4分

⇒20x771227010.50xxxx或⇒0x7710.539xx或

⇒3x7或7x10.5p, ·················································· 6分

即3x10.5p. ··································································· 7分

所以要使工厂盈利,产品数量应控制在大于300台小于1050台的范围内. ···· 8分

(II)当3x7时, 2f(x)0.5(6)4.5x

故当x=6时,f(x)有最大值4.5. ······················································ 10分

而当x>7时,f(x)10.573.5.

所以当工厂生产600台产品时,盈利最大. ·········································· 12分

21. (本小题满分12分)

解:(1)32fx=2xxaxQ() '2fx=34xxa() ············ 2分

对于xR恒有2'()224fxxx,即2240xxa对于xR恒成立

····································································································· 4分 44(4)0a 3a ······················································· 5分

max3a ··················································································· 6分

(2)a=3Fx=()fxkxQ当时()有三个零点

3224kxxx有三个不同的实根 ··············································· 7分

32()24gxxxx令,则2'()=3x4x4gx ···························· 8分

令'()0gx解得1222,3xx

,'(),()xgxgx情况如下表:

x (,2) 2 2(2,)3 23 2(,)3

'()gx + 0 - 0 +

()gx 单调递增 极大值8 单调递减 极小极4027 单调递增

······································································································ 10分

由上表知,当2x时()gx取得极大值(2)8g,当23x时()gx取得极小值240()327g

数形结合可知,实数k的取值范围为40(,8)27 ········································· 12分

22. (本小题满分14分)

解:(I)设双曲线C的方程为22221(00)xyabab,, ························ 1分

由题设得2295.2abba, ······························································· 3分

解得2245.ab,, ········································································· 5分 所以双曲线C的方程为22145xy; ··········································· 6分

(II)设直线l的方程为(0)ykxmk,点11()Mxy,,22()Nxy,的坐标满足方程组221.45ykxmxy, ① ②,将①式代入②式,得22()145xkxm,