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Zkkxx202sin即 Zkkx2 ······················· 5分
故方程)(xg=0的解集为Zkkxx2 ······································· 6分
(Ⅱ)12sin3cos21)2sin,1()3,cos2(1)(22xxxxbaxf ···· 7分
)62sin(22sin32cosxxx ········································· 9分
∴函数)(xf的最小周期22T ···················································· 10分
由Zkkxk226222得Zkkxk63
故函数)(xf的单调增区间为Zkkk6,3. ( 开区间也可以)
···································································································· 12分
18. (本小题满分12分) 解:(Ⅰ)1111,033nnnnaaaanQ ks5u
1111==n13n13nnaaagQ,又 ······················································ 2分
nna11为首项为,公比为的等比数列33 ····································· 4分
n1n11n==n333nnaa, ····························································· 6分
(Ⅱ) 1231233333nnnSL……① ················································· 7分
231112133333nnnnnSL……② ········································ 8分
①-② 得:123121111333333nnnnSL ·························· 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7x10.5p, ·················································· 6分
即3x10.5p. ··································································· 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=2xxaxQ() '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=()fxkxQ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,