09理数B-6.20

1、绝密启用前试卷类型：B2009年普通高等学校招生全国统一考试（广东卷）数学（理科）本试卷共4页，21小题，满分150分。考试用时120分钟。注意事项：1 答卷前，考生务必用黑色字迹的钢笔或签宇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（B）填涂在答题卡相应位置上。将条形码横贴在答题卡右上角“条形码粘贴处”。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 区域内相应位置上；如需改动，先划掉原来的答

2、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组号对应的信息点，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。1参考公式：锥体的体积公式 V sh，其中S是锥体的底面积，h是锥体的高3一、选择题：本大题共 8小题，每小题5分，满分40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巳知全集U =R，集合M =x 2兰x1兰2和N =xx = 2k1,k=1,2,|&的关系的韦恩（V enn）图如图1所示，则阴影部分所示的集合的元素共有A. 3个E.2个

3、C.1个D.无穷个 2 设z是复数，a（z）表示满足Zn =1的最小正整数n ,则对虚数单位 i , a（i）=A.8E.6C.4D.23若函数y=f（x）是函数y=ax（a0,且a=1）的反函数，其图像经过点（、-a,a），则f （x）二A. Iog2xb. log1 x2C.2xD.x24 巳知等比数列耳满足an 0, n = 1,2)1 ,且a5 a2_22n (n 3),则当n1时，logzar gas 川loga 1 =A. n(2n -1)b. (n 1)2d. (n-1)2C. n2数学（理科）试题 8第1页（共4页）5. 给定下列四个命题： 若一个平面内的两条直线与另一个平面都

4、平行，那么这两个平面相互平行； 若一个平面经过另一个平面的垂线，那么这两个平面相互垂直； 垂直于同一直线的两条直线相互平行； 若两个平面垂直，那么一个平面内与它们的交线不垂直的直线与另一个平面也不垂直.其中， 为真命题的是A.和E.和C.和D.和6. 一质点受到平面上的三个力F, F2, F3 （单位：牛顿）的作用而处于平衡状态. 已知F,F2成600角，且F, , F2的大小分别为2和4,贝y F3的大小为A.6B.2C. 2 5D. 2 77. 2010年广州亚运会组委会要从小张、小赵、小李、小罗、小王五名志愿者中选派四人分别从 事翻译、导游、礼仪、司机四项不同工作，若其中小张和小赵只能从

5、事前两项工作，其余三人均 能从事这四项工作，则不同的选派方案共有A. 36 种B. 12 种C. 18 种D. 48 种&已知甲、乙两车由同一起点同时出发，并沿同一路线假定为直线）行驶.甲车、乙车的速度曲线分别为V甲和V乙（如图2所示）.那么对于图中给定的t0和t1，下列判断中一定正确的是A .在匕时刻，甲车在乙车前面B . E时刻后，甲车在乙车后面C .在t时刻，两车的位置相同D . t0时刻后，乙车在甲车前面二、填空题：本大题共 7小题，考生作答 6小题，每小题5分， 满分30分.（一）必做题（912题）9 .随机抽取某产品n件，测得其长度分别为 a,a2,lil,an，则图3所示的程序框

6、图输出的 S，表示的样本的数字特征是.（注：框图中的赋值符号 “=”也可以写成=”）10 .若平面向量a,b满足a+b=1 , a+b平行于x轴，b =(2, 1)贝U a =11.巳知椭圆G的中心在坐标原点,长轴在x轴上，离心率为2且G上一点到G的两个焦点的距离之和为 12,则椭圆G的方程为 数学（理科）试题 B第2页（共4页）12 .已知离散型随机变量 X的分布列如右表.若 EX =0 ,DX =1，贝y a =, b =.（二）选做题（13 15题，考生只能从中选做两题）13 .（坐标系与参数方程选做题）若直线li：x =1 2tyaJ为参数）与直线I2：X=s( s为参数)y =1 -

7、 2s.I12垂直，则k =14.（不等式选讲选做题）不等式乂1 1的实数解为x 215 .（几何证明选讲选做题）如图4,点A, B,C是圆O上的点,AB = 4,ACB = 45，则圆O的面积等于三、解答题：本大题共 6小题，满分80分.解答须写出文字说明、证 过程和演算步骤，16.（本小题满分12分）已知向量a = （si n0,2）与b=（1,cosT）互相垂直，其中日壬（0,二）.2（1）求 sin 二禾口 cos 二的值;/10(2)若 sin(一)10,0 八103T，求cos的值.217.（本小题满分12分）根据空气质量指数 API （为整数） 对某城市一年（365天）的不同，可

8、将空气质量分级如下表：的空气质量进行监测，获得API数据按照0,50,(50,100,(100,150,(150,200,(200,250,(250,300进行分组,得到频率分布直方图如API05051 100101 2 150151 - 200201 - 250251 - 300 300级别I111112V狀况优良轻微污染轻度污染中度污染中度重污染重度污染 1（1）求直方图中X的值；（2）计算一年屮空气质量分别为良和轻微污染的天数；（3）求该城市某一周至少有 2天的空气质量为良或轻微污染的概率(结果用分数表示已知77327381235 =78125,2 =128,365 =73 5)1825

9、3651825182591259125数学理科)试题 B第3页(共4页)18 (本小题满分14分)如图6,已知正方体 ABCD-A1BGD1的棱长为2,点E是正方形 BCGB的中心，点f、g分别是棱GUAA的中点设点 E1,G1分别是点e、g在平面 DCC1D1内的正投影.02：(1) 求以E为顶点，以四边形 FGAE在平面DCC1D1内的 投影为底面边界的棱锥的体积；(2) 证明：直线 FG _平面FEE1;SI6(3) 求异面直线 巳与EA所成角的正统值19 (本小题满分14分) 已知曲线C : y = X2与直线丨：x-y 2 = 0交于两点A(Xa,a)和B(Xb,b)，且Xa ： X

10、b .记曲 线C在点A和点B之间那一段L与线段AB所围成的平面区域(含边界)为D 设点P(s,t)是L 上的任一点，且点 P与点A和点B均不重合.(1)若点Q是线段AB的中点，试求线段 PQ的中点M的轨迹方程;22251_(2)若曲线G : X -2aX y -4y a0与点D有公共点，试求a的最小值.2520 (本小题满分14分)已知二次函数y =g(x)的导函数的图像与直线 y =2x平行，且y=g(x)在x -1处取得极小值m -1(m =0) 设 f (x)x(1)若曲线y = f(X)上的点P到点Q(0, 2)的距离的最小值为.2，求m的值;(2) k(k R)如何取值时，函数 y=

11、 f(x)-kx存在零点，并求出零点.21 (本小题满分14分)已知曲线Cn:x2-2 nx，y2=0( n=1,2,ll|)从点P(-1,0)向曲线Cn引斜率为kn(k,0)的切线丨n，切点为Pn(Xn,yn) (1)求数列xn与 yn的通项公式;(正明：xzn*店亍屁吟Whe n you are old and grey and full of sleep,And no ddi ng by the fire, take dow n this book,And slowly read, and dream of the soft lookYour eyes had once, and of

12、their shadows deep;How many loved your mome nts of glad grace,And loved your beauty with love false or true,But one man loved the pilgrim soul in you,And loved the sorrows of your cha nging face;And bending dow n beside the glow ing bars,Murmur, a little sadly, how love fledAnd paced upon the mounta

13、ins overheadAnd hid his face amid a crowd of stars.The furthest dista nee in the worldIs not betwee n life and deathBut whe n I sta nd in front of youYet you dont know thatI love you.The furthest dista nee in the worldIs not whe n I sta nd in front of youYet you cant see my loveBut whe n un doubtedl

14、y knowing the love from bothYet cannot be together.The furthest dista nee in the worldIs not being apart while being in loveBut whe n I pla inly cannot resist the year ningYet prete nding you have n ever bee n in my heart.The furthest dista nee in the worldIs not struggli ng aga inst the tidesBut us

15、ing on es in differe nt heartTo dig an un crossable riverFor the one who loves you.Whe n you are old and grey and full of sleep,And no ddi ng by the fire, take dow n this book,And slowly read, and dream of the soft lookYour eyes had once, and of their shadows deep;How many loved your mome nts of gla

16、d grace,And loved your beauty with love false or true,But one man loved the pilgrim soul in you,And loved the sorrows of your cha nging face;And bending dow n beside the glow ing bars,Murmur, a little sadly, how love fledAnd paced upon the mountains overheadAnd hid his face amid a crowd of stars.The

17、 furthest dista nee in the worldIs not betwee n life and deathBut whe n I sta nd in front of youYet you dont know thatI love you.The furthest dista nee in the worldIs not whe n I sta nd in front of youYet you cant see my loveBut whe n un doubtedly knowing the love from bothYet cannot be together.The

18、 furthest dista nee in the worldIs not being apart while being in loveBut whe n I pla inly cannot resist the year ningYet prete nding you have n ever bee n in my heart.The furthest dista nee in the worldIs not struggli ng aga inst the tidesBut using on es in differe nt heartTo dig an un crossable ri

19、verFor the one who loves you.Whe n you are old and grey and full of sleep,And no ddi ng by the fire, take dow n this book,And slowly read, and dream of the soft lookYour eyes had once, and of their shadows deep;How many loved your mome nts of glad grace,And loved your beauty with love false or true,

20、But one man loved the pilgrim soul in you,And loved the sorrows of your cha nging face;And bending dow n beside the glow ing bars,Murmur, a little sadly, how love fledAnd paced upon the mountains overheadAnd hid his face amid a crowd of stars.The furthest dista nee in the worldIs not betwee n life a

21、nd deathBut whe n I sta nd in front of youYet you dont know thatI love you.The furthest dista nee in the worldIs not whe n I sta nd in front of youYet you cant see my loveBut whe n un doubtedly knowing the love from bothYet cannot be together.The furthest dista nee in the worldIs not being apart whi

22、le being in loveBut whe n I pla inly cannot resist the year ningYet prete nding you have n ever bee n in my heart.The furthest dista nee in the worldIs not struggli ng aga inst the tidesBut using on es in differe nt heartTo dig an un crossable riverFor the one who loves you.Whe n you are old and gre

23、y and full of sleep,And no ddi ng by the fire, take dow n this book,And slowly read, and dream of the soft lookYour eyes had once, and of their shadows deep;How many loved your mome nts of glad grace,And loved your beauty with love false or true,But one man loved the pilgrim soul in you,And loved the sorrows of your cha nging face;And bending dow