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1、2014年普通高等学校招生全国统一考试(新课标卷)理科数学注意事项:1本试卷分第卷(选择题)和第卷(非选择题)两部分。答卷前,考生务必将自己的姓名、准考证号填写在答题卡上。2回答第卷时,选出每小题答案后,用铅笔把答题卡上对应题目的答案标号涂黑。如需改动,用橡皮擦干净后,再选涂其他答案标号。写在本试卷上无效。3回答第卷时,将答案写在答题卡上,写在本试卷上无效。4考试结束后,将本试卷和答题卡一并交回。第卷一选择题:本大题共12小题,每小题5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)设集合,则(A) (B) (C) (D) 解析:,答案:D(2)设复数,在复平面内的对应点关于虚轴
2、对称,则(A) (B) (C) (D) 解析:,答案:A(3)设向量,满足,则(A) 1(B) 2(C) 3(D) 5解析:,由得:答案:A(4)钝角三角形的面积是,则(A) 5(B) (C) 2(D) 1解析:,即:,即或又或5,又为钝角三角形,即:答案:B(5)某地区空气质量监测资料表明,一天的空气质量为优良的概率是0.75,连续两天为优良的概率是0.6,已知某天的空气质量为优良,则随后一天的空气质量为优良的概率是(A) 0.8(B) 0.75(C) 0.6(D) 0.45解析:此题为条件概率,所以答案:A(6)如图,网格纸上正方形小格的边长为1(表示1cm),图中粗线画出的是某零件的三视
3、图,该零件有一个底面半径为3cm,高为6cm的圆柱体毛坯切削得到,则切削掉部分的体积与原来毛坯体积的比值为(A) (B) (C) (D) 解析:原来毛坯体积为:,由三视图得,该零件由左侧底面半径为2cm,高为4cm的圆柱和右侧底面半径为3cm,高为2cm的圆柱构成,所以该零件的体积为:,则切削掉部分的体积为,所以切削掉部分的体积与原来毛坯体积的比值为结束输出,开始输入,是否答案:C(7)执行右面的程序框图,如果输入的,均为2,则输出的(A) (B) (C) (D) 解析:输入的,均为2是,;是,否,输出答案:D(8)设曲线在点处的切线方程为,则(A) (B) (C) (D) 解析:,且在点处的
4、切线的斜率为2,即答案:D(9)设,满足约束条件,则的最大值为(A) (B) (C) (D) 解析:作出,满足约束条件表示的平面区域如图阴影部分:做出目标函数:,当的截距最小时,有最大值。当经过点时,有最大值。由得:此时:有最大值答案:B(10)设为抛物线:的焦点,过且倾斜角为的直线交于,两点,为坐标原点,则的面积为(A) (B) (C)(D)解析:,设、,直线的方程为,代入抛物线方程得:, 由弦长公式得 由点到直线的距离公式得:到直线的距离 答案:D(11)直三棱柱中,分别是,的中点, ,则与所成角的余弦值为(A) (B) (C) (D) 解析:如图所示,取的中点,连结、 ,分别是,的中点,
5、 四边形为平行四边形, 所求角的余弦值等于的余弦值 不妨令,则 ,答案:C(12)设函数若存在的极值点满足,则的取值范围是(A) (B) (C) (D)解析:,令得:,又,即:,故:,即:,故:或答案:C第卷本卷包括必考题和选考题两部分。第13题第21题为必考题,每个试题考生都必须做答。第22题第24题为选考题,考生根据要求做答。二填空题:本大题共4小题,每小题5分。(13)的展开式中,的系数为,则 (用数字填写答案)解析:,即,解之:答案:(14)函数的最大值为 解析:的最大值为1答案:1 (15)已知偶函数在单调递减,若,则的取值范围是 解析:是偶函数,又在单调递减,解之:答案: (16)
6、设点,若在圆:上存在点,使得,则的取值范围是 解析:由图可知点所在直线与圆相切,又,由正弦定理得:,即:又,即,解之:答案: 三解答题:解答应写出文字说明,证明过程或演算步骤。(17)(本小题满分12分)已知数列满足,()证明是等比数列,并求的通项公式;()证明解析:()证明:,即:又,是以为首项,3为公比的等比数列,即()证明:由()知,故:(18)(本小题满分12分)如图,四棱锥中,底面为矩形,平面,为的中点()证明:平面;()设二面角为,求三棱锥的体积解析:()证明:连结交于点,连结 底面为矩形,点为的中点,又为的中点,平面,平面,平面()以为原点,直线、分别为、轴建立空间直角坐标系,设
7、,则, ,设是平面的法向量, 则,解之:,令,得又是平面的一个法向量,解之(19)(本小题满分12分)某地区2007年至2013年农村居民家庭人均纯收入y(单位:千元)的数据如下表:年份2007200820092010201120122013年份代号1234567人均纯收入y2.93.33.64.44.85.25.9()求y关于的线性回归方程;()利用()中的回归方程,分析2007年至2013年该地区农村居民家庭人均纯收入的变化情况,并预测该地区2015年农村居民家庭人均纯收入附:回归直线的斜率和截距的最小二乘估计公式分别为:,解析:()由题意得:, 故:所求线性回归方程为:()由()中的回归
8、方程的斜率可知,2007年至2013年该地区农村居民家庭人均纯收入逐渐增加令得:,故:预测该地区2015年农村居民家庭人均纯收入为6.8千元。(20)(本小题满分12分)设,分别是椭圆:的左,右焦点,是上一点且与轴垂直直线与的另一交点为()若直线的斜率为,求的离心率;()若直线在轴上的截距为2,且,求,解析:()由题意得:,的斜率为 ,又,解之:或(舍) 故:直线的斜率为时,的离心率为()由题意知:点在第一象限,直线的斜率为:,则:;在直线上,得,且,又在椭圆上,联立、解得:,(21)(本小题满分12分) 已知函数()讨论的单调性;()设,当时,求的最大值;()已知,估计的近似值(精确到0.0
9、01)解析:(),在上单调递增请考生在第22、23、24题中任选一题做答,如果多做,则按所做的第一题给分。做答时请写清题号。(22)(本小题满分10分)选修:几何证明选讲如图,是外一点,是切线,为切点,割线与相交与,为的中点,的延长线交与点证明:()()(23)(本小题满分10分)选修:坐标系与参数方程 在直角坐标系中,以坐标原点为极点,轴正半轴为极轴建立极坐标系,半圆的极坐标方程为,()求的参数方程;()设点在上,在处的切线与直线:垂直,根据()中你得到的参数方程,确定的坐标解析:()设点是曲线上任意一点, ,即: 的参数方程为,为参数 ()设点,在处的切线与直线:垂直 ,又,解之:或 点的
10、坐标为:或(24)(本小题满分10分)选修:不等式选讲 设函数()证明:;()若,求的取值范围解析:(),且 ,当且仅当时,取“” 故: (), 即: 或 解之:6a*CZ7H$dq8KqqfHVZFedswSyXTy#&QA9wkxFyeQ!djs#XuyUP2kNXpRWXmA&UE9aQGn8xp$R#͑GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcU
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27、nD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3
28、tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&
29、MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&
30、MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$
31、vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z
32、89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%MadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*adNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!z
33、n%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnG
34、K8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEw
35、Z#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrW
36、wcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z8
37、9AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89A
38、mYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&g
39、TXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9
40、CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv
41、$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx
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