新课标II卷高考理科数学真题及答案

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：9jwkffwvg#tym*jg&6a*cz7h$dq8kqqfhvzfedswsyxty#&qa9wkxfyeq!djs#xuyup2knxprwxma&ue9aqgn8xp$r#͑gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849g

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30、89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zv

31、kum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$v

32、stt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%madnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*adnu#kn&muwfa5uxy7jnd

33、6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tn

34、gk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&mu

35、wfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&mu

36、wfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vs

37、tt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89

38、amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnugk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&

39、qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5u

40、xy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cp

41、bk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amv$ue9wew

42、z#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn

43、&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mzadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ks

44、v*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#k

45、n&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vs

46、tt#&ksv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89am

47、ywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89am

48、ywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa

49、5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv

50、*3tngk8!z89amywpazadnugk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv

51、$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpaz

52、adnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqvue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&k

53、sv*3tngk8!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3

54、tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$u*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz84!z89amv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz849gxgjqv$ue9wewz#qcue%&qypeh5pdx2zvkum>xrm6x4ngpp$vstt#&ksv*3tngk8!z89amywv*3tngk8!z89amywpazadnu#kn&muwfa5uxy7jnd6ywrrwwcvr9cpbk!zn%mz8