![](https://wkrtcs.bdimg.com/rtcs/image?w=542&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"542","h":"806","dataType":"png","c":"word/media/image1.png","imgOriW":2480,"imgOriH":3507})
2014年黄冈市3月份调研考试
数学试题答案（理科）
一、选择题
ABBAC CBDCD
10.提示：由题意得![](https://wkrtcs.bdimg.com/rtcs/image?w=28&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"28","h":"24","dataType":"png","c":"word/media/image5_1.png","imgOriW":672,"imgOriH":576})
只能在线段BC上取得最大值，即有，则由柯西不等式或均值不等式取最大值
二、填空题
11. 2![](https://wkrtcs.bdimg.com/rtcs/image?w=24&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"24","h":"24","dataType":"png","c":"word/media/image8_1.png","imgOriW":635,"imgOriH":635})
12. 13. 7 14. 15. 5 16.
14．提示：由题意有![](https://wkrtcs.bdimg.com/rtcs/image?w=100&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"100","h":"21","dataType":"png","c":"word/media/image12_1.png","imgOriW":2400,"imgOriH":512})
，。对于抛物线，不论a,b取何值，图形形状不变，所围成的面积为一定值。故令，则所围成图形面积为36，概率为
三、解答题
17. （1）由已知得![](https://wkrtcs.bdimg.com/rtcs/image?w=188&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"188","h":"41","dataType":"png","c":"word/media/image17_1.png","imgOriW":2909,"imgOriH":639})
![](https://wkrtcs.bdimg.com/rtcs/image?w=219&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"219","h":"67","dataType":"png","c":"word/media/image18_1.png","imgOriW":2498,"imgOriH":762})
化简得 ![](https://wkrtcs.bdimg.com/rtcs/image?w=281&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"281","h":"44","dataType":"png","c":"word/media/image19_1.png","imgOriW":3490,"imgOriH":546})
，……6分
![](https://wkrtcs.bdimg.com/rtcs/image?w=430&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"430","h":"270","dataType":"png","c":"word/media/image21_1.png","imgOriW":1796,"imgOriH":1129})
18 （1）![](https://wkrtcs.bdimg.com/rtcs/image?w=236&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"236","h":"24","dataType":"png","c":"word/media/image22_1.png","imgOriW":4204,"imgOriH":428})
两式相减得，所以，是公差为2的准等差数列当为奇数时，当为偶数时，
![](https://wkrtcs.bdimg.com/rtcs/image?w=198&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"198","h":"48","dataType":"png","c":"word/media/image29_1.png","imgOriW":2677,"imgOriH":655})
………………………………6分
（2） 在S63=![](https://wkrtcs.bdimg.com/rtcs/image?w=13&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"13","h":"14","dataType":"png","c":"word/media/image30_1.png","imgOriW":13,"imgOriH":14})
1+2+...+63中，有32个奇数项和31个偶数项，
![](https://wkrtcs.bdimg.com/rtcs/image?w=379&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"379","h":"24","dataType":"png","c":"word/media/image31_1.png","imgOriW":5410,"imgOriH":343})
![](https://wkrtcs.bdimg.com/rtcs/image?w=267&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"267","h":"24","dataType":"png","c":"word/media/image33_1.png","imgOriW":4489,"imgOriH":404})
....................................12分
19.(1)依题意，得
该同学恰有2所通过概率![](https://wkrtcs.bdimg.com/rtcs/image?w=251&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"251","h":"41","dataType":"png","c":"word/media/image34_1.png","imgOriW":3367,"imgOriH":555})
（2）设该同学共参加了i次考试的概率为![](https://wkrtcs.bdimg.com/rtcs/image?w=117&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"117","h":"29","dataType":"png","c":"word/media/image35_1.png","imgOriW":2816,"imgOriH":576})
![](https://wkrtcs.bdimg.com/rtcs/image?w=128&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"128","h":"107","dataType":"png","c":"word/media/image36_1.png","imgOriW":1508,"imgOriH":1257})
…………9分
![](https://wkrtcs.bdimg.com/rtcs/image?w=291&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"291","h":"85","dataType":"png","c":"word/media/image37_1.png","imgOriW":2583,"imgOriH":758})
利用错位相减法可得
![](https://wkrtcs.bdimg.com/rtcs/image?w=492&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"492","h":"128","dataType":"png","c":"word/media/image38_1.png","imgOriW":2779,"imgOriH":723})
答：该同学参加考试所需费用![](https://wkrtcs.bdimg.com/rtcs/image?w=13&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"13","h":"21","dataType":"png","c":"word/media/image39_1.png","imgOriW":352,"imgOriH":564})
期望为元................................12分
20.（1）由已知得![](https://wkrtcs.bdimg.com/rtcs/image?w=192&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"192","h":"25","dataType":"png","c":"word/media/image41_1.png","imgOriW":3697,"imgOriH":488})
在矩形![](https://wkrtcs.bdimg.com/rtcs/image?w=48&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"48","h":"19","dataType":"png","c":"word/media/image42_1.png","imgOriW":1270,"imgOriH":493})
中，……………………2分
![](https://wkrtcs.bdimg.com/rtcs/image?w=213&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"213","h":"41","dataType":"png","c":"word/media/image44_1.png","imgOriW":3109,"imgOriH":602})
在![](https://wkrtcs.bdimg.com/rtcs/image?w=47&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"47","h":"19","dataType":"png","c":"word/media/image45_1.png","imgOriW":1234,"imgOriH":493})
中，……………5分
![](https://wkrtcs.bdimg.com/rtcs/image?w=391&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"391","h":"23","dataType":"png","c":"word/media/image47_1.png","imgOriW":5685,"imgOriH":329})
………………6分
![](https://wkrtcs.bdimg.com/rtcs/image?w=172.46666666667&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"172.46666666667","h":"118.6","dataType":"jpeg","c":"word/media/image48.png","imgOriW":96,"imgOriH":66})
（2）。
以![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"16","h":"19","dataType":"png","c":"word/media/image50_1.png","imgOriW":423,"imgOriH":493})
为原点，建立如图所示的空间直角坐标系，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=199&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"199","h":"25","dataType":"png","c":"word/media/image51_1.png","imgOriW":3765,"imgOriH":480})
![](https://wkrtcs.bdimg.com/rtcs/image?w=211&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"211","h":"27","dataType":"png","c":"word/media/image52_1.png","imgOriW":3794,"imgOriH":480})
………………8分
设![](https://wkrtcs.bdimg.com/rtcs/image?w=161&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"161","h":"32","dataType":"png","c":"word/media/image53_1.png","imgOriW":3022,"imgOriH":599})
的法向量。则即
取![](https://wkrtcs.bdimg.com/rtcs/image?w=261&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"261","h":"32","dataType":"png","c":"word/media/image56_1.png","imgOriW":3906,"imgOriH":478})
为平面的一个法向量。
![](https://wkrtcs.bdimg.com/rtcs/image?w=283&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"283","h":"76","dataType":"png","c":"word/media/image58_1.png","imgOriW":2693,"imgOriH":724})
故二面角![](https://wkrtcs.bdimg.com/rtcs/image?w=79&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"79","h":"17","dataType":"png","c":"word/media/image59_1.png","imgOriW":2081,"imgOriH":458})
的余弦值为………………………………12分
21.设![](https://wkrtcs.bdimg.com/rtcs/image?w=51&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"51","h":"21","dataType":"png","c":"word/media/image61_1.png","imgOriW":1340,"imgOriH":564})
直线由坐标原点到的距离
为![](https://wkrtcs.bdimg.com/rtcs/image?w=235&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"235","h":"48","dataType":"png","c":"word/media/image65_1.png","imgOriW":3045,"imgOriH":623})
又![](https://wkrtcs.bdimg.com/rtcs/image?w=84&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"84","h":"45","dataType":"png","c":"word/media/image66_1.png","imgOriW":1811,"imgOriH":977})
所以……………………………………5分
（2）由（1）知椭圆的方程为![](https://wkrtcs.bdimg.com/rtcs/image?w=99&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"99","h":"44","dataType":"png","c":"word/media/image68_1.png","imgOriW":2002,"imgOriH":893})
。设由题意知的斜率一定不为0，故不妨设代入椭圆的方程中整理得，显然由韦达定理有①……………………………7分
假设存在点![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"16","h":"17","dataType":"png","c":"word/media/image75_1.png","imgOriW":423,"imgOriH":458})
，使成立，则其充要条件为：点的坐标为，点在椭圆上，即整理得
![](https://wkrtcs.bdimg.com/rtcs/image?w=265&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"265","h":"25","dataType":"png","c":"word/media/image80_1.png","imgOriW":4365,"imgOriH":417})
又![](https://wkrtcs.bdimg.com/rtcs/image?w=43&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"43","h":"19","dataType":"png","c":"word/media/image81_1.png","imgOriW":1128,"imgOriH":493})
在椭圆上，即故②
将![](https://wkrtcs.bdimg.com/rtcs/image?w=335&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"335","h":"24","dataType":"png","c":"word/media/image84_1.png","imgOriW":5096,"imgOriH":365})
及①代入②，解得……11分
所以![](https://wkrtcs.bdimg.com/rtcs/image?w=329&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"329","h":"45","dataType":"png","c":"word/media/image86_1.png","imgOriW":3736,"imgOriH":514})
，即
当![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"56","h":"45","dataType":"png","c":"word/media/image88_1.png","imgOriW":1481,"imgOriH":1199})
时，
当![](https://wkrtcs.bdimg.com/rtcs/image?w=68&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"68","h":"45","dataType":"png","c":"word/media/image90_1.png","imgOriW":1615,"imgOriH":1076})
时，……………………………13分
22.解：（1）![](https://wkrtcs.bdimg.com/rtcs/image?w=181&md5sum=62ba20868ed45f59d285e6331bcc54eb&sign=de11e1c12f&rtcs_flag=1&rtcs_ver=3&l=webapp&bucketNum=97&ipr={"t":"img","w":"181","h":"21","dataType":"png","c":"word/media/image92_1.png","imgOriW":3882,"imgOriH":456})
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