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元三维高中 2023 级第二次诊断性考试
数学参考答案及评分标准
一、选择题：本题共8小题，每小题5分，共40分．
1．B 2．A 3．C 4．D 5．D 6．A 7．C 8．B
二、选择题：本大题共3小题，每小题6分，共18分．全部选对的得6分，选对但不
全的得部分分，有选错的得0分．
9．BC 10．BCD 11．ABD
三、填空题：本题共3个小题，每小题5分，共15分．
12．1； 13．65； 14．
四、解答题：本题共5小题，第15题13分，第16、17小题15分，第18、19小题17
分，共77分．解答应写出文字说明、证明过程或演算步骤．
15．解：（1）∵ ，由正弦定理： ，····································1分
又 ，
∴ ，···············································································2分
∴ ，又 是三角形的内角，························································3分
∴ ，········································································6分
∴ ；··················································································7分
（2）由余弦定理： ，··············································8分
∴ ，·································································9分
∴ ，·············································································11分
∴ 或 ．···············································································13分
16．（1）设等比数列 的公比为 ，
第 1 页 共 7 页∵ ，则 ，·························································3分
化简得 ，解得 ，··························································5分
∴ ；·····································································7分
（2）∵ ，则 ，····························9分
∵ ，
∴ ，·································································11分
∴ ，··················································································12分
∴n可以为全体正的偶数及大于等于3的奇数，
∴n的取值范围为 ．···················································15分
17．解：（1）∵ ，·························································2分
∴ ，且 ，································································3分
∴所求切线方程为： ，即： ；···········5分
（2）∵ ，所以 ，
则 为 的最小值，也是 的极小值，········································7分
，·········································································8分
∴ ，即： ，·························································9分
下面验证： 时， 的最小值为e．
令 ，
∴令 ，则 ，··················································9分
第 2 页 共 7 页∴ ， ， 为减函数，且 ；
， ， 为增函数，······································10分
， ，·······································11分
∴ 必有唯一零点且等于1，························································12分
∴ ， ， 为减函数；
， ， 为增函数，··········································14分
∴ 时， ．·····································································15分
18．解：（1）∵ ，则 ，
又离心率e= ，则 ，
解得： ，
∴椭圆的标准方程为： ；························································4分
（2）设 ， ， ，
则由 ，且 ， ，························5分
∴ ，则 ，··················································6分
代入椭圆方程，可得： ……….①······························7分
又 ……………….②
联立①和②可解得： ，故P(2，1)，A( )，····················8分
第 3 页 共 7 页又N(2，0)，则B(2，−1)，故直线AB的斜率 ；································9分
（3）设 ， ， ，则 ，
故直线PA：
联立方程： ·································································10分
可得： ，······················11分
又 ，代入上式化简整理可得：
由韦达定理： ，解得： ，
代入直线PA： ，解得： ，
故 ，······································································12分
同理：直线PB斜率 ，故直线PB： ，
联立方程：
可得： ，
又 ，代入上式化简整理可得： ，······13分
由韦达定理： ，解得： ，
代入直线PA： ，解得： ，故 ，14分
第 4 页 共 7 页，
，
所以 ·····································15分
，·····16分
当且仅当 ，即 时，等号成立，
故 的最大值为 ．·····································································17分
19．解：（1）证明：旋转后AC //BC，所以A，B，C，C 四点共面；················3分
1 1 1 1
（2）一种思路是发现多面体 实际由两个完全相同的
A1 F1
四棱锥拼接而成，
D1 C1
即B 1 -A 1 BCC 1 与A-A 1 BCC 1 ，求其中任意一个四棱锥的体 B1 E1
积即可．
另一种思路是将该多面体补成一个上下底面均为正六棱 D A
柱，再减去6个全等的小三棱锥的体积，这个上下底面 B F
E C
均为正六边形的棱柱的体积为 ，每一个小三棱锥
的体积为 ，
∴这个多面体体积为 ；············································9分
（3）如图所示，以底面中心O作为坐标原点，建立空间直角坐标系，将平面
投影到底面 中，如下图所示.延长AO交BC于
G，
第 5 页 共 7 页则 . 易知 ， ， .
则 ， ， ，
且 ， ，
，············································································12分
故 ， ，
，
由平面 到底面的距离为4，
∴ ，
， ，
由P在多面体 的棱 ，可设 ，
其中 ，从而 ，·······································13分
故 ，
平面 一个法向量 ，
AP与平面 所成角的正弦值满足 ，
即 ，······································································14分
第 6 页 共 7 页令 ，
则 ，····································15分
故 在 单调递增，在 单调递减，
因此 ，································································16分
所以AP与平面 所成角的正弦值为 ，此时 ．··················17分
第 7 页 共 7 页
