2026届南京市高三数学高考二模模拟试卷（含答案详解与评分标准）

2026届南京市高三数学高考二模模拟试卷（含答案详解与评分标准）学校：__________班级：__________姓名：__________考号：__________考试时间：120分钟满分：150分考试节点：高考二模适用年级：高三注意事项：1.本试卷用于高三高考二模阶段综合检测，重点考查基础知识、关键能力、数学思想方法与规范表达。2.答题前，请将学校、班级、姓名和考号填写在指定位置；选择题答案写入作答栏，填空题答案写在横线上。3.解答题须写出必要的文字说明、推理过程或演算步骤；只写结论但缺少关键过程的，按评分标准扣分。4.作图、列表和计算应清晰规范；考试结束后按监考要求交卷。题型选择题填空题解答题合计题数10题6题6题22题分值30分18分102分150分一、选择题：本大题共10小题，每小题3分，共30分。在每小题给出的四个选项中，只有一项符合题目要求。1．设全集，集合，，则等于A．B．C．D．2．已知复数，则等于A．B．C．D．3．已知向量，，若与垂直，则的值为A．B．C．D．4．命题为“对任意，都有”，则命题为A．存在，使B．对任意，都有C．存在，使D．对任意，都有5．展开式中的系数为A．B．C．D．6．一组数据的平均数为，方差为。若，则数据的平均数与方差分别为A．B．C．D．7．已知，且，则等于A．B．C．D．8．直线截圆所得弦长为A．B．C．D．9．一个盒中有3个红球、2个蓝球、1个白球，除颜色外完全相同。从中不放回地任取2个球，则至少取到1个红球的概率为A．B．C．D．10．若对任意，不等式恒成立，则实数的取值范围为A．B．C．D．选择题作答栏：12345678910二、填空题：本大题共6小题，每小题3分，共18分。请把答案填写在题中横线上。11．等比数列满足，，且公比为正数，则＝________。12．方程的解为＝________。13．在中，，，，则三角形面积为________。14．曲线在处的切线方程为________。15．椭圆的离心率为，则＝________。16．正方体的棱长为，直线与平面所成角为，则＝________。三、解答题：本大题共6小题，每小题17分，共102分。解答应写出文字说明、证明过程或演算步骤。17．已知函数。

（1）化简并求其最小正周期；

（2）求在区间上的值域；

（3）解不等式在区间上的解集。（17分）作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．已知数列满足，。

（1）证明数列为等差数列，并求；

（2）求前项和；

（3）若，求正整数的最大值。（17分）作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．某校高三二模前设置5个专项模块：函数、数列、立体几何、解析几何、概率统计。随机抽取3个模块组成一次限时训练；其中函数、数列、解析几何记为代数综合模块，共3个，立体几何、概率统计记为应用模块，共2个。设为抽取的代数综合模块个数。

（1）求的分布列与数学期望；

（2）若抽中的3个模块按随机顺序讲评，求前两次讲评中恰有1个代数综合模块的概率；

（3）在抽到的3个模块中恰有2个代数综合模块的条件下，甲、乙两名学生独立完成训练。两人在每个代数综合模块达标的概率均为，在每个应用模块达标的概率均为，各模块达标情况相互独立。求甲三项全部达标且乙至少两项达标的概率。（17分）作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．已知四棱锥的底面为边长为2的正方形，平面，且。

（1）证明：平面；

（2）求点到平面的距离；

（3）求直线与平面所成角的正弦值。（17分）作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．已知椭圆，点在椭圆上。过点的直线与椭圆交于另一点。

（1）求椭圆的焦点坐标和离心率；

（2）用表示点的坐标；

（3）若弦的长为，求的值。（17分）作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．已知函数，其中为实数。

（1）讨论函数的单调性；

（2）若对任意，都有，求的取值范围；

（3）当时，证明方程在内有两个不同实根的充要条件为。（17分）作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析一、选择题答案与解析12345678910CDBCBACBDA1．答案：C。由得；由对数定义域和单调性得且，即。交集为。2．答案：D。，所以。3．答案：B。由垂直得。其中，，故，即。4．答案：C。全称命题的否定为存在命题，并把结论否定，因此得到“存在，使”。5．答案：B。通项为。令得，系数为。6．答案：A。线性变换使平均数变为，方差变为。7．答案：C。因为，又，所以，即，故。8．答案：B。圆心到直线的距离为，圆半径，弦长为。9．答案：D。反面事件为“2个球都不是红球”。非红球有3个，因此所求概率为。10．答案：A。令。若，则，且，所以恒成立；若，则，在0右侧可取到负值，不恒成立。选择题评分标准：每小题3分，共30分；选对得3分，未选、错选