2026届重庆市高三数学高考三模模拟试卷（含答案详解与评分标准）

2026届重庆市高三数学高考三模模拟试卷（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：高考三模适用年级：高三注意事项1．本试卷分第Ⅰ卷和第Ⅱ卷，满分150分，考试时间120分钟。请按高考三模考前检测要求独立完成。2．选择题每小题只有一个正确选项；填空题只需填写最终结果；解答题须写出必要的文字说明、推理过程或演算步骤。3．作答前请填写学校、班级、姓名和考号；保持卷面整洁，书写规范。选择题答题栏题号12345678910答案填空题答题区111213141516________________________________________________________________________________________________第Ⅰ卷选择题（共30分）一、选择题：本题共10小题，每小题3分，共30分。每小题只有一个选项符合题目要求。1．已知集合，，则为A．B．C．D．2．复数的虚部为A．B．C．D．3．若，且，则的值为A．B．C．D．4．设向量，。若与垂直，且，则A．B．C．D．5．函数在处取得最大值，则实数的值为A．B．C．D．6．在的展开式中，的系数为A．B．C．D．7．某小组有5名男生和4名女生，从中随机选出3人参加高考三模数学讲评展示，则选出的3人中至少有2名女生的概率为A．B．C．D．8．已知数列的前项和为，则A．B．C．D．9．若直线与抛物线相切，则的值为A．B．C．D．10．在棱长为2的正方体中，点到平面的距离为A．B．C．D．第Ⅱ卷非选择题（共120分）二、填空题：本题共6小题，每小题3分，共18分。11．不等式组，的解集为____________。12．曲线在点处的切线方程为____________。13．一组数据的方差为____________。14．在中，，，，则边的长为____________。15．等比数列的公比为正数，若，，则前5项和____________。16．若随机变量，则____________。三、解答题：本题共6小题，共102分。解答应写出文字说明、证明过程或演算步骤。17．（12分）已知函数。（1）求函数的最小正周期和取值范围；（2）求方程在区间上的所有解。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（12分）重庆某校高三年级在高考三模后，从全体学生中随机抽取100名学生的数学成绩，整理成如下频数分布表。[90,100)[100,110)[110,120)[120,130)[130,140)[140,150]51530251510（1）用组中值估计这100名学生数学成绩的平均数；（2）若从这100名学生中不放回地随机抽取2人，求两人成绩均不低于130分的概率；（3）若按成绩低于130分与不低于130分两层进行分层抽样抽取20人，求两层应分别抽取的人数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（17分）在四棱锥中，底面是边长为2的正方形，平面，且。点为的中点。（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。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（17分）已知椭圆的焦点为，，且椭圆过点。（1）求椭圆的方程；（2）若直线与椭圆交于两点，求弦长；（3）设过点的动直线与椭圆交于两点，弦的中点为，求点的轨迹方程。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（22分）已知数列满足，。（1）求数列的通项公式；（2）求前项和；（3）设，证明：。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（22分）已知函数，其中。（1）当时，求的单调区间与最大值；（2）讨论方程的实根个数；（3）利用（1）的结论证明：对任意，都有，并指出等号成立的条件。作答区：___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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