2026届福建省高三数学高考冲刺模拟试卷（含答案详解与评分标准）

2026届福建省高三数学高考冲刺模拟试卷（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分试卷类型：高考冲刺模拟作答方式：闭卷注意事项：1．本试卷面向2026届福建省高三数学高考冲刺阶段使用，覆盖函数、导数、三角、数列、立体几何、解析几何、概率统计等核心内容。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；解答题应写出必要的推理、计算过程和结论。3．所有答案均须写在相应作答区内，超出区域作答仍应保持步骤清楚、符号规范。4．考试结束后，建议按参考答案与评分标准进行自评，重点检查概念迁移、运算准确率和压轴题分层得分。一、选择题：本题共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．设复数，则等于（）。A．B．C．D．2．已知集合，，则为（）。A．B．C．D．3．函数在点处的切线方程为（）。A．B．C．D．4．等差数列满足，。若前项和，则最小的正整数为（）。A．B．C．D．5．某高三学生独立完成4道同类基础题，每题答对的概率均为。则他至少答对3道题的概率为（）。A．B．C．D．6．已知，且，则的值为（）。A．B．C．D．7．圆被直线截得的弦长为（）。A．B．C．D．8．若函数在区间上单调递增，则实数的取值范围为（）。A．B．C．D．9．一个正四棱锥的底面边长为，高为，则该棱锥的体积为（）。A．B．C．D．10．一组数据为，按总体方差计算，其方差为（）。A．B．C．D．二、填空题：本题共6小题，每小题3分，共18分。请把答案填写在题中横线上。11．函数在处的导数值为__________。12．若向量与满足，，夹角为，则__________。13．展开式中项的系数为__________。14．椭圆的离心率为__________。15．已知，则的最小值为__________。16．数列满足，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．（17分）已知函数。（1）将化为一个角的三角函数形式；（2）求在区间上的最大值及取得最大值时的；（3）求区间内满足的。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

18．（17分）某校在高考冲刺阶段进行一次5题限时训练，随机抽取一名学生，其答对题数的分布列如下表。答对题数X012345概率P0.040.100.240.320.200.10（1）求这名学生至少答对4题的概率；（2）求随机变量的数学期望与方差；（3）若全年级有120名学生参加同类训练，按该分布估计，答对4题及以上的学生约有多少人？作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19．（17分）在长方体中，，，，点为的中点。（1）证明：；（2）求直线与平面所成角的正弦值；（3）求点到平面的距离。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

20．（17分）已知椭圆，其左、右焦点分别为。（1）求椭圆的离心率和焦点坐标；（2）过点作斜率为的直线，若与椭圆相切，求；（3）直线与椭圆交于不同两点，若线段的中点横坐标为，求的值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

21．（17分）已知函数，定义域为。（1）当时，求的单调区间；（2）若对任意，都有，求实数的取值范围；（3）利用（2）的结论证明：对任意，都有。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

22．（17分）已知数列满足，。（1）证明：；（2）设，求；（3）设，求数列的最小项。作答区：_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________