2026届福建省高三数学高考三模模拟试卷（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分题量：22题适用：2026届高三高考三模注意事项：1．本试卷分第Ⅰ卷和第Ⅱ卷，共22题，满分150分，考试时间120分钟。请认真审题，规范书写，保持卷面整洁。2．选择题每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文字说明、证明过程或演算步骤。3．本卷用于福建省高三高考三模考前检测，考查基础掌握、综合迁移、运算求解和数学表达能力。4．主观题请在对应作答区作答，超出作答区的内容应标明题号；作图可用铅笔，答案最终书写应清楚。第Ⅰ卷（选择题，共30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．（3分）已知集合，，其中。若，则正整数的最小值为（）。A．1B．2C．3D．42．（3分）复数满足，则的值为（）。A．7B．10C．13D．173．（3分）已知向量，，且，则（）。A．B．C．1D．24．（3分）函数的值域为（）。A．B．C．D．5．（3分）在展开式中，项的系数为（）。A．B．C．15D．206．（3分）一个袋中有5个红球、3个蓝球，除颜色外完全相同。从中不放回地随机取出2个球，则取出的2个球颜色不同的概率为（）。A．B．C．D．7．（3分）直线与圆相切，且，则（）。A．1B．2C．D．8．（3分）等差数列的前项和为，若，，则（）。A．10B．12C．14D．169．（3分）函数的最大值为（）。A．B．1C．D．210．（3分）已知双曲线的离心率为，焦距为4，则其实轴长为（）。A．2B．C．4D．第Ⅱ卷（非选择题，共120分）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．（3分）__________。12．（3分）展开式中项的系数为__________。13．（3分）已知等比数列满足，，且公比，则前4项和__________。14．（3分）函数在上的最小值为__________。15．（3分）圆被直线截得的弦长为__________。16．（3分）若对任意实数，不等式恒成立，则实数的最大值为__________。学生客观题答案填写区12345678910111213141516三、解答题：本大题共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．（15分）在中，角所对的边分别为，且满足，已知，。（1）证明，并求；（2）求的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

18．（17分）已知数列满足，。（1）证明数列为等比数列，并求；（2）设，求；（3）求满足的最小正整数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19．（17分）某校高三年级在三模前进行一次数学专项检测，从中随机抽取100名学生的成绩，按分数段统计如下表。（1）用各组中点估计这100名学生数学专项检测成绩的平均数；（2）若按分层抽样从这100名学生中抽取25人参加错题回访，求各分数段应抽取的人数；（3）若从这100名学生中随机选2人，求两人成绩都不低于80分的概率。分数段[60,70)[70,80)[80,90)[90,100]人数12283624作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

20．（17分）如图意可取空间直角坐标理解：四棱锥的底面为矩形，，，平面，且。点为的中点。（1）证明平面；（2）求直线与平面所成角的正弦值；（3）求四棱锥的体积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

21．（18分）已知椭圆，直线，其中，直线与椭圆交于两点。（1）求两点横坐标的和与积；（2）若弦的中点在直线上，求的值与弦长；（3）在第（2）问条件下，求的面积，其中为坐标原点。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

22．（18分）已知函数，其中，。（1）讨论的单调性；（2）求实数的值，使得对任意，恒有；（3）利用第（2）问所得结论，证明对任意正整数，都有。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析一、选择题答案12345678910CCCBCBBCBB选择题评分标准：每小题3分，共30分；选对得3分，选错、多选或不选均得0分。1．答案C。由得；由得。要使，需，即。正整数的最小值为3。2．答案C。，所以，，所求为13。3．答案C。利用恒等式，得，即，所以。4．答案B。函数定义域为，。故在上递减，在上递增，且。两端极限均趋于，值域为。5．答案C。通项为。令，得，系数为。6．答案B。不同色取法有种，总取法有种，概率为。7．答案B。圆心为，半径为。圆心到直线的距离为。相切得，又，故。8．答案C。设