2026届苏州市高三数学高考冲刺模拟试卷（含答案详解与评分标准）

2026届苏州市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分本卷用于高考冲刺阶段综合检测。请在规定时间内完成作答，保持卷面整洁，过程题写出必要步骤。题型题量每题分值合计选择题10题3分30分填空题6题3分18分解答题6题按小问给分102分总计22题—150分注意事项1．本试卷共22题，满分150分，考试时间120分钟。2．选择题每题只有一个选项符合题意；填空题只需填写最终结果；解答题应写出文字说明、证明过程或演算步骤。3．客观题答案填写在题后横线或括号内；主观题在题后作答区书写，若空间不足可接续书写但须标明题号。4．允许使用规定的作图工具，不得使用具有符号运算或联网功能的设备。一、选择题：本大题共10小题，每小题3分，共30分。每小题给出的四个选项中，只有一项符合题目要求。1．已知集合，，则（）A．B．C．D．2．复数，其中为虚数单位，则（）A．B．C．D．3．设向量，。若，且，则（）A．B．C．D．4．若，且，则（）A．B．C．D．5．二项式的展开式中常数项为（）A．B．C．D．6．函数在上有两个不同零点，则实数的取值范围是（）A．B．C．D．7．某射手每次命中目标的概率为，连续独立射击次，恰好命中次的概率为（）A．B．C．D．8．椭圆的离心率为（）A．B．C．D．9．数列满足，，则（）A．B．C．D．10．关于的方程有三个不同实根的充要条件是（）A．B．C．D．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．在区间内，方程的解为________________。12．________________。13．若，，且，则的最大值为________________。14．圆的半径为________________。15．已知，，则与夹角的余弦值为________________。16．用数字组成没有重复数字的三位偶数，共有________________个。三、解答题：本大题共6小题，共102分。解答应写出必要的文字说明、证明过程或演算步骤。17．（本小题满分14分）在中，角所对的边分别为。已知，，。

（1）求角与角；

（2）求的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

18．（本小题满分16分）已知数列满足，。

（1）证明数列是等比数列，并求；

（2）记，求；

（3）若，，求。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19．（本小题满分18分）苏州市某校为了解高三学生高考冲刺阶段数学复习效果，随机抽取名学生的模拟检测成绩，按区间分组得到下表。

（1）用各组中点估计这名学生的平均成绩；

（2）把成绩不低于分记为“冲刺达标”。从该校高三学生中随机抽取人，按样本频率估计每人达标概率，求恰有人达标的概率；

（3）若从这名样本学生中不放回地随机抽取人，求两人成绩均不低于分的概率。成绩区间[80,90)[90,100)[100,110)[110,120)[120,130)[130,150]人数51225301810作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

20．（本小题满分18分）如图形关系所述：四棱锥的底面是边长为的正方形，平面，且。

（1）证明平面；

（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：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

21．（本小题满分18分）已知函数，其中，。

（1）当时，求的单调区间与最小值；

（2）讨论函数在上零点的个数；

（3）若对任意恒成立，求实数的取值范围。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

22．（本小题满分18分）已知抛物线，焦点为。过的直线与抛物线交于不同两点。

（1）设，，证明；

（2）过分别作抛物线的切线，设两切线交于点，证明点在定直线上；

（3）若弦的长为，求的值。作答区：______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________