2026届成都市高三数学高考冲刺模拟试卷（含答案详解与评分标准）

2026届成都市高三数学高考冲刺模拟试卷2026届成都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分注意事项：1．本卷用于2026届成都市高三数学高考冲刺阶段综合检测，满分150分，考试时间120分钟。2．选择题、填空题请把答案填入答题栏；解答题必须写出必要的文字说明、演算步骤和推理过程。3．答题过程保持卷面整洁，作图可使用符合考试规定的直尺、圆规等工具。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，，则等于（）。A．B．C．D．2．复数，则等于（）。A．B．C．D．3．函数的定义域内，不等式的解集为（）。A．B．C．D．4．已知向量，，且，则的值为（）。A．B．C．D．5．袋中有3个红球、2个蓝球，任取2个球，记取到红球个数为随机变量，则为（）。A．B．C．D．6．若，且，则等于（）。A．B．C．D．7．椭圆的离心率为（）。A．B．C．D．8．数列的前项和为，则等于（）。A．B．C．D．9．曲线在点处的切线方程为（）。A．B．C．D．10．已知随机变量，，且，则为（）。A．B．C．D．二、填空题：本大题共6小题，每小题3分，共18分。请把答案填在题中横线上。11．二项式展开式中的常数项为________。12．函数的最大值为________。13．已知双曲线的中心在原点、焦点在轴上，渐近线方程为，且半焦距，则该双曲线的标准方程为________。14．若实数满足，，，，则的最大值为________。15．若直线与圆有公共点，则实数的取值范围是________。16．已知数列满足，，则________。选择题、填空题答题栏题号12345678答案题号910111213141516答案三、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．（17分）在中，角所对的边分别为。已知，，。（1）求边的长和的面积；（2）求的值；（3）点在边上，且，求线段的长。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

18．（17分）已知数列满足，（）。（1）求数列的通项公式；（2）设，求；（3）求满足的最小正整数。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19．（17分）某校在2026届高三数学高考冲刺阶段组织一次模拟检测，从120名学生中得到成绩分布如下表。分数段、、、、、、的人数依次为6、12、24、36、24、12、6。（1）用各组中点估计本次成绩的平均数与方差；（2）估计从该校同层次学生中任取1人，成绩不低于130分的概率；（3）在最高分数段的6人中有男生4人、女生2人，随机选2人参加考后答疑展示，求至少选到1名女生的概率。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

20．（17分）四棱锥的底面是边长为2的正方形，平面，且，点为的中点。（1）证明平面；（2）求四棱锥的体积；（3）求直线与平面所成角的余弦值；（4）求点到平面的距离。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

21．（17分）已知抛物线，焦点为。过点的直线可写成，与抛物线交于两点。（1）写出焦点的坐标和准线方程；（2）用参数表示弦长；（3）若，求弦的中点的坐标，并求的面积，其中为坐标原点。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

22．（17分）已知函数，定义域为，其中为实数。（1）当时，求的单调区间和最大值；（2）求所有实数，使得对任意，恒有；（3）当时，证明方程除外还有唯一一个根，并写出不等式的解集形式。解答：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________备用作答区（可用于解答题续写、作图或验算）________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析评分说明：选择题每题3分，填空题每题3分；解答题按步骤给分。若方法正确但表述不完整，可参照关键步骤酌情给分；若前一步计算结果错误而后续推理方法正确，可在不重复扣分的原则下给后续相应步骤分。题号12345678910答案DBCCB