2026届成都市高三数学高考三模模拟试卷（含答案详解与评分标准）

2026届成都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分试卷类型：高考三模适用年级：2026届高三注意事项：1．本试卷用于2026届成都市高三数学高考三模考前检测，全卷共22题，满分150分，考试时间120分钟。2．答题前，请将学校、班级、姓名、考号填写清楚；选择题请在答题卡相应位置填涂，非选择题写在规定作答区内。3．解答题应写出必要的文字说明、证明过程或演算步骤；只写答案不得相应步骤分。4．考试结束后，按监考要求交卷。选择题答题卡12345678910一、单项选择题（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．已知复数，则等于（）A．B．C．D．2．设集合，，则为（）A．B．C．D．3．若，且，则等于（）A．B．C．D．4．已知向量，，若，则的值为（）A．B．C．D．5．某学习小组有5名女生、3名男生，从中随机选出3人参加三模考前错题讲评，所选3人中至少有2名女生的概率为（）A．B．C．D．6．函数在实数集上为增函数且为奇函数。若，则的取值范围是（）A．B．C．D．7．数列满足，。若为前项和，且，则等于（）A．B．C．D．8．函数有三个不同的实数零点，则实数的取值范围为（）A．B．C．D．9．已知双曲线的一条渐近线方程为，则该双曲线的离心率为（）A．B．C．D．10．过抛物线的焦点作直线交于两点。若，则该直线斜率的平方为（）A．B．C．D．二、填空题（本大题共6小题，每小题3分，共18分）11．二项式展开式中的常数项为__________。12．曲线在点处的切线方程为__________。13．在棱长为2的正方体中，点到平面的距离为__________。14．一组数据的平均数为4，则这组数据的方差为__________。15．等比数列的前项和为，则为__________。16．椭圆在点处的切线方程为__________。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．（14分）已知函数。（1）求的最小正周期和最大值；（2）在锐角三角形中，若，，，求三角形的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（16分）已知等差数列的前项和为，且，。（1）求数列的通项公式；（2）设，，求。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（16分）某校高三年级在三模前组织一次数学专项检测，随机抽取100名学生成绩，按分数段整理如下表：分数段[60,70)[70,80)[80,90)[90,100]人数12244024（1）用各组中点估计该样本的平均成绩；（2）按分层抽样从这100名学生中抽取25人，再从这25人中随机抽取3人进行解题过程复核。设抽到的90分及以上学生人数为X，求X的分布列和数学期望。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（18分）如图形条件所述，在直三棱柱中，，，，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：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（18分）已知椭圆，其左右焦点分别为。（1）求椭圆C的离心率；（2）过右焦点的直线与椭圆交于两点，若线段的中点横坐标为，求；（3）在（2）的条件下，求弦长。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（20分）已知函数，其中，为实数。（1）当时，求的单调区间和最大值；（2）若对任意恒成立，求的取值范围；（3）当时，设方程的两个正根为，且。证明：，并证明。作答区：___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________