2026届南京市高三数学高考三模模拟试卷（含答案详解与评分标准）

2026届南京市高三数学高考三模模拟试卷（含答案详解与评分标准）南京市2026届高三考前第三次模拟检测数学试卷学校班级姓名考号________________________________________________________________考试时间：120分钟满分：150分适用范围：高三数学高考三模考前检测注意事项：•请在规定时间内完成全部题目，选择题和填空题答案填写在指定位置，解答题写出必要的文字说明、演算步骤或证明过程。•本试卷分为选择题、填空题和解答题三部分，共22题。考试结束后，请将试卷和答题纸一并整理。•本卷面向高考三模阶段复习检测，重点考查基础落实、综合迁移、运算规范和临场表达。12345678910111213141516一、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．1－iB．－1－iC．1＋iD．－1＋i2．已知集合，，则为（）A．[1，3]B．(2，3)C．[2，3]D．(1，2)3．函数的单调递增区间是（）A．(－∞，－1)和(1，＋∞)B．(－1，1)C．(－∞，1)D．(－1，＋∞)4．某三模训练中，学生独立完成4道判断题，每题答对的概率均为。若各题作答相互独立，则恰好答对3题的概率为（）A．1/8B．1/4C．3/8D．1/25．在平行四边形ABCD中，，，M为BC的中点，则等于（）A．a＋bB．a＋(1/2)bC．(1/2)a＋bD．(1/2)(a＋b)6．已知，且，则的值为（）A．3/5B．4/5C．2/5D．1/57．等差数列的前5项和，则的值为（）A．2B．3C．4D．58．直线与圆相切，则等于（）A．1/2B．1C．2D．39．若函数在处有驻点，则实数的值为（）A．－1B．0C．1D．e10．若函数有两个不同的零点，则实数的取值范围是（）A．(0，1/e)B．(1/e，＋∞)C．(－∞，0)D．(0，＋∞)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．展开式中项的系数是__________。12．已知向量，，且，则__________。13．方程的解为__________。14．一个底面半径为3、高为4的圆锥，其全面积为__________。15．椭圆的离心率为__________。16．曲线在处的切线方程为__________。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．（本小题满分15分）已知函数。（1）求函数的最小正周期和最大值；（2）求方程在区间内的全部解。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（本小题满分17分）已知等差数列的前项和为。（1）求数列的通项公式；（2）设，，求；（3）求满足的最小正整数。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（本小题满分17分）在直三棱柱中，底面为等腰直角三角形，，，。D为的中点。（1）证明：平面垂直于平面；（2）求直线与平面所成角的正弦值；（3）求二面角的余弦值。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（本小题满分17分）某校在高考三模后对高三数学成绩进行专题复盘，从全年级随机抽取100名学生，将成绩按区间分组，得到如下频数表。成绩区间[80,90)[90,100)[100,110)[110,120)[120,130)[130,140)频数81225281710（1）用各组中点估计这100名学生的平均成绩；（2）若从成绩低于100分和不低于130分的学生中，按分层抽样抽取6人参加访谈，分别应抽取多少人？（3）在（2）抽取的6人中再随机选2人，求这2人来自两个不同层的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（本小题满分18分）已知椭圆，右焦点为。过点F的直线的斜率为，且与椭圆E交于A、B两点，O为坐标原点。（1）求椭圆E的离心率；（2）若，求的值；（3）在（2）的条件下，求弦的长。作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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