2026届深圳市高三数学高考二模模拟试卷（含答案详解与评分标准）

2026届深圳市高三数学高考二模模拟试卷（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分注意事项：1．本卷用于2026届高三数学高考二模考前综合检测，试题分为选择题、填空题和解答题三部分。2．作答前请将学校、班级、姓名和考号填写清楚；选择题答案请按题号填涂，填空题和解答题写在相应作答区内。3．解答题应写出必要的文字说明、证明过程或演算步骤；只写最后答案不得满分。4．全卷满分150分，其中选择题30分，填空题18分，解答题102分。一、单项选择题：本题共10小题，每小题3分，共30分。每小题只有一个选项符合题意。1．已知复数，则等于（）A．-3/2B．-1/2C．1/2D．3/22．设集合，，则为（）A．(-∞，1]B．[1，2]C．∅D．(1，2)3．已知向量，，若，则的值为（）A．-1B．-1/2C．1/2D．14．数列满足，，则等于（）A．22B．26C．30D．325．袋中有大小和质地完全相同的3个红球、2个白球和1个蓝球，从中不放回随机取出2个球，则两球颜色不同的概率为（）A．7/15B．2/3C．11/15D．4/56．二项式的展开式中常数项为（）A．160B．-160C．80D．-807．若，且，则等于（）A．√3B．1C．√3/3D．-√3/38．函数在区间上的最大值为（）A．3/e³B．1/eC．1D．09．抛物线的焦点为，直线与抛物线交于两点，则等于（）A．2B．2√2C．4D．610．已知，关于的方程在内恰有两个不同实根，则的取值范围是（）A．(0，1)B．{1}C．(1，+∞)D．(0，1)∪(1，+∞)二、填空题：本题共6小题，每小题3分，共18分。请把答案填在题中横线上。11．一组数据的平均数为，则这组数据的方差为__________。12．等比数列的前项和为，则__________。13．若，且，则的最大值为__________。14．双曲线的离心率为，则__________。15．将函数的图象向右平移个单位后得到偶函数图象，则最小正数__________。16．函数在区间上有三个不同零点，则实数的取值范围为__________。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12分）在△ABC中，角A，B，C所对的边分别为a，b，c。已知（1）求角A，B的大小；（2）若A>B，求a，b及△ABC的面积。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（本小题满分14分）某校为了解2026届高三学生二模前阶段复习情况，从一次数学模拟检测中抽取50名学生成绩，分组统计如下表。[80，90)[90，100)[100，110)[110，120)[120，130)[130，150]261014126（1）用各组中点估计这50名学生的平均分；（2）从成绩不低于120分的学生中随机抽取2人，求恰有1人成绩在[130，150]内的概率；（3）若将成绩不低于120分记为“达标”，从这50名学生中随机抽取3人，记达标人数为X，求X的分布列和数学期望。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（本小题满分16分）在长方体ABCD-A₁B₁C₁D₁中，AB=4，AD=3，AA₁=2，E为BB₁的中点，F为DD₁的中点。（1）证明EF∥BD；（2）求直线AE与平面A₁C₁D所成角的正弦值；（3）求点B到平面A₁C₁D的距离。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（本小题满分16分）已知椭圆C的方程为右焦点为，直线与椭圆交于两点，其中的横坐标小于的横坐标。（1）求的坐标；（2）求的面积；（3）若直线与椭圆交于两点，且该公共弦中点在直线上，求实数。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（本小题满分20分）已知函数（1）当a=2时，求函数的单调区间和极小值；（2）讨论方程eˣ=ax+1的实根个数；（3）若对任意x≥0，不等式eˣ≥1+x+bx²恒成立，求实数b的最大值。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（本小题满分24分）对正整数n，记（1）证明：对任意，有（2）证明数列单调递减；（3）证明；（4）设，证明。【作答区】__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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