2026届广东省高三数学高考三模模拟试卷（含答案详解与评分标准）-a1

2026届广东省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分适用对象：2026届广东省高三高考三模考前综合检测注意事项：1．本卷分选择题、填空题和解答题三部分，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。3．选择题每题只有一个正确选项；填空题只填写最终结果；解答题应写出必要的文字说明、证明过程或演算步骤。4．本卷用于高考三模阶段复习质量检测，答题时请注意计算准确、书写规范。题型选择题填空题解答题分值10题×3分=30分6题×3分=18分6题=102分一、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．设复数，则等于（）A．B．C．D．2．已知集合，，则等于（）A．B．C．D．3．在二项式的展开式中，的系数为（）A．10B．15C．20D．304．已知向量，，且。若，则的值为（）A．B．0C．1D．25．若，，则等于（）A．B．C．D．6．某次三模前限时训练中，一组成绩近似服从均值为72、标准差为8的分布。若某同学的标准分为，则其原始分约为（）A．80B．82C．84D．887．圆与点的最短距离为（）A．1B．2C．3D．58．函数的零点个数为（）A．0B．1C．2D．39．袋中有4个红球、3个蓝球，除颜色外完全相同。不放回任取2个球，2个球颜色相同的概率为（）A．B．C．D．10．正四棱锥的底边长为2，高为，则其侧棱与底面所成角为（）A．B．C．D．二、填空题：本题共6小题，每小题3分，共18分。11．等差数列中，，公差。若，则__________。12．函数在处有极值，则__________。13．方程的解为__________。14．椭圆的离心率为__________。15．5名同学排成一排照相，其中甲、乙两人不相邻的排法共有__________种。16．已知向量满足，，，则__________。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．（14分）在中，内角的对边分别为。已知，，。（1）求边长a和三角形ABC的面积；（2）求的值，并判断本题所得三角形中最大角是哪一个。作答区________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（16分）已知数列满足，，。（1）求的通项公式；（2）设，求；（3）在三模复习任务量按累计的模型中，求使的最小正整数。作答区________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（16分）某校高三年级在高考三模前进行错题复盘专项训练。随机抽取120名学生，按“是否完成错题复盘”和“三模前一次综合检测数学成绩是否较上次提高10分及以上”统计如下：类别进步10分及以上未达10分完成错题复盘4832未完成错题复盘1228（1）从这120名学生中随机抽取1人，求该生“完成错题复盘且进步10分及以上”的概率，并求在已完成错题复盘条件下进步10分及以上的概率；（2）判断事件“完成错题复盘”与事件“进步10分及以上”是否相互独立，并说明理由；（3）从完成错题复盘的学生中进行有放回抽取，每次随机抽取1人，共抽取3次。记3次中“进步10分及以上”的人数为X，求X的分布列与数学期望。作答区________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（18分）如图形条件所述，四棱锥P-ABCD的底面ABCD是边长为2的正方形，O为底面中心，PO垂直于底面ABCD，且PO=2。点E、F分别为PA、PC的中点。（1）证明EF∥AC，并说明EF与平面ABCD的位置关系；（2）求四棱锥P-ABCD的体积；（3）求二面角P-AB-D的余弦值。作答区________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（18分）已知抛物线的焦点为。直线（）与交于两点，其中在第一象限。（1）用表示弦长；（2）若的面积为12，求；（3）在（2）的条件下，求抛物线在A、B两点处切线的交点Q，并证明O、F、Q三点共线，其中O为坐标原点。作答区________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（20分）已知函数，其中为实数。（1）当a=1时，求f₁(x)在实数集上的单调区间与极值；（2）若对任意，都有，求的取值范围；（3）当时，证明方程在内有两个不同零点。作答区______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________