2026届河北省高三数学高考冲刺模拟试卷（含答案详解与评分标准）

2026届河北省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分适用对象：2026届高三考试节点：高考冲刺选择题10题30分填空题6题18分解答题6题102分总分核验：150分注意事项：1．本试卷依据河北省高三数学高考冲刺阶段复习检测要求命制，重点考查主干知识、综合运算、逻辑推理、空间想象、数据分析与函数思想。2．答题前填写学校、班级、姓名和考号；选择题须将答案填写在答题栏中，非选择题写出必要的推理、计算过程和结论。3．全卷共22题。考试时间120分钟，满分150分。书写应规范，图形可按题意自行补充辅助线。客观题答题栏题号12345678910答案题号111213141516答案一、单项选择题（本大题共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．（3分）设集合，，则等于（）A．B．C．D．2．（3分）复数，则的值为（）A．5B．C．3D．73．（3分）一个不透明袋中有5个红球、3个蓝球，除颜色外完全相同。从中不放回随机取出2个球，则两球颜色相同的概率为（）A．B．C．D．4．（3分）函数在区间上取得最大值时，x可以为（）A．0B．C．D．5．（3分）已知向量a，b均为单位向量，且夹角为120°，则等于（）A．B．C．D．36．（3分）函数的单调递减区间为（）A．B．C．D．7．（3分）数列满足，，则的值为（）A．28B．54C．82D．8．（3分）在棱长为2的正方体中，直线与底面所成角为θ，则等于（）A．B．C．D．9．（3分）椭圆的离心率为（）A．B．C．D．10．（3分）若函数有两个不同零点，则实数a的取值范围为（）A．B．C．D．二、填空题（本大题共6小题，每小题3分，共18分。请把答案填写在答题栏相应位置。）11．（3分）二项式展开式中的常数项为__________。12．（3分）数列的前n项和，则__________。13．（3分）方程在上的解为__________。14．（3分）某次高三数学冲刺检测中，成绩X近似服从正态分布。已知，，则__________。15．（3分）抛物线在点处的切线方程为__________。16．（3分）若函数在处取得最小值，则实数__________。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△ABC中，角的对边分别为。已知，，且。（1）求以及角；（2）求△ABC的面积。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（17分）为检验高考冲刺阶段复习效果，某高三班对“函数与导数专题A”和“解析几何专题B”进行达标检测。全班60人，其中A专题达标36人，B专题达标42人，两个专题均达标30人。（1）完成下列列联表，并求从全班随机抽取1名学生，该学生恰好只有一个专题达标的概率；B达标B未达标合计A达标3036A未达标合计4260（2）从全班不放回随机抽取3名学生，设X为其中两个专题均达标的学生人数，求X的分布列与数学期望；（3）在B专题达标的学生中随机抽取1人，求其A专题也达标的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（17分）如图意：四棱锥中，底面是边长为2的正方形，PA⊥平面ABCD，且。点E为PC的中点。（1）证明：BD⊥平面PAC；（2）求点B到平面的距离；（3）求二面角的大小。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（17分）已知数列满足，。（1）求的通项公式；（2）设，求；（3）证明：对任意，都有。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（17分）已知椭圆，其右焦点为。过F的直线与椭圆交于两点。（1）求椭圆E的离心率；（2）当时，求弦AB的长；（3）证明：当直线l过右焦点变化时，弦AB的最大值为4，并指出取得最大值时的直线。作答区：_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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