2026届上海市浦东新区高三数学高考冲刺模拟试卷（含答案详解与评分标准）

2026届高三数学高考冲刺模拟试卷2026届上海市浦东新区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分注意事项：1．本卷用于高三数学高考冲刺阶段综合检测，重点考查基础落实、通性通法、综合建模与规范表达。2．答题前请将学校、班级、姓名、考号填写清楚；选择题答案填入指定位置，主观题写出必要的推理与计算过程。3．全卷共22题，选择题10题30分，填空题6题18分，解答题6题102分，合计150分。4．作答时保持卷面整洁，结论须与条件相符；不得使用计算器。一、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，只有一项符合题意。请将所选答案填在题后的括号内。1．已知集合，，则（）。A．B．C．D．2．复数满足，则等于（）。A．B．C．D．3．展开式中项的系数为（）。A．B．C．D．4．向量，。若，则（）。A．B．C．D．5．袋中有3个红球、2个蓝球，除颜色外完全相同。从中不放回地任取2个球，则两球同色的概率为（）。A．B．C．D．6．等比数列各项均为正，且，，则等于（）。A．B．C．D．7．函数的最小正周期与最大值分别为（）。A．B．C．D．8．圆锥的底面半径为3，高为4，则该圆锥的侧面积为（）。A．B．C．D．9．若函数在实数集上存在最小值0，则实数的值为（）。A．B．C．D．10．椭圆的左、右焦点分别为。点在椭圆上，且，则（）。A．B．C．D．二、填空题（本大题共6小题，每小题3分，共18分）请将正确答案直接填写在题中横线上。11．不等式的解集为____________________。12．抛物线的焦点坐标为____________________，准线方程为____________________。13．曲线在点处的切线方程为____________________。14．方程的实根个数为____________________。15．一组数据的方差为____________________。16．若，且，则____________________。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。请在每题下方作答区内完成。17．（14分）已知函数。（1）将化为的形式，其中，；（2）求在区间上的取值范围；（3）求方程在区间上的全部解。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18．（16分）设等差数列的前项和为，已知，。（1）求数列的通项公式；（2）设，求；（3）设，求。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19．（16分）某校在高三数学高考冲刺阶段进行两次综合检测，随机抽取60名学生，统计第二次相对第一次的提升分数，得到如下分组表。61224126（1）用组中值估计本次样本提升分数的平均数，并估计中位数；（2）提升分数不低于20分的18名学生中，女生10人、男生8人。若从这18人中随机抽取2人分享复习方法，求至少有1名女生的概率；（3）以样本频率估计全校学生提升分数不低于20分的概率，若独立抽查5名学生，求恰有2名学生提升分数不低于20分的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20．（18分）如图形关系所示，三棱锥中，，，平面，。点分别为的中点。（1）证明；（2）求平面与平面所成锐二面角的余弦值；（3）求点到平面的距离。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21．（18分）已知抛物线，焦点为。过的直线与抛物线交于不同的两点。设，。（1）求、，并证明；（2）求弦长的最小值及此时的值；（3）设为坐标原点，求的取值范围。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22．（20分）设函数，其中为实数，。（1）当时，证明，并指出等号成立的条件；（2）讨论函数在上的单调性，并据此判断方程的正根个数；（3）当且时，设方程的两个正根中异于1的根为，证明：若，则；若，则。作答区：_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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