2026学年高三数学高考一模模拟试卷（含答案详解与评分标准）

2025-2026学年高三数学高考一模模拟试卷（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分考试节点：高考一模适用年级：高三注意事项：本卷用于2025-2026学年高三高考一模阶段综合检测，重点考查函数与导数、三角、数列、立体几何、解析几何、概率统计等主干知识。请将客观题答案填写在答题栏内，解答题须写出必要的文字说明、演算步骤或证明过程。试卷结构：选择题10题，每题3分，共30分；填空题6题，每题3分，共18分；解答题6题，共102分。全卷满分150分。题型题号每题分值小计选择题1—103分30分填空题11—163分18分解答题17—2217分102分一、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）已知集合，，则等于（）。A．[2，3]B．[2，3)C．(2，3]D．(-∞，3)2．（3分）若复数，则的值为（）。A．√10/2B．√5/2C．√10D．5/23．（3分）已知向量，，若，则的取值为（）。A．2B．-2C．0D．±24．（3分）等差数列中，，，则前10项和为（）。A．90B．100C．110D．1205．（3分）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，则恰有1个红球的概率为（）。A．1/2B．3/5C．2/5D．3/106．（3分）函数在区间上的值域为（）。A．[-4，4]B．[-3，3]C．[-2，2]D．[-1，1]7．（3分）二项式展开式中的系数为（）。A．6B．15C．20D．308．（3分）若，则的值为（）。A．16/25B．-12/25C．-16/25D．24/259．（3分）圆外一点到该圆的切线长为（）。A．3B．√7C．4D．510．（3分）棱长为2的正四面体的体积为（）。A．√2/3B．2√2/3C．4√2/3D．8√2/3二、填空题：本大题共6小题，每小题3分，共18分。11．（3分）方程的解为________。12．（3分）曲线在点处的切线方程为________。13．（3分）已知，则的值为________。14．（3分）向量，所张成的平行四边形面积为________。15．（3分）数列满足，则=________。16．（3分）从1，2，3，4，5中随机选取两个不同的数，则这两个数之和为偶数的概率为________。客观题答题栏（学生填写）题号12345678答案题号910111213141516答案三、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）已知函数。（1）将化为形如的形式；（2）求方程在上的解；（3）求在上的最大值与最小值。____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

18．（17分）已知数列的前项和为。（1）求通项公式；（2）求和；（3）求使成立的最小正整数。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

19．（17分）某高三数学一模阶段训练题库中有8道代数题、6道几何题、6道概率统计题。现从20道题中不放回随机抽取5道组成专项检测小卷。（1）求恰有2道代数题且恰有2道几何题的概率；（2）设抽到的代数题道数为，写出的分布列形式并求；（3）在已知恰有2道代数题的条件下，求其余3道题中至少有1道几何题的概率。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

20．（17分）已知抛物线，点在上，焦点为。（1）求抛物线在点处的切线方程；（2）过焦点作斜率为的直线交于两点，证明；（3）若，求。____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

21．（17分）如图形关系所述，四边形为边长4的正方形，平面，且。（1）求的长；（2）证明平面与平面垂直；（3）求点到平面的距离。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

22．（17分）已知函数，其中为实数。（1）当时，求的单调区间和最大值；（2）讨论方程的实根个数；（3）求使对一切恒成立的的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析评分标准总则：本答案按照高考一模阶段综合检测的阅卷要求编制。客观题以最终答案为准；解答题以过程、方法和结论共同赋分，关键公式、关键推理、代入计算和结论呈现均纳入评分。若学生使用与参考解法不同但逻辑严密、计算正确的方法，可参照相同采分点给分；若中间计算失误但后续步骤思路清楚，按该题评分标准扣除相应计算分并保留已完成的推理分；若只有答案没有必要过程，解答题不得超过该小问结论分。书写不影响数学意义时不扣分，符号含义混乱、条件遗漏或结论与题意不符时按采分点扣分。单位、区间端点、取值范围和概率结果应写完整；涉及分类讨论的题目，分类依据清楚、覆盖全面且互不重复方可获得相应过程分。一、选择题答案与关键理由1．B由，得；又，故。评分标准：选对得3分，错选、多选或不选得0分。2．A将复数化简：，所以。评分标准：选对得3分。3．D由垂直得，即，所以。评分标准：选对得3分。4．C设公差为，由，，得，故。评分标准：选对得3分。5．B恰有1个红球的选法为，总选法为，概率。评分标准：选对得3分。6．C求导，临界点为；比较处函数值，最大值为2，最小值为-2，值域为。评分标准：选对得3分。7．B通项为，令得，系数为。评分标准：选对得3分。8．C两边平方得，所以。评分标准：选对得3分。9．C圆化为，圆心，半径3。，切线长为。评分标准：选对得3分。10．B正四面体棱长为的体积为。代入，得。评分标准：选对得3分。二、填空题答案与关键理由11．3由对数定义域得，原方程化为，即，故。评分标准：填对得3分。12．y=x由，得，在处斜率为1，切线方程为。评分标准：填对得3分。13．-3/5利用公式，代入得。评分标准：填对得3分。14．√3平行四边形面积为。，其模为。评分标准：填对得3分。15．31递推式可化为，故，，所以。评分标准：填对得3分。16．2/5两数和为偶数即同奇同偶。共有种，符合条件的有种，概率为。评分标准：填对得3分。三、解答题答案详解与评分标准17．答案详解（1）。因为，，所以（2）由得。当时，，故。（3）区间覆盖正弦函数的最大、最小取值点，所以最大值为1，最小值为-1。评分标准：第（1）问6分，能正确使用二倍角公式得3分，化为得3分；第（2）问6分，列出方程得2分，给出区间范围得1分，解出3个