2026年杭州高三数学高考三模冲刺卷：数列三角与解析几何联动（名校协作体第3套）含参考答案、逐题解析与评分细则

2026年杭州高三数学高考三模冲刺卷：数列三角与解析几何联动（名校协作体第3套）含参考答案、逐题解析与评分细则杭州名校协作体·2026届高三数学高考三模冲刺模拟卷（第3套）考试科目：数学满分：150分考试时间：120分钟适用：临考冲刺、查漏补缺、教师讲评学校/班级：________________姓名：________________准考证号：________________成绩：________________卷型：名校协作体第3套重点：数列、三角、解析几何联动考试节点：2026届高考三模冲刺满分/时间：150分/120分钟注意事项•本卷分为单项选择题、多项选择题、填空题、解答题四部分。请将客观题答案填写在答题栏，主观题写在指定作答区。•单项选择题每题只有一个正确选项；多项选择题每题有两个或两个以上正确选项，全部选对得满分，少选得部分分，有错选得0分。•解答题应写出必要的文字说明、演算步骤或证明过程。书写规范、逻辑清楚、符号统一是评分的重要依据。•本卷突出函数与导数、圆锥曲线、立体几何、概率统计、数列三角联动，适合作为杭州高三数学三模冲刺卷使用。一、单项选择题（本大题共8小题，每小题5分，共40分。每小题只有一个正确选项）1.（5分）设复数，其中为虚数单位，则等于A.B.C.D.2.（5分）函数的定义域为A.B.C.D.3.（5分）已知向量，。若与垂直，则A.B.C.D.4.（5分）若，且，则A.B.C.D.5.（5分）等差数列中，，，则A.B.C.D.6.（5分）袋中有3个红球、2个蓝球，从中不放回任取2个，设随机变量表示取到红球的个数，则A.B.C.D.7.（5分）椭圆的两焦点间距离为4，离心率为，则该椭圆的标准方程为A.B.C.D.8.（5分）若函数在区间上单调递增，则实数的取值范围为A.B.C.D.单项选择题答题栏：12345678二、多项选择题（本大题共4小题，每小题5分，共20分。全部选对得5分，少选得2分，有错选得0分）9.（5分）设，则下列结论正确的是A.B.对任意实数成立C.在上递减，在上递增D.方程有两个负根10.（5分）在中，角的对边分别为。若，，，则下列说法正确的是A.B.C.D.角是中最大的角11.（5分）圆，直线。下列结论正确的是A.当时，直线过圆心B.当时，直线与圆相交于两点C.直线与圆相切当且仅当D.对任意实数，直线都与圆有公共点12.（5分）设事件满足，，，则下列结论正确的是A.与相互独立B.C.D.多项选择题答题栏：9101112三、填空题（本大题共4小题，每小题5分，共20分）13.（5分）方程的解为：________________。14.（5分）二项式的展开式中的系数为：________________。15.（5分）在中，已知，，，则的面积为：________________。16.（5分）数列满足，，则________________。填空题作答区：13.________________________________14.________________________________15.________________________________16.________________________________四、解答题（本大题共6小题，共70分。解答应写出文字说明、证明过程或演算步骤）17.（10分）已知函数。（1）求的最小正周期和单调递增区间；（2）求在区间上的所有解；（3）在中，，且。若，（分别为角的对边），求边的长。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（12分）已知数列满足，（）。（1）设，证明为等差数列，并求；（2）记，求，并求使的最小正整数；（3）设，证明。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（12分）某校高三数学三模冲刺训练后，对50名学生的“数列、三角、解析几何联动”专题得分作统计，得到频数分布表如下。[50，60)[60，70)[70，80)[80，90)[90，100]4818146（1）用各组中点估计该专题得分的平均数与方差；（2）现按“80分及以上”和“80分以下”两层抽取10人进行面批，请确定两层各抽取多少人；（3）已知组中有2名男生、4名女生，从该组随机抽取2人进行讲题展示，求至少有1名男生的概率。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（12分）如图形关系用文字描述：在四面体中，平面，为直角三角形，，，，。（1）证明：平面；（2）求点到平面的距离；（3）求直线与平面所成角的正弦值。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（12分）已知抛物线，焦点为。过作直线，直线与抛物线交于两点，且与轴正方向所成角为。（1）当的斜率存在时，证明；（2）若，求的可能取值；（3）若一列焦点弦的倾斜角满足，记弦长为，求，并说明的单调性与极限。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（12分）已知函数。（1）若对任意恒成立，求实数的取值范围；（2）取，证明，并指出等号成立条件；（3）设，（）。证明数列单调递减且收敛，并求其极限。【作答区】________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________备用作答区与检查栏用于第21题、第22题续写、草稿整理或订正检查。正式交卷时，请确认所有答案均已写在指定位置。______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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