2026年丽水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1套）含参考答案、逐题解析与评分细则

2026年丽水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1套）含参考答案、逐题解析与评分细则丽水名校协作体2026届高三数学高考三模冲刺卷（第1套）数学·数列三角与解析几何联动满分150分考试时间120分钟学校：____________班级：____________姓名：____________准考证号：____________座位号：____________注意事项：1.答题前请填写学校、班级、姓名、准考证号和座位号；2.选择题答案必须写在答题栏，单项选择题只有一个正确选项，多项选择题全部选对得5分，部分选对得2分，有错选或不选得0分；3.填空题只写最终结果，解答题须写出必要文字说明、证明过程或演算步骤；4.本卷突出数列、三角、解析几何联动，兼顾函数与导数、立体几何、概率统计，适用于2026年高考三模冲刺讲评与查漏补缺。一、单项选择题（本大题共8小题，每小题5分，共40分。每小题只有一个选项符合题意。）1.已知复数z=3−4iA.−1/2B.7/2C.1/2D.−7/22.设集合A={x｜ln(2−x)有意义}，B={x｜x2−3x+2>0}A.(−∞,1)B.(1,2)C.(2,+∞)D.(−∞,2)3.若α∈(π2,3π4A.−√3/2B.√3/2C.−1/2D.1/24.已知向量a=(1,2)，b=(m,−1)，m>0，且(2a−b)⊥(a+b)，则m=（）A.(1+√29)/2B.(1−√29)/2C.√29−1D.75.二项式(x+1xA.10B.15C.20D.306.函数f(x)=ex−axA.1B.eC.ln2D.27.数列{an}满足aA.4+2√3B.3+2√3C.2+4√3D.68.椭圆x29+y2A.29/4B.31/4C.33/4D.35/4二、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，部分选对得2分，有错选或不选得0分。）9.已知f(x)=lnx+1A.定义域为(0,+∞)B.f′(x)=(x−1)/x²C.f在(0,1)上递减，在(1,+∞)上递增D.f的最小值为110.函数g(x)=2sin(x+πA.g(x)=sin2x+√3/2B.g的最小正周期为πC.g的值域为[√3/2−1,√3/2+1]D.g在x=π/2+kπ处取最大值11.等差数列{an}满足aA.公差d=2B.a₁=1C.Sₙ=n²D.点(n,Sₙ)恒在直线y=2x−1上12.圆C：(x−1)2+(y+2)A.k=1时，l与C无公共点B.k=0时，l与C相切C.k=2时，l与C有两个公共点D.对任意实数k，l与C都有公共点选择题答题栏（请将所选字母填入对应题号下方）：123456789101112三、填空题（本大题共4小题，每小题5分，共20分。）13.函数f(x)=e2xcosx14.在△ABC中，内角A，B，C的对边分别为a，b，c。若a=√3，b=2，C=30°，则c=__________。15.设an=sinnπ4，16.圆心在x轴上的圆过点A(0,1)，且与直线x+y−3=0相切。若圆心为(t,0)，则t的所有可能取值为__________。填空题答题栏（每空5分）：13141516填空题演算区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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）在△ABC中，角A，B，C所对的边分别为a，b，c，已知A=π3，b=2，c=1。

（1）求边a与△ABC的面积；

（2）令un=2sin(nA)+cos(nA)作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（12分）如图形关系用文字描述：四边形ABCD为边长2的正方形，PA⊥平面ABCD，PA=2，M为PB的中点，N为CD的中点。

（1）证明MN∥平面PAD；

（2）求直线MN与平面ABCD所成角的正弦值；

（3）求点C到平面PMN的距离。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（12分）某校高三三模复盘时，从“数列、三角、解析几何”错题卡中随机抽取题签做讲评。袋中共有9张题签，其中红色4张、蓝色3张、白色2张，从中不放回随机抽取3张。

（1）求恰有2张红色题签的概率；

（2）设X为抽到红色题签的张数，写出X的分布列并求E(X)；

（3）求抽到的3张题签中至少有1张白色题签的概率。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（12分）已知数列{an}满足a1=1，an+1=a作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（12分）已知椭圆E：x24+y2作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（12分）设函数fa(x)=ex−1−ax。

（1）讨论fa(x)的单调性；

（2）若直线y=ax与曲线y=作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________试题部分到此结束，请检查答题栏与解答题作答区。

参考答案与详解本部分按题号逐题给出答案、关键解析、易错点与评分细则。客观题以选项为准，解答题按步骤采分；若学生采用等价方法且逻辑完整，可参照对应关键结论给分。一、答案速查12345678AAAABBAB910111213141516ABCDABCABCABC211+√2/21或−7二、逐题解析与评分细则1.答案：A。z=3−4i2.答案：A。ln(2−x)有意义要求x<2；x2−3x+2=(x−1)(x−2)>0，得x<1或x>2。交集为3.答案：A。由(sinα+cosα)2=1+sin2α=12，得sin2α=−14.答案：A。2a−b=(2−m,5)，a+b=(1+m,1)。垂直得(2−m)(1+m)+5=0，即m2−m−7=0。因m>0，故5.答案：B。展开式通项为C(6,k)x6−2k，令6−2k=2，得k=2，系数6.答案：B。f′(x)=ex−a。若有最小值0，必有a>0，极小点x=lna。最小值为a−alna=a(1−lna)7.答案：A。a1到a6依次为8.答案：B。椭圆长轴2a=6，所以PF₁+PF₂=6；焦距c=√5。由PF₁²−PF₂²=4cx=6√5，得PF₁−PF₂=√5。故PF₁·PF₂=(PF₁+PF₂)²−(PF₁−PF₂)²9.答案：ABCD。定义域由x>0给出；f′(x)=1x−10.答案：ABC。由2sinAcosB=sin(A+B)+sin(A−B)，得g(x)=sin2x+√32，周期π，值域[√32−1,11.答案：ABC。由a7−a3=4d=8，得d=212.答案：ABC。圆心为(1,−2)，半径2。直线y=kx到圆心距离为|k+2|/sqrt(k²+1)。k=1时距离3/√2>2，无公共点；k=0时距离2，相切；k=2时距离4/√5<2，有两个公共点。D说任意k均有公共点错误。评分按多选规则给分。13.答案：2。f′(x)=2e2xcosx−14.答案：1。由余弦定理c2=a15.答案：1+√22。an=sinnπ16.答案：1或−7。圆心(t,0)，半径r=sqrt(t²+1)。与直线x+y−3=0相切，故|t−3|/√2=sqrt(t²+1)。平方得(t−3)²=2(t²+1)，即t2+6t−7=0，17.参考答案、解析与评分细则答案：a=√3，面积为√32，u解析：（1）由余弦定理a2=b2+（2）因为A=π3，对任意n，有三角递推关系。又2cosA=1，将正弦、余弦两式按2与1作线性组合，得（3）u1=√3+12，u2评分细则：第（1）问4分，其中余弦定理列式2分，边长与面积各1分；第（2）问3分，能写出三角递推关系2分，线性组合完成证明1分；第（3）问3分，列出周期或前6项2分，代入余数求值1分。易错点：把2026除以6的余数误写为2。替代解法提示：可直接用三角函数周期性求值。18.参考答案、解析与评分细则答案：MN∥平面PAD；直线MN与平面ABCD所成角的正弦值为1√5；点C到平面PMN的距离为2解析：建立空间直角坐标系，令A(0,0,0)，B(2,0,0)，C(2,2,0)，D(0,2,0)，P(0,0,2)。M为PB中点，M(1,0,1)；N为CD中点，N(1,2,0)。（1）MN=(0,2,−1)，平面PAD为x=0的平面，其方向向量均可写成(0,y,z)形式。因MN方向向量与平面PAD平行且M不在平面PAD内，故MN∥平面PAD。（2）MN在平面ABCD上的投影向量为(0,2,0)，竖直分量长度为1，|MN|=√5，故线面角θ满足sinθ=1（3）平面PMN中，PM=(1,0,−1)，PN=(1,2,−2)，法向量可取n=(2,1,2)。平面方程为2x+y+2z−4=0。点C(2,2,0)到该平面的距离为|4+2−4|/sqrt(4+1+4)=2/3。评分细则：建系并写出关键点坐标3分；第（1）问3分，方向向量判断2分，说明线不在面内1分；第（2）问3分，投影或法向量方法正确2分，结果1分；第（3）问3分，平面方程或法向量2分，距离公式1分。易错点：把线面角误算成MN与PA的夹角。19.参考答案、解析与评分细则答案：P(恰有2张红色)=514；E(X)=4解析：总取法数为C(9,3)=84。（1）恰有2红1非红，取法数