2026届常州市高三数学考前压轴QS01黑白可打印标准付费预览仿真卷B1第0380套（含参考答案、逐题解析、评分细则、压轴题讲评与学生作答空间）

2026届常州市高三数学考前压轴QS01黑白可打印标准付费预览仿真卷B1第0380套（含参考答案、逐题解析、评分细则、压轴题讲评与学生作答空间）考试时间：120分钟满分：150分适用：2026届常州市高三考前整卷训练注意事项：本卷为QS01黑白可打印标准付费预览仿真卷B1第0380套。试题区与参考答案区分开排版；主观题下方设学生作答空间；教师讲评时可依据答案区的逐题解析、评分细则和压轴题讲评组织订正。1.作答前请检查页码、题号与答题栏，确认无缺页后再开始限时训练。2.选择题请在答题栏填写选项；填空题写出规范结果；解答题须写出必要推理、运算过程和结论。3.全卷突出考前压轴训练节奏，基础题力求稳准，中档题强调综合迁移，压轴题重在审题、建模、转化与步骤得分。一、选择题：本大题共10小题，每小题4分，共40分。每小题只有一个选项符合题意。1.（4分）已知集合A={x|x²-5x+6≤0}，B={x|ln(x-1)≥0}，则A∩B为A.[2，3]B.(1，3]C.[1，2]D.[3，+∞)2.（4分）复数z=(1+2i)/(1-i)，则|z|等于A.√5B.√10/2C.3/2D.5/23.（4分）已知α为第一象限角，且cosα=3/5，则sin(α+π/3)等于A.(3+4√3)/10B.(4+3√3)/10C.(4-3√3)/10D.7/104.（4分）等差数列{aₙ}满足a₁+a₃+a₅=15，a₂+a₄+a₆=21，则a₁₀=A.17B.18C.19D.215.（4分）二项式(x-2/x)⁶展开式中x²项的系数为A.-60B.15C.60D.1206.（4分）函数f(x)=eˣ+x-2的零点所在区间为A.(-1，0)B.(0，1)C.(1，2)D.(2，3)7.（4分）从5名女生和3名男生中任取2人，则所取2人中至少有1名男生的概率为A.5/14B.9/14C.3/7D.11/148.（4分）在平行四边形ABCD中，设向量AB=a，向量AD=b，M为BC的中点，N在CD上且CN:ND=1:2，则向量MN=A.-a/3+b/2B.a/3+b/2C.-a/2+b/3D.a/2-b/39.（4分）椭圆x²/9+y²/4=1的离心率为A.√5/2B.√5/3C.2/3D.5/910.（4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选择题答题栏12345678910二、填空题：本大题共5小题，每小题4分，共20分。请把答案填写在横线上。11.（4分）函数f(x)=x³-3x，则f′(2)=________。答：____________________________________________________________________12.（4分）数列{aₙ}满足a₁=1，aₙ₊₁=2aₙ+1，则a₅=________。答：____________________________________________________________________13.（4分）点P(3，-1)到直线2x-y+1=0的距离为________。答：____________________________________________________________________14.（4分）若tanθ=2，则sin2θ=________。答：____________________________________________________________________15.（4分）若x≥0，y≥0，x+y≤4，2x+y≤5，则z=x+2y的最大值为________。答：____________________________________________________________________三、解答题：本大题共10小题，共90分。解答应写出文字说明、证明过程或演算步骤。16.（8分）在△ABC中，已知A为锐角，sinA=3/5，b=6，c=8。

（1）求△ABC的面积；

（2）求边a的长。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________17.（8分）已知数列{aₙ}满足a₁=2，aₙ₊₁=3aₙ-2。

（1）证明{aₙ-1}是等比数列；

（2）求数列{aₙ}的前n项和Sₙ。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（8分）某班考前限时训练后，抽取10名学生的数学小题得分如下：76，82，88，90，92，84，86，88，94，100。

（1）求这10个数据的平均数和中位数；

（2）从得分不低于90分的学生中随机抽取2人，求恰有1人的得分为90分的概率。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（8分）已知函数f(x)=x³-3ax。

（1）当a=1时，求f(x)的单调区间与极值；

（2）若f(x)在区间[0，2]上的最大值为2，求实数a的值。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（10分）已知椭圆C：x²/4+y²=1，直线l：y=kx+1与椭圆C相交于点P(0，1)和另一点Q。

（1）求点Q的坐标；

（2）若弦PQ的长为2，求k的值。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（10分）如图形情境所示：在正方体ABCD-A1B1C1D1中，棱长为2，E为BB1的中点，F为CD的中点。

（1）证明EF平行于平面A1BD；

（2）求直线EF与底面ABCD所成角的正弦值。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（10分）某同学进行3道同类型综合小题训练，每道题独立作答且做对的概率均为0.7。设随机变量X表示做对的题数。

（1）写出X的分布列；

（2）求E(X)与P(X≥2)；

（3）若老师要求至少做对2题才算达标，求该同学达标后恰好做对2题的概率。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（10分）已知函数phi(x)=lnx-x+1（x>0）。

（1）证明phi(x)≤0，并指出等号成立的条件；

（2）利用（1）的结论证明：对任意正数a，b，有ln((a+b)/2)≥(lna+lnb)/2；

（3）说明第（2）问中等号成立的条件。答：__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（9分）已知抛物线C：y²=4x，焦点为F。过F的直线与C交于A，B两点，直线斜率为m（m≠0），AB的中点为M。

（1）求弦长AB关于m的表达式；

（2）求点M的轨迹方程；

（3）结合计算结果说明本题的转化关键。答：______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（9分）已知函数f(x)=eˣ-ax-1。

（1）当a=1时，求f(x)的零点个数；

（2）讨论f(x)在实数集上的单调性；

（3）若f(x)≥0对一切x≥0恒成立，求实数a的取值范围。答：______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案、逐题解析、评分细则与压轴题讲评本区按题号给出参考答案、解析思路和步骤得分点。选择题、填空题的评分以答案正确为主；解答题按关键步骤给分，结论正确但缺少必要过程时应扣除相应过程分。一、选择题答案与解析12345678910ABBCCBBABB1.答案：A。解析：A=[2，3]，B=[2，+∞)，交集为[2，3]。评分细则：选对得4分，错选、多选或不选得0分。2.答案：B。解析：z=(1+2i)(1+i)/2=(-1+3i)/2，故|z|=√(1+9)/2=√10/2。评分细则：选对得4分，错选、多选或不选得0分。3.答案：B。解析：由α为第一象限角得sinα=4/5，所以sin(α+π/3)=sinαcosπ/3+cosαsinπ/3=(4+3√3)/10。评分细则：选对得4分，错选、多选或不选得0分。4.答案：C。解析：a₁+a₃+a₅=3a₃=15，得a₃=5；a₂+a₄+a₆=3a₄=21，得a₄=7，公差d=2，a₁₀=a₄+6d=19。评分细则：选对得4分，错选、多选或不选得0分。5.答案：C。解析：通项为C(6,k)x⁶⁻ᵏ(-2/x)ᵏ=C(6,k)(-2)ᵏx⁶⁻²ᵏ。令6-2k=2，得k=2，系数为C(6,2)*4=60。评分细则：选对得4分，错选、多选或不选得0分。6.答案：B。解析：f′(x)=eˣ+1>0，函数严格递增；f(0)=-1，f(1)=e-1>0，所以唯一零点在(0，1)。评分细则：选对得4分，错选、多选或不选得0分。7.答案：B。解析：反面为取到2名女生，概率为C(5,2)/C(8,2)=10/28=5/14，所求概率为1-5/14=9/14。评分细则：选对得4分，错选、多选或不选得0分。8.答案：A。解析：取A为原点，则M=a+b/2，N=2a/3+b，所以向量MN=N-M=-a/3+b/2。评分细则：选对得4分，错选、多选或不选得0分。9.答案：B。解析：a=3，b=2，c=√(a²-b²)=√5，离心率e=c/a=√5/3。评分细则：选对得4分，错选、多选或不选得0分。10.答案：B。解析：正四面体体积V=(√2/12)a³，代入a=2得V=2√2/3。评分细则：选对得4分，错选、多选或不选得0分。二、填空题答案与解析11.答案：9。解析：f′(x)=3x²-3，故f′(2)=12-3=9。评分细则：结果化简等价得4分；计算过程正确但最终化简有轻微失误可给2分。12.答案：31。解析：令bₙ=aₙ+1，则bₙ₊₁=2bₙ，b₁=2，所以b₅=2⁵=32，a₅=31。评分细则：结果化简等价得4分；计算过程正确但最终化简有轻微失误可给2分。13.答案：8√5/5。解析：距离d=|2*3-(-1)+1|/√(2²+(-1)²)=8/√5=8√5/5。评分细则：结果化简等价得4分；计算过程正确但最终化简有轻微失误可给2分。14.答案：4/5。解析：sin2θ=2tanθ/(1+tan²θ)=4/5。评分细则：结果化简等价得4分；计算过程正确但最终化简有轻微失误可给2分。15.答案：8。解析：可行域顶点为(0，0)，(5/2，0)，(1，3)，(0，4)，代入z=x+2y，最大值为8。评分细则：结果化简等价得4分；计算过程正确但最终化简有轻微失误可给2分。三、解答题参考答案、逐题解析与评分细则16.答案与解析（1）因为A为锐角，sinA=3/5，所以cosA=4/5。三角形面积S=1/2*bc*sinA=1/2*6*8*3/5=72/5。（2）由余弦定理，a²=b²+c²-2bccosA=36+64-96*4/5=116/5，所以a=√(116/5)=2√145/5。步骤得分点分值1由锐角和正弦值得到cosA=4/52分2正确写出并计算面积公式2分3正确运用余弦定理列式2分4求出边长并化简2分17.答案与解析（1）由aₙ₊₁=3aₙ-2得aₙ₊₁-1=3(aₙ-1)。又a₁-1=1，所以{aₙ-1}是首项为1、公比为3的等比数列。（2）aₙ-1=3ⁿ⁻¹，故aₙ=3ⁿ⁻¹+1。于是Sₙ=(1+3+…+3ⁿ⁻¹)+n=(3ⁿ-1)/2+n。步骤得分点分值1将递推式转化为aₙ₊₁-1=3(aₙ-1)3分2说明首项与公比，完成等比证明2分3写出通项aₙ=3ⁿ⁻¹+12分4求出前n项和Sₙ=(3ⁿ-1)/2+n1分18.答案与解析（1）10个数据和为880，平均数为88。按从小到大排列为76，82，84，86，88，88，90，92，94，100，中位数为(88+88)/2=88。（2）得分不低于90分的有90，92，94，100，共4人。从4人中任取2人共有C(4,2)=6种；恰有1人得90分时，另一人从其余3人中取，共3种，概率为3/6=1/2。步骤得分点分值1求出总分和平均数2分2排序并求出中位数2分3确定不低于90分的样本个数2分4正确列组合数并求概率2分19.答案与解析（1）当a=1时，f(x)=x³-3x，f′(x)=3x²-3=3(x-1)(x+1)。因此f(x)在(-∞，-1)和(1，+∞)上单调递增，在(-1，1)上单调递减。x=-1处取得极大值f(-1)=2；x=1处取得极小值f(1)=-2。（2）在[0，2]上，f(0)=0，f(2)=8-6a。当a≤0时，f′(x)=3x²-3a≥0，最大值f(2)=8-6a=2，得a=1，与a≤0矛盾。当a>0时，需比较端点与可能的极值点。f′(x)=0得x=√a。若0<√a<2，则x=√a为极小点，最大值只能在端点取得；因f(0)=0，故最大值为max{0，8-6a}。要使最大值为2，必须8-6a=2，得a=1。此时f(2)=2，符合条件。故a=1。步骤得分点分值1求导并判断单调区间3分2给出极大值、极小值2分3分析[0，2]端点与驻点的最大值来源2分4求得并检验a=11分20.答案与解析把y=kx+1代入x²/4+y²=1，得x²/4+(kx+1)²=1，即(k²+1/4)x²+2kx=0。一个根为x=0，对应P(0，1)，另一根对应的Q点横坐标=-2k/(k²+1/4)=-8k/(4k²+1)。于是Q点纵坐标=kQ点横坐标+1=1-8k²/(4k²+1)=(1-4k²)/