2026届吉林省长春市高三统编版数学高考二轮冲刺作文范文升级专项训练卷B010（含参考答案解析与考点清单）

吉林省长春市高三数学高考二轮冲刺原创试卷B010黑色可打印本卷满分150分；考试时间120分钟；含学生作答区、答案逐题解析与评分标准2026届吉林省长春市高三统编版数学高考二轮冲刺作文范文升级专项训练卷B010（含参考答案解析与考点清单）考试卷头项目1适用范围：吉林省长春市2026届高三二轮复习；科目版本：统编版数学复习体系项目2考试节点：高考二轮冲刺；试卷性质：原创试卷；批次：B010项目3考试时间：120分钟；满分：150分；答题形式：闭卷，黑色可打印项目4特色题型：数学作文范文升级与规范表达；交付内容：答案逐题解析、评分标准、学生作答区注意事项1.本卷共23题。选择题、填空题与解答题均按题号顺序作答，答案写在相应作答区内；书写须清楚、步骤须完整。2.单项选择题每题只有一个正确选项；多项选择题每题至少有两个正确选项，全部选对得满分，漏选且无错选得部分分，有错选或不选得0分。3.解答题应写出必要的文字说明、演算步骤或证明过程；仅写结论但无关键过程，按评分标准酌情扣分。4.本卷围绕高考二轮冲刺的综合提升目标命题，突出函数与导数、三角与向量、解析几何、立体几何、概率统计、数列与数学表达升级。5.第23题为数学作文范文升级题，重点考查“把思路写成高考可评分证明”的能力。所有题目均为黑色版式，适合直接打印。二轮冲刺定位本卷难度定位为“基础保分+综合提升+表达规范”三段式：第1—16题用于排查常见概念、公式与运算漏洞；第17—22题用于训练高考二轮阶段的多步骤推理、参数讨论、空间向量与概率统计综合；第23题聚焦数学作文范文升级，即把草稿式思路改写为阅卷可识别、逻辑可追踪、结论可核验的规范证明。建议学生完成后按“答案正确、过程完整、理由充分、书写清楚、等号条件明确”五项自评。教师可据考点清单定位薄弱模块，安排函数导数、圆锥曲线、立体几何和概率统计四类二轮专题回炉。学生作答区选择题答题卡题号123456789101112答案题号13141516填空题作答13题14题15题16题解答题作答提示：第17—23题在各题题干下方作答区作答。若空间不足，可在本页空白处补写并标明题号。试题部分一、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.已知集合A={x|x²-5x+6≤0}，B={1，2，4}，则A∩B=（）。A.{1}B.{2}C.{4}D.{2，4}2.复数z=(1+i)²/(1-i)，则z=（）。A.-1+iB.1+iC.-1-iD.1-i3.向量a=(2，-1)，b=(1，3)，则a在b方向上的数量投影为（）。A.-1/√10B.1/√10C.-1/10D.1/104.已知α∈(0，π/2)，tanα=2，则sin2α=（）。A.2/5B.3/5C.4/5D.5/45.函数f(x)=lnx-x+1（x>0）的零点个数为（）。A.0B.1C.2D.无穷多个6.二项式(2x-1/x)^5的展开式中x的系数为（）。A.40B.60C.80D.1007.数列{a_n}满足a₁=1，a_{n+1}=2a_n+1，则a₅=（）。A.15B.31C.32D.638.一个盒中有3个红球、2个蓝球，任取2个球，则至少取到1个红球的概率为（）。A.3/5B.4/5C.9/10D.1/10二、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，漏选且无错选得2分，有错选或不选得0分。9.抛物线C：y²=4x的有关结论中，正确的是（）。A.焦点为(1，0)B.准线方程为x=-1C.点(1，-2)不在C上D.过焦点且垂直于对称轴的弦长为410.函数f(x)=x³-3x的性质中，正确的是（）。A.f(x)为奇函数B.f(x)在(-∞，-1)和(1，+∞)上单调递增C.f(x)在x=-1处取得极大值2D.方程f(x)=0只有两个实根11.直三棱柱ABC-A₁B₁C₁的底面△ABC为直角三角形，AB=AC=2，AA₁=3，则正确的是（）。A.体积为6B.侧面积为12+6√2C.底面外接圆半径为1D.线段BC₁的长为√1712.关于函数g(x)=|x-1|+|x+1|，下列说法正确的是（）。A.g(x)的最小值为2B.g(x)为偶函数C.当x≥1时，g(x)=2xD.不等式g(x)≤3的解集为[-3/2，3/2]三、填空题：本大题共4小题，每小题5分，共20分。请把答案填写在题后横线上。13.一组数据2，3，4，6，m的平均数为4，则m=__________。14.方程log₂(x-1)+log₂(x+1)=3的解为x=__________。15.已知数列{a_n}的前n项和S_n=n²+2n，则a₁₀=__________。16.圆(x-1)²+(y+2)²=25上的点满足x+2y的最大值为__________。四、解答题：本大题共7小题，每小题10分，共70分。解答应写出文字说明、证明过程或演算步骤。17.在△ABC中，已知a=√3，b=1，夹角C=30°。

（1）求边c；（2）求△ABC的面积和角A的大小。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________18.已知正项数列{a_n}满足a₁=2，a_{n+1}=a_n/(a_n+1)。

（1）证明数列{1/a_n}为等差数列；（2）求a_n，并求Σ_{k=1}^n1/a_k。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________19.设函数f(x)=lnx-ax+a，定义域为(0，+∞)。

（1）当a=1时，求f(x)的单调区间并证明f(x)≤0；（2）若对任意x>0都有f(x)≤0，求实数a的取值。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________20.椭圆E：x²/4+y²=1。

（1）求椭圆E的离心率；（2）取椭圆上一点P(1，√3/2)，求E在P处的切线方程，并求该切线与坐标轴围成三角形的面积。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________21.某篮球运动员进行3次独立投篮，每次命中概率均为0.6。设X为命中次数。

（1）写出X的分布列并求E(X)；（2）若规则规定“至少命中2次”为达标，求达标概率；（3）已知已经至少命中1次，求恰好命中2次的条件概率。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________22.如图意描述：在棱长为2的正方体ABCD-A₁B₁C₁D₁中，取常规空间直角坐标系A(0，0，0)，B(2，0，0)，D(0，2，0)，A₁(0，0，2)。

（1）求直线AC₁与底面ABCD所成角的正弦值；（2）求点D₁到平面AB₁C的距离。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________23.数学作文范文升级题。已知学生甲要证明不等式lnx≤x-1（x>0），其草稿写法为：“因为函数h(x)=x-1-lnx的导数h′(x)=1-1/x，所以h(x)≥0，故lnx≤x-1。”

（1）指出这段草稿在高考规范作答中的4处不足；（2）把它升级为一段完整、严谨、可评分的证明范文；（3）说明等号成立条件。作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________作答区：________________________________________________________________________________________

学生补充作答纸本页为第17—23题续写区。请在补写处标明题号，保持步骤清楚、结论明确。17题续写区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18题续写区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19题续写区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20题续写区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21题续写区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22题续写区：_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与逐题解析说明：选择题与填空题给出唯一答案或全部正确选项；解答题给出关键步骤、评分标准与易错提醒。阅卷评分通则1.结果与过程并重。选择题按答案给分；填空题只看最终结果，但若结果含多个值、区间或条件，缺任一关键部分不得满分。解答题必须体现核心依据，计算中间值虽有误但方法正确，可按后续步骤酌情给分。2.规范表达计入主观题评分。涉及函数单调性、最值、几何位置关系、概率条件事件、参数范围时，应写明定义域、事件含义、变量限制、等号成立条件或分类依据。只写“显然”“可得”而未给出关键理由的，原则上不超过该小问一半分。3.二轮冲刺重点看迁移能力。对于第19、20、22、23题，若学生能建立正确模型但计算有轻微误差，应保留建模、公式、逻辑分；若模型选择错误导致后续结论失真，则按评分点逐项扣除。客观题答案速查题号123456789101112答案BAACBCBCABDABCABDABCD题号13141516题号13141516答案5321-3+5√51.答案：B。解析：x²-5x+6≤0，即(x-2)(x-3)≤0，得2≤x≤3，所以A=[2，3]。集合B中落在该区间内的元素只有2，故A∩B={2}。评分点：能求出A的区间得3分，完成交集判断得2分。易错点：把不等式方向判反，或把3误认为B中元素。2.答案：A。解析：(1+i)²=1+2i+i²=2i，z=2i/(1-i)=2i(1+i)/[(1-i)(1+i)]=2i(1+i)/2=i+i²=-1+i。评分点：正确化简平方2分，有理化2分，写出结果1分。易错点：漏写i²=-1。3.答案：A。解析：a·b=2×1+(-1)×3=-1，|b|=√(1²+3²)=√10，a在b方向上的数量投影为(a·b)/|b|=-1/√10。评分点：点积2分，模长1分，投影公式2分。易错点：把向量投影写成(a·b)/|b|²。4.答案：C。解析：sin2α=2tanα/(1+tan²α)=2×2/(1+4)=4/5。因α在第一象限，结果为正。评分点：使用二倍角正切形式4分，符号判断1分。易错点：把sin2α与tan2α混淆。5.答案：B。解析：f′(x)=1/x-1。x∈(0，1)时f′(x)>0，x∈(1，+∞)时f′(x)<0，所以f在x=1处取得最大值。f(1)=0，且两侧均小于0，故仅有一个零点x=1。评分点：导数与单调性3分，最大值判断1分，零点结论1分。易错点：看到对数函数与直线相交就误判为两个零点。6.答案：C。解析：通项为T_{k+1}=C(5,k)(2x)^{5-k}(-1/x)^k=C(5,k)2^{5-k}(-1)^kx^{5-2k}。令5-2k=1，得k=2，因此x项系数为C(5,2)2³(-1)²=10×8=80。评分点：通项3分，指数方程1分，系数计算1分。易错点：把常数项与x项混淆。7.答案：B。解析：由a_{n+1}+1=2(a_n+1)，且a₁+1=2，可得a_n+1=2^n，所以a_n=2^n-1，a₅=32-1=31。评分点：构造a_n+1得3分，通项1分，代入1分。易错点：把a₅写成2⁵。8.答案：C。解析：从5个球中取2个共有C(5,2)=10种。至少一个红球的反面是两个全为蓝球，概率为C(2,2)/C(5,2)=1/10，所求概率为1-1/10=9/10。评分点：总数2分，反面事件2分，结果1分。易错点：误把“至少一个”按恰好一个计算。9.答案：ABD。解析：y²=2px中2p=4，p=2，焦点为(p/2，0)=(1，0)，准线x=-1，A、B正确；(1，-2)满足(-2)²=4×1，在抛物线上，C错误；通径长为2p=4，D正确。评分点：每个选项判断有依据；漏选无错选得部分分。易错点：混淆y²=2px与y²=4ax的参数记法。10.答案：ABC。解析：f(-x)=-x³+3x=-(x³-3x)，所以是奇函数，A正确。f′(x)=3x²-3=3(x-1)(x+1)，在(-∞，-1)、(1，+∞)上为正，B正确；在x=-1处由增到减，f(-1)=2，为极大值，C正确。f(x)=x(x²-3)=0有三个实根，D错误。易错点：只看导数零点个数而忽视x=0也是零点。11.答案：ABD。解析：底面积S=1/2×2×2=2，体积V=2×3=6，A正确。底面周长为2+2+2√2，直三棱柱侧面积为周长×高=3(4+2√2)=12+6√2，B正确。直角三角形外接圆半径为斜边一半，即√2，C错误。BC=2√2，BC₁=√[(2√2)²+3²]=√17，D正确。12.答案：ABCD。解析：g(x)表示数轴上点x到1和-1距离之和，最小值为两定点距离2，A正确；函数关于原点对称，B正确。当x≥1时，g(x)=x-1+x+1=2x，C正确。分段可得-1≤x≤1时g=2；x≥1时2x≤3得x≤3/2；x≤-1时-2x≤3得x≥-3/2，故解集[-3/2，3/2]，D正确。13.答案：5。解析：平均数为4表示(2+3+4+6+m)/5=4，15+m=20，m=5。评分点：列式3分，求解2分。14.答案：3。解析：定义域要求x>1。由对数运算得log₂[(x-1)(x+1)]=3，即x²-1=8，x²=9。结合定义域x>1，得x=3。易错点：保留x=-3而未检验定义域。15.答案：21。解析：a_n=S_n-S_{n-1}。S₁₀=10²+20=120，S₉=9²+18=99，故a₁₀=21。也可由n≥2时a_n=2n+1得a₁₀=21。16.答案：-3+5√5。解析：圆心为(1，-2)，半径为5。线性函数x+2y在圆上的最大值等于圆心处取值加上半径乘以方向向量(1，2)的模，即1+2(-2)+5√(1²+2²)=-3+5√5。评分点：圆心半径2分，线性函数极值公式或几何投影2分，结果1分。解答题参考答案、评分标准与易错提醒17.答案与解析：由余弦定理，c²=a²+b²-2abcosC=3+1-2√3×1×cos30°=4-2√3×