2026届高一数学高一学业水平模拟模拟试卷（广东专用版·学生练习版含答案详解与评分标准）

2026届高一数学高一学业水平模拟模拟试卷（广东专用版·学生练习版，含答案详解与评分标准）试卷卷头学校：班级：姓名：考号：考试时间：120分钟满分：120分范围：广东专用版题量：26题注意事项：1.本卷为学生练习版，满分120分，考试时间120分钟。2.选择题请在答题栏填写选项，填空题请把结果写在横线上，解答题写出必要的公式、运算过程和结论。3.作图、计算和推理题须保持步骤清楚；书写答案时不得把不同题号答案混写。4.允许使用常规作图工具，不使用计算器。题型与分值题型题号题量每题分值合计选择题1—10103分30分填空题11—1663分18分解答题17—26106—8分72分全卷1—2626—120分选择题（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）选择题答题栏：123456789101.（3分）已知集合A={x｜x²−5x+6=0}，B={1，2，3，4}，则A∩B等于（）A.{1，2}B.{2，3}C.{3，4}D.{1，4}2.（3分）若0<a<1，则下列关系正确的是（）A.log_a2<log_a3B.log_a2=log_a3C.log_a2>log_a3D.log_a2与log_a3无法比较3.（3分）函数y=√(2x−1)+1/(x−3)的定义域是（）A.[1/2，+∞)B.(3，+∞)C.[1/2，3)∪(3，+∞)D.(−∞，3)∪(3，+∞)4.（3分）若角α的终边在第二象限，且sinα=3/5，则cosα的值为（）A.4/5B.−4/5C.3/5D.−3/55.（3分）已知向量a=(2，−1)，b=(−1，3)，则2a+b等于（）A.(3，1)B.(1，5)C.(5，−5)D.(−3，1)6.（3分）指数函数f(x)=aˣ（a>0，a≠1）的图象经过点(2，9)，则a+f(−1)等于（）A.8/3B.10/3C.4D.28/37.（3分）等差数列{a_n}中，a₁=4，a₇=22，则公差d为（）A.2B.3C.4D.68.（3分）函数y=2sinx在区间[0，2π]上的最大值是（）A.−2B.0C.1D.29.（3分）一个圆柱的底面半径为3，高为5，则它的体积为（）A.15πB.30πC.45πD.90π10.（3分）某班5名同学一分钟跳绳个数分别为108，116，121，121，134，则这组数据的中位数是（）A.116B.121C.123D.134填空题（本大题共6小题，每小题3分，共18分。请把正确结果写在对应横线上）11.（3分）方程log₂(x−1)=3的解为x=。答：______________________________________________________________12.（3分）等差数列{a_n}中，a₁=3，d=2，则前10项和S₁₀=。答：______________________________________________________________13.（3分）若sinα=5/13，cosα=12/13，sinβ=3/5，cosβ=4/5，且α，β均为锐角，则cos(α+β)=。答：______________________________________________________________14.（3分）已知向量u=(t，2)，v=(4，−1)，若u⊥v，则t=。答：______________________________________________________________15.（3分）袋中有大小、质地相同的红球3个、白球2个，随机取出1个球，取到红球的概率为。答：______________________________________________________________16.（3分）函数f(x)=x²−4x+1在区间[0，3]上的最小值为。答：______________________________________________________________解答题（本大题共10小题，共72分。解答应写出文字说明、证明过程或演算步骤）17.（6分）已知集合A={x｜−1≤x≤5}，B={x｜x²−4x≤0}。

（1）求A∩B；

（2）若C={x｜x<m}，且B⊆C，求实数m的取值范围。作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（6分）某校为布置校园读书角，计划用一段长为20m的围栏靠墙围成一个矩形区域（靠墙一边不用围栏）。设垂直于墙的两边长均为xm，围成面积为Sm²。

（1）写出S关于x的函数关系式，并给出x的取值范围；

（2）当x取何值时，面积S最大？最大面积是多少？作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（7分）已知角θ满足0<θ<π，且cosθ=−1/2。

（1）求sinθ的值；

（2）求2sinθ−tanθ的值；

（3）判断θ所在象限，并说明理由。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（7分）在平面直角坐标系中，A(1，2)，B(5，4)，C(3，−2)。

（1）求向量AB与AC的坐标；

（2）求|AB|；

（3）求证：三角形ABC为直角三角形，并指出直角顶点。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（7分）某学习小组记录连续5天完成数学练习的题量，构成等差数列{a_n}。已知第1天完成18题，第5天完成34题。

（1）求这5天每天完成题量；

（2）求5天总题量；

（3）若第6天起继续按同样规律增加，求第8天完成题量。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（8分）如图形情境所述：一个正四棱柱的底面是边长为4cm的正方形，高为6cm。

（1）求该正四棱柱的体积；

（2）求它的表面积；

（3）若从一个顶点A到与其不在同一底面的相对顶点C₁连接一条空间线段，求AC₁的长度。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________23.（8分）为了解高一学生每天课外体育锻炼时间，某校随机抽取40名学生，统计结果如下表。

（1）补全频数合计，并求样本中锻炼时间不少于60分钟的学生人数；

（2）估计该校高一学生每天课外体育锻炼时间不少于60分钟的比例；

（3）若从“30≤t<60”组中随机选2人参加访谈，请说明这是抽样调查中的哪类简单随机方式，并写出保证公平的做法。锻炼时间t（分钟）0≤t<3030≤t<6060≤t<9090≤t<120合计人数614155作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（8分）某水库在一次降雨后开始排水，监测到t小时后水位相对安全水位的高度h（单位：m）可近似表示为h(t)=3×0.8ᵗ（t≥0）。

（1）求开始排水时的h(0)；

（2）求排水2小时后的高度h(2)；

（3）若要求h(t)<1.5，结合0.8³=0.512，0.8⁴=0.4096，判断至少需要排水多少小时。作答区：____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________25.（8分）已知函数f(x)=2sin(x+π/6)，x∈[0，2π]。

（1）写出函数的最小正周期；

（2）求f(x)的最大值和取得最大值时x的一个取值；

（3）求f(x)=1在[0，2π]内的所有解。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________26.（7分）已知函数f(x)=x²−2ax+1，其中a为实数。

（1）当a=2时，求f(x)在区间[0，3]上的最小值；

（2）若f(x)在区间[1，3]上的最小值为−3，求a的值；

（3）若对任意x∈[0，2]，都有f(x)≥−2，求a的取值范围。作答区：________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

参考答案与解析说明：本答案给出参考结果、关键步骤、易错点和采分点。解答题评分时，过程正确但结果因非关键算术失误受到影响的，按对应步骤酌情给分；未写必要过程直接给出结论的，通常不得取得该题满分。一、选择题答案与解析题号12345678910答案BCCBABBDCB1.答案：B。解析：x²−5x+6=(x−2)(x−3)，所以A={2，3}，与B={1，2，3，4}的公共元素为2，3。A、C、D都把不满足方程或不在公共部分的元素放入结果。2.答案：C。解析：当0<a<1时，对数函数y=log_ax在(0，+∞)上单调递减。因为2<3，所以log_a2>log_a3。A把递减关系看成递增关系，是常见误判。3.答案：C。解析：根式要求2x−1≥0，得x≥1/2；分式要求x−3≠0，得x≠3。合并为[1/2，3)∪(3，+∞)。A未排除x=3，B缺少[1/2，3)部分。4.答案：B。解析：sin²α+cos²α=1，得|cosα|=4/5。α在第二象限，余弦为负，所以cosα=−4/5。5.答案：A。解析：2a+b=2(2，−1)+(−1，3)=(4，−2)+(−1，3)=(3，1)。6.答案：B。解析：f(2)=a²=9且a>0，所以a=3。f(−1)=3⁻¹=1/3，因此a+f(−1)=3+1/3=10/3。7.答案：B。解析：等差数列a₇=a₁+6d，22=4+6d，得d=3。8.答案：D。解析：sinx在[0，2π]上的最大值为1，所以y=2sinx的最大值为2。9.答案：C。解析：圆柱体积V=πr²h=π×3²×5=45π。10.答案：B。解析：数据已从小到大排列：108，116，121，121，134。共有5个数，中间第3个数为121。二、填空题答案与解析题号111213141516答案912033/651/23/5−311.解析：log₂(x−1)=3，得x−1=2³=8，所以x=9。需注意真数x−1>0，所得x=9满足条件。12.解析：S₁₀=10/2×[2×3+(10−1)×2]=5×24=120。13.解析：cos(α+β)=cosαcosβ−sinαsinβ=(12/13)(4/5)−(5/13)(3/5)=48/65−15/65=33/65。14.解析：u⊥v时u·v=0，即4t+2×(−1)=0，4t−2=0，t=1/2。15.解析：共有3+2=5个球，红球3个，取到红球的概率为3/5。16.解析：f(x)=x²−4x+1=(x−2)²−3。顶点x=2在[0，3]内，所以最小值为−3。三、解答题答案、解析与评分标准17.答案：（1）A∩B=[0，4]；（2）m>4。解析：B={x｜x²−4x≤0}={x｜x(x−4)≤0}=[0，4]。A=[−1，5]，故A∩B=[0，4]。若C={x｜x<m}且B⊆C，则区间[0，4]中每个数都要小于m，尤其端点4也要小于m，所以m>4。易错点：把B⊆C误写成m≥4会使x=4不属于C，因此不满足包含关系。评分标准：求出B=[0，4]给2分；求出A∩B=[0，4]给2分；由端点包含关系得到m>4给2分。18.答案：（1）S=x(20−2x)=20x−2x²，0<x<10；（2）x=5时，S最大为50m²。解析：两条垂直于墙的边共用2xm围栏，平行于墙的另一边为20−2xm。面积S=x(20−2x)=20x−2x²。由边长为正，得x>0，20−2x>0，所以0<x<10。二次函数S=−2(x−5)²+50，开口向下，故x=5时取得最大值50。易错点：不要把靠墙一边也算入围栏；取值范围必须保证矩形两边都为正。评分标准：正确列出面积函数给2分；写出0<x<10给1分；完成配方或顶点计算给2分；写出最大面积及单位给1分。19.答案：（1）sinθ=√3/2；（2）2sinθ−tanθ=2√3；（3）θ在第二象限。解析：0<θ<π且cosθ=−1/2，说明θ在第二象限。由sin²θ+cos²θ=1，得sinθ=√(1−1/4)=√3/2。tanθ=sinθ/cosθ=(√3/2)/(−1/2)=−√3，所以2sinθ−tanθ=√3−(−√3)=2√3。易错点：在0<θ<π内，余弦为负时角在第二象限，正弦应取正值。评分标准：判断象限给1分；求sinθ给2分；求tanθ给2分；代入并化简给2分。20.答案：（1）AB=(4，2)，AC=(2，−4)；（2）|AB|=2√5；（3）AB·AC=0，直角顶点为A。解析：AB=(5−1，4−2)=(4，2)，AC=