2026年山西太原高一数学月考阶段检测原创模拟试卷第068套（含参考答案与分步解析）

学校：________________班级：________姓名：________考号：________________高一数学月考阶段检测原创模拟试卷满分：120分考试时间：120分钟本卷复习范围：集合与函数基础、指数与对数、三角函数、平面向量、解三角形及阶段综合应用。请将选择题和填空题答案填写在答题栏内，解答题写出必要步骤。答题要求：选择题须选出唯一正确选项；填空题只写最终结果；解答题应体现模型建立、公式代入、化简计算和结论回扣，涉及长度或面积的实际问题应标明单位。选择题答题栏题号12345678910答案填空题答题栏题号111213141516答案一、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）1.已知集合A={x｜x²-5x+6≤0}，B={x｜1≤x<4}，则A∩B为（）A.[1,3]B.[2,3]C.(2,3)D.[2,4)2.若α∈(π,3π/2)，且sinα=-3/5，则tanα的值为（）A.-4/3B.4/3C.-3/4D.3/43.已知向量a=(2,-1)，b=(m,3)，且a⊥(a+b)，则m的值为（）A.-2B.-1C.0D.14.函数f(x)=log₂(x+1)-1的零点是（）A.0B.1C.2D.35.在△ABC中，已知边a=4，b=5，夹角C=60°，则边c的长为（）A.√21B.3C.√41D.76.函数y=2sin(2x-π/3)的最小正周期及图象平移描述正确的是（）A.周期为2π，向右平移π/3B.周期为π，向右平移π/6C.周期为π，向左平移π/6D.周期为π/2，向右平移π/37.方程3^(2x)-4·3^x+3=0的实数解集为（）A.{0}B.{1}C.{0,1}D.{-1,1}8.在平行四边形ABCD中，设AB=a，AD=b，点E在BD上且BE:ED=1:2，则AE可表示为（）A.1/3a+2/3bB.2/3a+1/3bC.1/2a+1/2bD.a+1/3b9.函数f(x)=x²-2ax+1在区间[0,2]上的最小值为-3，则a的值为（）A.1B.2C.3D.-210.已知x∈(0,π)，且sinx+cosx=√2/2，则sin2x的值为（）A.1/2B.-1/2C.√3/2D.-√3/2二、填空题（本大题共6小题，每小题3分，共18分）11.150°化为弧度制为____________。12.已知a=(1,2)，b=(3,-1)，若a+λb与a垂直，则λ=____________。13.函数f(x)=sin(ωx+π/4)(ω>0)的最小正周期为π/3，则ω=____________。14.在△ABC中，若AB=6，AC=8，∠A=120°，则△ABC的面积为____________。15.方程log₂(x-1)+log₂(5-x)=2的解为____________。16.已知|a|=2，|b|=3，a·b=3，则|2a-b|=____________。三、解答题（本大题共6小题，每小题8分，共48分。解答应写出文字说明、演算步骤或证明过程）17.（8分）已知α∈(π/2,π)，tanα=-2。（1）求sinα、cosα的值；（2）解方程2sin²x-3sinx+1=0，x∈[0,2π)。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________18.（8分）在平面直角坐标系中，A(1,2)，B(5,0)，C(2,6)。（1）求AB、AC的坐标及AB·AC的值；（2）点P在线段BC上，且BP:PC=t:1。若AP⊥BC，求t的值。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________19.（8分）函数f(x)=Asin(ωx+φ)+k（A>0，ω>0，|φ|<π/2）的最大值为3，最小值为-1，最小正周期为π，且f(0)=1，图象在x=0处递增。（1）求f(x)的解析式；（2）求f(x)=2在区间[0,π]内的所有解。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________20.（8分）在△ABC中，记a=BC，b=CA，c=AB。已知A=45°，B=60°，a=6。（1）求角C以及边b、c的长；（2）求△ABC的面积。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________21.（8分）校园开展测量实践活动。旗杆CD垂直于水平地面，A、B、C在同一条直线上，A在B与C之间，AB=20m。在A处测得旗杆顶端D的仰角为60°，在B处测得仰角为30°。（1）求AC的长；（2）求旗杆CD的高度。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________22.（8分）已知函数f(x)=log₂(x+1)-log₂(3-x)。（1）写出函数定义域，并证明f(x)在定义域上单调递增；（2）解不等式f(x)≤1；（3）若方程f(x)=m在区间[0,2]上有解，求实数m的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________四、综合压轴题（本大题共2小题，每小题12分，共24分。请写出完整推理过程）23.（12分）设函数F(x)=sinx+cosx+a·sinxcosx，x∈[0,π]，其中a为实数。（1）当a=0时，求方程F(x)=1的解；（2）当a=2时，求F(x)在[0,π]上的最大值和最小值；（3）若对任意x∈[0,π]，恒有F(x)≥-1，求a的取值范围。________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________24.（12分）在△ABC中，AB=AC=2，∠BAC=120°。点P在线段BC上，设BP:PC=x:(1-x)，其中0≤x≤1。过P作PQ∥AB，交AC于点Q。（1）用x表示AP²，并求AP的最小值；（2）求使AP⊥BC的x的值；（3）用x表示△APQ的面积，并求当该面积不小于√3/5时x的取值范围。_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________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参考答案与解析一、选择题答案题号12345678910答案BDBBABCBBB客观题解析采用“确定范围—代入条件—排除干扰”的顺序给出。每题选项均按唯一答案设置，计算题给出关键式，概念题给出判定依据。1.解析：x²-5x+6≤0可化为(x-2)(x-3)≤0，所以A=[2,3]；与B=[1,4)取交集得[2,3]，选B。2.解析：α在第三象限，sinα<0，cosα<0。由sinα=-3/5得cosα=-4/5，所以tanα=sinα/cosα=3/4，选D。3.解析：a+b=(m+2,2)，由a⊥(a+b)得2(m+2)+(-1)·2=0，化简为2m+2=0，m=-1，选B。4.解析：零点满足log₂(x+1)-1=0，即log₂(x+1)=1，得x+1=2，x=1，选B。5.解析：由余弦定理c²=a²+b²-2abcosC=16+25-40·1/2=21，故c=√21，选A。6.解析：y=2sin[2(x-π/6)]，最小正周期T=2π/2=π，可看成y=2sin2x向右平移π/6个单位，选B。7.解析：设u=3^x(u>0)，则u²-4u+3=0，解得u=1或u=3，因此x=0或x=1，选C。8.解析：取A为起点，则B=a，D=b。点E在BD上且BE:ED=1:2，所以E=B+1/3(D-B)=2/3a+1/3b，选B。9.解析：f(x)=(x-a)²+1-a²。当a∈[0,2]时最小值为1-a²；由1-a²=-3得a=2。若a>2，端点x=2的最小值为5-4a，等于-3仍得a=2；若a<0最小值为1，不合。选B。10.解析：两边平方得(sinx+cosx)²=1+sin2x=1/2，因此sin2x=-1/2。结合x∈(0,π)可知该值可取，选B。二、填空题答案与解析11.答案：5π/6。解析：150°=150·π/180=5π/6。12.答案：-5。解析：a+λb=(1+3λ,2-λ)，与a=(1,2)垂直，则(1+3λ)+2(2-λ)=0，得5+λ=0，所以λ=-5。13.答案：6。解析：sin(ωx+π/4)的最小正周期为2π/ω，由2π/ω=π/3得ω=6。14.答案：12√3。解析：面积S=1/2·AB·AC·sinA=1/2·6·8·sin120°=24·√3/2=12√3。15.答案：3。解析：定义域为1<x<5。由对数运算得log₂[(x-1)(5-x)]=2，故(x-1)(5-x)=4，化简为(x-3)²=0，x=3。16.答案：√13。解析：|2a-b|²=4|a|²+|b|²-4a·b=4·4+9-12=13，所以|2a-b|=√13。三、解答题参考答案与分步解析17.参考答案与解析（1）因为α∈(π/2,π)，所以sinα>0，cosα<0。由tanα=-2可设sinα=2k，cosα=-k（k>0）。利用sin²α+cos²α=1，得4k²+k²=1，即k=√5/5。因此sinα=2√5/5，cosα=-√5/5。（2）2sin²x-3sinx+1=0可分解为(2sinx-1)(sinx-1)=0，所以sinx=1/2或sinx=1。在x∈[0,2π)内，sinx=1/2的解为x=π/6，5π/6；sinx=1的解为x=π/2。故解集为{π/6，π/2，5π/6}。评分点：象限符号判断2分，三角值求解2分，方程因式分解2分，区间内解集2分。规范要点：求三角值时先写象限，解三角方程时必须把给定区间内的所有角列全。18.参考答案与解析（1）AB=(5-1,0-2)=(4,-2)，AC=(2-1,6-2)=(1,4)。AB·AC=4·1+(-2)·4=4-8=-4。（2）BC=(2-5,6-0)=(-3,6)。由于BP:PC=t:1，点P的坐标为P=((5+2t)/(t+1),6t/(t+1))。于是AP=((5+2t)/(t+1)-1,6t/(t+1)-2)=((t+4)/(t+1),(4t-2)/(t+1))。由AP⊥BC得AP·BC=0，即-3(t+4)+6(4t-2)=0，化简为21t-24=0，所以t=8/7。评分点：向量坐标2分，数量积2分，点P坐标2分，垂直条件与求参2分。规范要点：线段定比分点应说明权重设置，垂直条件应写成数量积为0后再化简。19.参考答案与解析（1）最大值为3、最小值为-1，因此A=(3-(-1))/2=2，k=(3+(-1))/2=1。最小正周期为π，所以2π/ω=π，得ω=2。由f(0)=1得2sinφ+1=1，即sinφ=0。又|φ|<π/2，所以φ=0。图象在x=0处递增，因为f′(0)=2·2·cos0=4>0，符合条件。故f(x)=2sin2x+1。（2）令2sin2x+1=2，得sin2x=1/2。当x∈[0,π]时，2x∈[0,2π]，所以2x=π/6或5π/6，解得x=π/12或5π/12。评分点：由最值定A、k2分，由周期定ω2分，由初值与递增定φ2分，解方程2分。规范要点：三角函数解析式不能只凭图象形状猜测，需要逐一落实振幅、中线、周期和初相。20.参考答案与解析（1）C=180°-45°-60°=75°。由正弦定理a/sinA=b/sinB=c/sinC，得b=6·sin60°/sin45°=6·(√3/2)/(√2/2)=3√6。又sin75°=sin(45°+30°)=(√6+√2)/4，故c=6·sin75°/sin45°=3(√3+1)。（2）面积S=1/2·a·c·sinB=1/2·6·3(√3+1)·√3/2=(27+9√3)/2。评分点：角C1分，正弦定理使用2分，求b2分，求c1分，面积计算2分。规范要点：边角对应关系要一致，面积公式中选用的两边必须与所夹角配套。21.参考答案与解析设AC=xm，旗杆高度CD=hm，则BC=x+20。在直角三角形ACD中，tan60°=h/x，所以h=√3x。在直角三角形BCD中，tan30°=h/(x+20)，所以h=(x+20)/√3。联立√3x=(x+20)/√3，得3x=x+20，解得x=10。因此AC=10m。代入h=√3x，得CD=10√3m。评分点：设未知量1分，两个仰角关系各2分，联立求AC2分，求高度1分。规范要点：实际测量题应画出或描述直角三角形，最后结论写清米这一单位。22.参考答案与解析（1）由x+1>0且3-x>0，得定义域为(-1,3)。将函数写成f(x)=log₂[(x+1)/(3-x)]。设g(x)=(x+1)/(3-x)，在(-1,3)上分母为正，且g(x)随x增大而增