2026年高考数学核心知识点精讲及试题

2026年高考数学核心知识点精讲及试题考试时长：120分钟满分：100分班级：__________姓名：__________学号：__________得分：__________一、单选题（总共10题，每题2分，总分20分）1.函数f(x)=log₃(x²-2x+3)的定义域为（）A.(-∞,1)∪(1,+∞)B.[1,3]C.RD.(-1,3)2.若向量a=(1,k),b=(2,-1),且a⊥b，则k的值为（）A.-2B.2C.-1/2D.1/23.抛掷两枚均匀的骰子，记事件A为“点数之和为7”，事件B为“点数之和为偶数”，则P(B|A)等于（）A.1/6B.1/3C.1/2D.2/34.已知数列{aₙ}满足a₁=1,aₙ₊₁=2aₙ+1，则a₅的值为（）A.31B.63C.127D.2555.直线y=kx+1与圆(x-1)²+y²=1相交于两点，则k的取值范围是（）A.(-∞,-1)∪(1,+∞)B.(-1,1)C.[-1,1]D.(-∞,-1)∪(1,+∞)6.已知f(x)=sin(x+α)，若f(π/4)=√2/2且f(π/2)>0，则α的取值范围是（）A.[0,π/4]B.[π/4,π/2]C.[π/2,3π/4]D.[3π/4,π]7.某几何体的三视图如图所示（主视图为正方形，左视图为矩形，俯视图为矩形），则该几何体的体积为（）A.8B.16C.24D.328.已知函数f(x)=x³-ax²+bx在x=1处取得极值，且f(1)=0，则a+b的值为（）A.3B.4C.5D.69.在△ABC中，角A、B、C的对边分别为a、b、c，若a²+b²-c²=ab，则cosC的值为（）A.1/2B.1/3C.1/4D.1/510.已知椭圆C:x²/9+y²/4=1的焦点为F₁、F₂，P为C上一点，若∠F₁PF₂=90°，则|PF₁|+|PF₂|的值为（）A.6B.8C.10D.12二、填空题（总共10题，每题2分，总分20分）11.若函数f(x)=x²+px+q在x=1处取得最小值，且f(0)=3，则p+q的值为______。12.已知向量a=(3,1),b=(m,2)，且|a+b|=5，则m的值为______。13.从6名男生和4名女生中随机选出3人参加比赛，则恰好选出2名女生的概率为______。14.已知数列{aₙ}的前n项和为Sₙ，若aₙ=Sₙ-Sₙ₋₁（n≥2），且a₁=2，则S₅的值为______。15.圆(x-2)²+(y+1)²=r²与直线y=x+1相切，则r的值为______。16.已知f(x)=cos²(x-π/4)-sin²(x-π/4)，则f(π/2)的值为______。17.在△ABC中，角A、B、C的对边分别为a、b、c，若a=3,b=4,c=5，则cosB的值为______。18.已知双曲线C:x²/16-y²/9=1的焦点为F₁、F₂，P为C右支上一点，若|PF₁|-|PF₂|=2，则|PF₁|的值为______。19.已知函数f(x)=e^x-ax在x=1处取得极值，则a的值为______。20.在直三棱柱ABC-A₁B₁C₁中，底面△ABC的面积为6，侧棱长为3，则该三棱柱的体积为______。三、判断题（总共10题，每题2分，总分20分）21.若函数f(x)=x³-3x+1在区间[-2,2]上的最大值与最小值分别为4和-2，则f(x)在[-2,2]上单调递增。22.已知直线l₁:ax+by=c与直线l₂:mx+ny=p相交于点P，则a/m=b/n。23.在等差数列{aₙ}中，若a₃+a₇=12，则a₅的值为6。24.若f(x)=sin(x+α)是偶函数，则α=kπ+π/2（k∈Z）。25.在圆锥中，若底面半径为r，母线长为l，则侧面积为πrl。26.已知椭圆C:x²/9+y²/4=1的焦点为F₁、F₂，P为C上一点，则|PF₁|+|PF₂|=6。27.若函数f(x)=x²-2x+3在区间[1,3]上单调递增，则f(x)在[1,3]上的最小值为1。28.在直角坐标系中，点A(1,2)与点B(3,0)的距离为√10。29.已知函数f(x)=log₃(x²-2x+3)在区间(1,+∞)上单调递增。30.在△ABC中，若a²+b²>c²，则cosC>0。四、简答题（总共4题，每题4分，总分16分）31.已知函数f(x)=x³-3x²+2，求f(x)的单调区间。32.在△ABC中，角A、B、C的对边分别为a、b、c，若a=5,b=7,cosC=1/2，求c的值。33.已知数列{aₙ}满足a₁=1,aₙ₊₁=aₙ+n（n∈N），求aₙ的通项公式。34.求函数f(x)=sin(2x-π/4)在区间[0,π]上的最大值和最小值。五、应用题（总共4题，每题6分，总分24分）35.某工厂生产一种产品，固定成本为10万元，每件产品的可变成本为20元，售价为50元。若销售量为x件，求该工厂的利润函数P(x)的表达式，并求销售量为1000件时的利润。36.在直三棱柱ABC-A₁B₁C₁中，底面△ABC的边长分别为AB=3,AC=4,BC=5，侧棱长为4。求该三棱柱的体积。37.已知函数f(x)=x²-mx+4在x=2处取得最小值，且f(1)=0，求m的值。38.某学校组织一次数学竞赛，共有100名参赛选手，其中男生60名，女生40名。若随机抽取3名选手，求恰好抽到2名男生的概率。【标准答案及解析】一、单选题1.C解析：x²-2x+3>0对任意x∈R恒成立，故定义域为R。2.A解析：a•b=1×2+k×(-1)=0，解得k=2。3.C解析：点数之和为7的组合为(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)，共6种；点数之和为偶数的组合为(1,1),(1,3),(1,5),(2,2),(2,4),(3,3),(3,5),(4,4),(4,6),(5,5),(5,7),(6,6),(6,8),(7,7),(7,9),(8,8),(8,10),(9,9),(9,11),(10,10),(10,12),(11,11),(11,13),(12,12),(12,14),(13,13),(13,15),(14,14),(14,16),(15,15),(15,17),(16,16),(16,18),(17,17),(17,19),(18,18),(18,20),(19,19),(19,21),(20,20),(20,22),(21,21),(21,23),(22,22),(22,24),(23,23),(23,25),(24,24),(24,26),(25,25),(25,27),(26,26),(26,28),(27,27),(27,29),(28,28),(28,30),(29,29),(29,31),(30,30),(30,32),(31,31),(31,33),(32,32),(32,34),(33,33),(33,35),(34,34),(34,36),(35,35),(35,37),(36,36),(36,38),(37,37),(37,39),(38,38),(38,40),(39,39),(39,41),(40,40),(40,42),(41,41),(41,43),(42,42),(42,44),(43,43),(43,45),(44,44),(44,46),(45,45),(45,47),(46,46),(46,48),(47,47),(47,49),(48,48),(48,50),(49,49),(49,51),(50,50),(50,52),(51,51),(51,53),(52,52),(52,54),(53,53),(53,55),(54,54),(54,56),(55,55),(55,57),(56,56),(56,58),(57,57),(57,59),(58,58),(58,60),(59,59),(59,61),(60,60),(60,62),(61,61),(61,63),(62,62),(62,64),(63,63),(63,65),(64,64),(64,66),(65,65),(65,67),(66,66),(66,68),(67,67),(67,69),(68,68),(68,70),(69,69),(69,71),(70,70),(70,72),(71,71),(71,73),(72,72),(72,74),(73,73),(73,75),(74,74),(74,76),(75,75),(75,77),(76,76),(76,78),(77,77),(77,79),(78,78),(78,80),(79,79),(79,81),(80,80),(80,82),(81,81),(81,83),(82,82),(82,84),(83,83),(83,85),(84,84),(84,86),(85,85),(85,87),(86,86),(86,88),(87,87),(87,89),(88,88),(88,90),(89,89),(89,91),(90,90),(90,92),(91,91),(91,93),(92,92),(92,94),(93,93),(93,95),(94,94),(94,96),(95,95),(95,97),(96,96),(96,98),(97,97),(97,99),(98,98),(98,100),(99,99),(99,101),(100,100),(100,102),(101,101),(101,103),(102,102),(102,104),(103,103),(103,105),(104,104),(104,106),(105,105),(105,107),(106,106),(106,108),(107,107),(107,109),(108,108),(108,110),(109,109),(109,111),(110,110),(110,112),(111,111),(111,113),(112,112),(112,114),(113,113),(113,115),(114,114),(114,116),(115,115),(115,117),(116,116),(116,118),(117,117),(117,119),(118,118),(118,120),(119,119),(119,121),(120,120),(120,122),(121,121),(121,123),(122,122),(122,124),(123,123),(123,125),(124,124),(124,126),(125,125),(125,127),(126,126),(126,128),(127,127),(127,129),(128,128),(128,130),(129,129),(129,131),(130,130),(130,132),(131,131),(131,133),(132,132),(132,134),(133,133),(133,135),(134,134),(134,136),(135,135),(135,137),(136,136),(136,138),(137,137),(137,139),(138,138),(138,140),(139,139),(139,141),(140,140),(140,142),(141,141),(141,143),(142,142),(142,144),(143,143),(143,145),(144,144),(144,146),(145,145),(145,147),(146,146),(146,148),(147,147),(147,149),(148,148),(148,150),(149,149),(149,151),(150,150),(150,152),(151,151),(151,153),(152,152),(152,154),(153,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