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2023年高考数学二轮复习(新高考版)第2部分思想方法 第3讲 分类讨论思想_2.2025数学总复习_2023年新高考资料_二轮复习_2023年高考数学二轮复习讲义+课件(新高考版)

  • 2026-03-10 04:11:47 2026-03-10 04:11:47

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2023年高考数学二轮复习(新高考版)第2部分思想方法 第3讲 分类讨论思想_2.2025数学总复习_2023年新高考资料_二轮复习_2023年高考数学二轮复习讲义+课件(新高考版)
2023年高考数学二轮复习(新高考版)第2部分思想方法 第3讲 分类讨论思想_2.2025数学总复习_2023年新高考资料_二轮复习_2023年高考数学二轮复习讲义+课件(新高考版)
2023年高考数学二轮复习(新高考版)第2部分思想方法 第3讲 分类讨论思想_2.2025数学总复习_2023年新高考资料_二轮复习_2023年高考数学二轮复习讲义+课件(新高考版)
2023年高考数学二轮复习(新高考版)第2部分思想方法 第3讲 分类讨论思想_2.2025数学总复习_2023年新高考资料_二轮复习_2023年高考数学二轮复习讲义+课件(新高考版)

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第 3 讲 分类讨论思想 思想概述 分类讨论思想是当问题的对象不能进行统一研究时,需对研究的对象按某个标 准进行分类,然后对每一类分别研究,给出每一类的结论,最终综合各类结果得到整个问题 的解答.实质上分类讨论就是“化整为零,各个击破,再集零为整”的数学思想. 方法一 由概念、公式、法则、计算性质引起的讨论 概念、定理分类整合即利用数学中的基本概念、定理对研究对象进行分类,如绝对值的 定义、不等式的转化、等比数列{a}的前n项和公式等,然后分别对每类问题进行解决. n 例1 (1)(2022·滁州质检)已知过点P(0,1)的直线l与圆x2+y2+2x-6y+6=0相交于A,B两 点,则当|AB|=2时,直线l的方程为( ) A.x=0 B.15x-8y-8=0 C.3x-4y+4=0或x=0 D.3x+4y-4=0或x=0 思路分析 设直线方程→k存在,l:y=kx+1→圆心到直线l的距离d=1求解→斜率不存在, l:x=0. 答案 D 解析 因为圆x2+y2+2x-6y+6=0化为(x+1)2+(y-3)2=4, 所以圆心为(-1,3),半径为r=2, 因为|AB|=2, 所以圆心到直线的距离为d=1, 当直线l的斜率不存在时,直线l的方程为x=0, 此时圆心(-1,3)到直线的距离为1,满足条件; 当直线l的斜率存在时,设斜率为k,直线l的方程为y=kx+1, 圆心到直线l的距离为d=, 所以=1,解得k=-, 此时直线方程为3x+4y-4=0, 综上,所求直线方程为3x+4y-4=0或x=0. (2)已知数列{a}满足a=-2,a=2,a -2a=1-(-1)n,则下列选项不正确的是( ) n 1 2 n+2 n A.{a }是等比数列 2n-1 B.(a +2)=-10 2i-1C.{a }是等比数列 2n D.=52 i 思路分析 a -2a=1--1n→n为奇,{a }为等比数列;n为偶,{a }为等比数列 n+2 n 2n-1 2n 答案 B 解析 对于A,当n是奇数时,a -2a=2, n+2 n 所以a +2=2(a+2), n+2 n 又因为a=-2,所以a+2=0, 1 1 所以当n是奇数时,a+2=0,即a=-2, n n 即{a }是以首项为-2,公比为1的等比数列, 2n-1 即选项A正确; 对于B,由A知,当n是奇数时,a+2=0, n 所以(a +2)=0, 2i-1 即选项B错误; 对于C,当n为偶数时,a -2a=0, n+2 n 即a =2a, n+2 n 又因为a=2,所以=2, 2 所以{a }是以首项为2,公比为2的等比数列, 2n 故选项C正确; 对于D, =(a+a+a+a+a)+(a+a+a+a+a ) i 1 3 5 7 9 2 4 6 8 10 =-10+=52,即选项D正确. 批注 涉及数列中(-1)n的问题,一般需分奇偶讨论,当n为奇数时,首项是a ,a 是第个 1 n 奇数项;当n为偶数时,首项是a,a 是第个偶数项. 2 n 规律方法 解题时应准确把握数学概念的本质,根据需要对所有情形分类.本例中,设直线 方程需分斜率存在和不存在两种情况,数列中含(-1)n需分奇偶两种情况,要注意分类讨论 要有理有据、不重不漏. 方法二 由图形位置或形状引起的讨论 图形位置、形状分类整合是指由几何图形的不确定性而引起的分类讨论,这种方法适用 于对几何图形中点、线、面的位置关系以及解析几何中直线与圆锥曲线的位置关系的研究. 例2 设F ,F 为椭圆+=1的两个焦点,点P为椭圆上一点,已知点P,F ,F 是一个直 1 2 1 2 角三角形的三个顶点,且|PF|>|PF|,则=________. 1 2答案 或2 解析 若∠PFF=90°, 2 1 则|PF|2=|PF|2+|FF|2, 1 2 1 2 又|PF|+|PF|=6,|FF|=2, 1 2 1 2 解得|PF|=,|PF|=,∴=. 1 2 若∠FPF=90°,则|FF|2=|PF|2+|PF|2, 1 2 1 2 1 2 ∴|PF|2+(6-|PF|)2=20, 1 1 又|PF|>|PF|,∴|PF|=4,|PF|=2, 1 2 1 2 ∴=2. 综上可知,=或2. 批注 P,F ,F 是一个直角三角形的三个顶点,并没有说明哪个点是直角顶点,所以需分 1 2 类讨论,仔细审题,理解题意是关键. 规律方法 圆锥曲线的形状、焦点位置不确定时要分类讨论;立体几何中点、线、面的位置 变化,三角形和平行四边形的不确定性都要进行分类讨论. 方法三 由参数变化引起的分类讨论 某些含有参数的问题,由于参数的取值不同会导致所得的结果不同,需对参数进行讨论, 如含参数的方程、不等式、函数等.解决这类问题要根据需要合理确定分类标准,讨论中做 到不重不漏,结论整合要周全. 例3 (2022·湖北七市(州)联考)已知函数f(x)=x+(x>0),若的最大值为,则正实数a= ________. 答案 1 解析 令t=x+(x>0),则t≥2, 则==, 令y=t+(a>0,t≥2), 当04时,t+≥2(当且仅当t=时,等号成立),则0<≤, 即的最大值为,则=, 解得a=(舍), 综上,所求正实数a=1. 规律方法 若遇到题目中含有参数的问题,常常结合参数的意义和对结果的影响进行分类讨 论,此类题目为含参型,应全面分析参数变化引起的结论的变化情况,在分类讨论时要遵循 分类的原则:一是分类的标准要一致,二是分类时要做到不重不漏,三是能不分类的要尽量 避免分类,杜绝无原则的分类讨论.
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  1. CONNECT:[ UseTime:0.000983s ] mysql:host=127.0.0.1;port=3306;dbname=wenku;charset=utf8mb4
  2. SHOW FULL COLUMNS FROM `fenlei` [ RunTime:0.001702s ]
  3. SELECT * FROM `fenlei` WHERE `fid` = 0 [ RunTime:0.000735s ]
  4. SELECT * FROM `fenlei` WHERE `fid` = 63 [ RunTime:0.000660s ]
  5. SHOW FULL COLUMNS FROM `set` [ RunTime:0.001305s ]
  6. SELECT * FROM `set` [ RunTime:0.000527s ]
  7. SHOW FULL COLUMNS FROM `wendang` [ RunTime:0.001490s ]
  8. SELECT * FROM `wendang` WHERE `id` = 369212 LIMIT 1 [ RunTime:0.000961s ]
  9. UPDATE `wendang` SET `liulancishu` = 1 WHERE `id` = 369212 [ RunTime:0.001451s ]
  10. UPDATE `wendang` SET `lasttime` = 1773087129 WHERE `id` = 369212 [ RunTime:0.001214s ]
  11. SELECT * FROM `wendang` WHERE `id` < 369212 ORDER BY `id` DESC LIMIT 1 [ RunTime:0.001029s ]
  12. SELECT * FROM `wendang` WHERE `id` > 369212 ORDER BY `id` ASC LIMIT 1 [ RunTime:0.001075s ]
  13. SELECT * FROM `wendang` WHERE `id` <> 369212 ORDER BY `id` DESC LIMIT 10 [ RunTime:0.002450s ]
  14. SELECT * FROM `wendang` WHERE `id` <> 369212 ORDER BY `id` DESC LIMIT 10,10 [ RunTime:0.002869s ]
  15. SELECT * FROM `wendang` WHERE `id` <> 369212 ORDER BY `id` DESC LIMIT 20,10 [ RunTime:0.006650s ]
0.191109s