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            【福建成人高考】高起點數學文科難點一

            福建成人高考信息網www.laiwobuy.com 發布時間: 2018年11月10日

            難點1:集合思想及應用


            集合是高中數學的基本知識,為歷年必考內容之一,主要考查對集合基本概念的認識和理解,以及作為工具,考查集合語言和集合思想的運用.本節主要是幫助考生運用集合的觀點,不斷加深對集合概念、集合語言、集合思想的理解與應用.


            ●難點磁場


            (★★★★★)已知集合A={(x,y)|x2+mx-y+2=0},B={(x,y)|x-y+1=0,且0≤x≤2},如果A∩B≠,求實數m的取值范圍.


            ●案例探究


            [例1]設A={(x,y)|y2-x-1=0},B={(x,y)|4x2+2x-2y+5=0},C={(x,y)|y=kx+b},是否存在k、b∈N,使得(A∪B)∩C=,證明此結論.


            命題意圖:本題主要考查考生對集合及其符號的分析轉化能力,即能從集合符號上分辨出所考查的知識點,進而解決問題.屬★★★★★級題目.


            知識依托:解決此題的閃光點是將條件(A∪B)∩C=轉化為A∩C=且B∩C=,這樣難度就降低了.


            錯解分析:此題難點在于考生對符號的不理解,對題目所給出的條件不能認清其實質內涵,因而可能感覺無從下手.


            技巧與方法:由集合A與集合B中的方程聯立構成方程組,用判別式對根的情況進行限制,可得到b、k的范圍,又因b、k∈N,進而可得值.


            解:∵(A∪B)∩C=,∴A∩C=且B∩C=∵∴k2x2+(2bk-1)x+b2-1=0


            ∵A∩C=∴Δ1=(2bk-1)2-4k2(b2-1)<0


            ∴4k2-4bk+1<0,此不等式有解,其充要條件是16b2-16>0,即b2>1①


            ∵∴4x2+(2-2k)x+(5+2b)=0


            ∵B∩C=,∴Δ2=(1-k)2-4(5-2b)<0


            ∴k2-2k+8b-19<0,從而8b<20,即b<2.5②


            由①②及b∈N,得b=2代入由Δ1<0和Δ2<0組成的不等式組,得


            ∴k=1,故存在自然數k=1,b=2,使得(A∪B)∩C=.


            [例2]向50名學生調查對A、B兩事件的態度,有如下結果:贊成A的人數是全體的五分之三,其余的不贊成,贊成B的比贊成A的多3人,其余的不贊成;另外,對A、B都不贊成的學生數比對A、B都贊成的學生數的三分之一多1人.問對A、B都贊成的學生和都不贊成的學生各有多少人?


            命題意圖:在集合問題中,有一些常用的方法如數軸法取交并集,韋恩圖法等,需要考生切實掌握.本題主要強化學生的這種能力.屬★★★★級題目.


            知識依托:解答本題的閃光點是考生能由題目中的條件,想到用韋恩圖直觀地表示出來.


            錯解分析:本題難點在于所給的數量關系比較錯綜復雜,一時理不清頭緒,不好找線索.


            技巧與方法:畫出韋恩圖,形象地表示出各數量關系間的聯系.


            解:贊成A的人數為50×=30,贊成B的人數為30+3=33,如上圖,記50名學生組成的集合為U,贊成事件A的學生全體為集合A;贊成事件B的學生全體為集合B.


            設對事件A、B都贊成的學生人數為x,則對A、B都不贊成的學生人數為+1,贊成A而不贊成B的人數為30-x,贊成B而不贊成A的人數為33-x.


            依題意(30-x)+(33-x)+x+(+1)=50,解得x=21.


            所以對A、B都贊成的同學有21人,都不贊成的有8人.


            ●錦囊妙計


            1.解答集合問題,首先要正確理解集合有關概念,特別是集合中元素的三要素;對于用描述法給出的集合{x|x∈P},要緊緊抓住豎線前面的代表元素x以及它所具有的性質P;要重視發揮圖示法的作用,通過數形結合直觀地解決問題.


            2.注意空集的特殊性,在解題中,若未能指明集合非空時,要考慮到空集的可能性,如AB,則有A=或A≠兩種可能,此時應分類討論.


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