再讀教材2025高考數(shù)學專項復(fù)習第六章數(shù)列第六章數(shù)列[知識網(wǎng)絡(luò)][命題方向]1．本章高考試題難度以中檔為主，題型一般為一小(選擇題或填空題)一大(解答題)，總分值約為17分．(1)在高考試題中多以等差數(shù)列、等比數(shù)列的基本量運算為載體，以數(shù)列遞推關(guān)系形式出現(xiàn)，考查數(shù)列求和及數(shù)列最值等綜合問題．(2)在處理等差、等比數(shù)列基本量運算，遞推關(guān)系求通項，數(shù)列求和等問題時，常用公式法．(3)本章重點考查的學科核心素養(yǎng)為數(shù)學運算和邏輯推理．2．考查內(nèi)容也較為穩(wěn)定，主要是以下幾個方面：(1)以等差、等比數(shù)列基本量的運算為載體，考查等差數(shù)列、等比數(shù)列的概念、性質(zhì)、通項公式的求解與應(yīng)用；(2)考查數(shù)列求和的綜合問題，涉及數(shù)列的最值及解決方法；(3)考查數(shù)學文化、實際應(yīng)用為背景的數(shù)列問題．探究1(人教A版選擇性必修第二冊P14)觀察等差數(shù)列的通項公式，你認為它與我們熟悉的哪一類函數(shù)有關(guān)？_______________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版選擇性必修第二冊P22)已知數(shù)列{an}的前n項和為Sn＝pn2＋qn＋r，其中p，q，r為常數(shù)，且p≠0.觀察數(shù)列{an}的特點，研究它是一個怎樣的數(shù)列，并證明你的結(jié)論．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究3(人教A版選擇性必修第二冊P24)已知等差數(shù)列{an}的前n項和為Sn，若a1＝10，當d＝－3.5時，Sn有最大值嗎？考慮更一般的等差數(shù)列前n項和的最大值問題．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究4(人教A版選擇性必修第二冊P32)已知b＞0且b≠1，如果數(shù)列{an}是等差數(shù)列，那么數(shù)列{ban}是否一定是等比數(shù)列？如果數(shù)列{an}是各項均為正的等比數(shù)列，那么數(shù)列{logban}是否一定是等差數(shù)列？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題1(人教A版選擇性必修第二冊P9習題4.1T7)已知函數(shù)f(x)＝eq\f(2x－1,2x)(x∈R)，設(shè)數(shù)列{an}的通項公式為an＝f(n)(n∈N*)．(1)求證an≥eq\f(1,2).(2){an}是遞增數(shù)列還是遞減數(shù)列？為什么？______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題2(人教A版選擇性必修第二冊P16例3)某公司購置了一臺價值為220萬元的設(shè)備，隨著設(shè)備在使用過程中老化，其價值會逐年減少．經(jīng)驗表明，每經(jīng)過一年其價值就會減少d(d為正常數(shù))萬元．已知這臺設(shè)備的使用年限為10年，超過10年，它的價值將低于購進價值的5%，設(shè)備將報廢．請確定d的取值范圍．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題3(人教A版選擇性必修第二冊P17例5)已知數(shù)列{an}是等差數(shù)列，p，q，s，t∈N*，且p＋q＝s＋t.求證ap＋aq＝as＋at.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題4(人教A版選擇性必修第二冊P18練習T4)已知數(shù)列{an}，{bn}都是等差數(shù)列，公差分別為d1，d2，數(shù)列{cn}滿足cn＝an＋2bn.(1)數(shù)列{cn}是否是等差數(shù)列？若是，證明你的結(jié)論；若不是，請說明理由．(2)若{an}，{bn}的公差都等于2，a1＝b1＝1，求數(shù)列{cn}的通項公式．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題5(人教A版選擇性必修第二冊P18練習T5)已知一個無窮等差數(shù)列{an}的首項為a1，公差為d.(1)將數(shù)列中的前m項去掉，其余各項組成一個新的數(shù)列，這個新數(shù)列是等差數(shù)列嗎？如果是，它的首項和公差分別是多少？(2)取出數(shù)列中的所有奇數(shù)項，組成一個新的數(shù)列，這個新數(shù)列是等差數(shù)列嗎？如果是，它的首項和公差分別是多少？(3)取出數(shù)列中所有序號為7的倍數(shù)的項，組成一個新的數(shù)列，它是等差數(shù)列嗎？你能根據(jù)得到的結(jié)論作出一個猜想嗎？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題6(人教A版選擇性必修第二冊P21例6)已知數(shù)列{an}是等差數(shù)列．(1)若a1＝7，a50＝101，求S50；(2)若a1＝2，a2＝eq\f(5,2)，求S10；(3)若a1＝eq\f(1,2)，d＝－eq\f(1,6)，Sn＝－5，求n._________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題7(人教A版選擇性必修第二冊P21例7)已知一個等差數(shù)列{an}前10項的和是310，前20項的和是1220.由這些條件能確定這個等差數(shù)列的首項和公差嗎？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題8(人教A版選擇性必修第二冊P23練習T4)在等差數(shù)列{an}中，若S15＝5(a2＋a6＋ak)，求k.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·全國甲卷)記Sn為等差數(shù)列{an}的前n項和．若a2＋a6＝10，a4a8＝45，則S5＝()A．25 B．22C．20 D．15點評本題與教材習題都是考查等差數(shù)列中的基本量的計算．典題9(人教A版選擇性必修第二冊P23練習T5)已知一個等差數(shù)列的項數(shù)為奇數(shù)，其中所有奇數(shù)項的和為290，所有偶數(shù)項的和為261.求此數(shù)列中間一項的值以及項數(shù)．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版選擇性必修第二冊P23例9)已知等差數(shù)列{an}的前n項和為Sn，若a1＝10，公差d＝－2，則Sn是否存在最大值？若存在，求Sn的最大值及取得最大值時n的值；若不存在，請說明理由．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·全國甲卷)記Sn為數(shù)列{an}的前n項和，已知eq\f(2Sn,n)＋n＝2an＋1.(1)證明：{an}是等差數(shù)列；(2)若a4，a7，a9成等比數(shù)列，求Sn的最小值．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________點評本題與教材例題都是考查前n項和Sn的最值問題．典題11(人教A版選擇性必修第二冊P24練習T5)已知數(shù)列{an}的通項公式為an＝eq\f(n－2,2n－15)，前n項和為Sn，求Sn取得最小值時n的值．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題12(人教A版選擇性必修第二冊P25習題4.2T2)已知{an}為等差數(shù)列，a1＋a3＋a5＝105，a2＋a4＋a6＝99.求a20.______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2020·全國Ⅰ卷)設(shè){an}是等比數(shù)列，且a1＋a2＋a3＝1，a2＋a3＋a4＝2，則a6＋a7＋a8＝()A．12 B．24C．30 D．32點評本題在所給條件的形式上與教材習題很類似，只是把等差數(shù)列改為了等比數(shù)列，解題的難點和關(guān)鍵點分別是求公差、公比．典題13(人教A版選擇性必修第二冊P25習題4.2T7)已知Sn是等差數(shù)列{an}的前n項和．(1)證明eq\b\lc\{\rc\}(\a\vs4\al\co1(\f(Sn,n)))是等差數(shù)列；(2)設(shè)Tn為數(shù)列eq\b\lc\{\rc\}(\a\vs4\al\co1(\f(Sn,n)))的前n項和，若S4＝12，S8＝40，求Tn._________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題14(人教A版選擇性必修第二冊P25習題4.2T8)已知兩個等差數(shù)列2，6，10，…，190及2，8，14，…，200，將這兩個等差數(shù)列的公共項按從小到大的順序組成一個新數(shù)列．求這個新數(shù)列的各項之和．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2020·新高考全國Ⅰ卷)將數(shù)列{2n－1}與{3n－2}的公共項從小到大排列得到數(shù)列{an}，則{an}的前n項和為__________．點評本題和教材習題考查角度完全相同，由于不用判斷求和數(shù)列的項數(shù)，所以其難度要小于教材習題，解決此類問題的關(guān)鍵是清楚新組合的數(shù)列的公差是兩個數(shù)列公差的最小公倍數(shù)．典題15(人教A版選項性必修第二冊P25習題4.2T10)已知等差數(shù)列{an}的公差為d，求證eq\f(am－an,m－n)＝d.你能從直線的斜率角度來解釋這個結(jié)果嗎？______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題16(人教A版選擇性必修第二冊P26習題4.2T12)如圖的形狀出現(xiàn)在南宋數(shù)學家楊輝所著的《詳解九章算法·商功》中，后人稱為“三角垛”．“三角垛”的最上層有1個球，第二層有3個球，第三層有6個球……設(shè)各層球數(shù)構(gòu)成一個數(shù)列{an}．(1)寫出數(shù)列{an}的一個遞推公式；(2)根據(jù)(1)中的遞推公式，寫出數(shù)列{an}的一個通項公式．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題17(人教A版選擇性必修第二冊P30例2)已知等比數(shù)列{an}的公比為q，試用{an}的第m項am表示an.______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題18(人教A版選擇性必修第二冊P31例4)用10000元購買某個理財產(chǎn)品一年．(1)若以月利率0.400%的復(fù)利計息，12個月能獲得多少利息(精確到1元)?(2)若以季度復(fù)利計息，存4個季度，則當每季度利率為多少時，按季結(jié)算的利息不少于按月結(jié)算的利息(精確到10－5)?______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題19(人教A版選擇性必修第二冊P34練習T5)已知數(shù)列{an}的通項公式為an＝eq\f(n3,3n)，求使an取得最大值時的n的值．___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題20(人教A版選擇性必修第二冊P35例7)已知數(shù)列{an}是等比數(shù)列．(1)若a1＝eq\f(1,2)，q＝eq\f(1,2)，求S8；(2)若a1＝27，a9＝eq\f(1,243)，q＜0，求S8；(3)若a1＝8，q＝eq\f(1,2)，Sn＝eq\f(31,2)，求n.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題21(人教A版選擇性必修第二冊P36例8)已知等比數(shù)列{an}的首項為－1，前n項和為Sn.若eq\f(S10,S5)＝eq\f(31,32)，求公比q.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題22(人教A版選擇性必修第二冊P37例9)已知等比數(shù)列{an}的公比q≠－1，前n項和為Sn，證明Sn，S2n－Sn，S3n－S2n成等比數(shù)列，并求這個數(shù)列的公比．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·新高考Ⅱ卷)記Sn為等比數(shù)列{an}的前n項和，若S4＝－5，S6＝21S2，則S8＝()A．120 B．85C．－85 D．－120點評本題是教材例題結(jié)論的直接應(yīng)用，考查了等比數(shù)列的前n項和的性質(zhì)．典題23(人教A版選擇性必修第二冊P38例10)如圖，正方形ABCD的邊長為5cm，取正方形ABCD各邊的中點E，F(xiàn)，G，H，作第2個正方形EFGH，然后再取正方形EFGH各邊的中點I，J，K，L，作第3個正方形IJKL，依此方法一直繼續(xù)下去．(1)求從正方形ABCD開始，連續(xù)10個正方形的面積之和；(2)如果這個作圖過程可以一直繼續(xù)下去，那么所有這些正方形的面積之和將趨近于多少？________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·新高考Ⅰ卷)某校學生在研究民間剪紙藝術(shù)時，發(fā)現(xiàn)剪紙時經(jīng)常會沿紙的某條對稱軸把紙對折．規(guī)格為20dm×12dm的長方形紙，對折1次共可以得到10dm×12dm，20dm×6dm兩種規(guī)格的圖形，它們的面積之和S1＝240dm2，對折2次共可以得到5dm×12dm，10dm×6dm，20dm×3dm三種規(guī)格的圖形，它們的面積之和S2＝180dm2，以此類推，則對折4次共可以得到不同規(guī)格圖形的種數(shù)為__________；如果對折n次，那么eq\o(∑,\s\up6(n),\s\do4(k＝1))Sk＝________dm2.點評本題是在教材習題基礎(chǔ)上的深化和拓展，考查角度完全相同，都要在具體的應(yīng)用情境中探尋數(shù)列的規(guī)律，然后求數(shù)列的和．典題24(人教A版選擇性必修第二冊P39例12)某牧場今年初牛的存欄數(shù)為1200，預(yù)計以后每年存欄數(shù)的增長率為8%，且在每年年底賣出100頭牛．設(shè)牧場從今年起每年年初的計劃存欄數(shù)依次為c1，c2，c3，….(1)寫出一個遞推公式，表示cn＋1與cn之間的關(guān)系；(2)將(1)中的遞推公式表示成cn＋1－k＝r(cn－k)的形式，其中k，r為常數(shù)；(3)求S10＝c1＋c2＋c3＋…＋c10的值(精確到1)．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題25(人教A版選擇性必修第二冊P41習題4.3T7)已知數(shù)列{an}的首項a1＝1，且滿足an＋1＋an＝3·2n.(1)求證：{an－2n}是等比數(shù)列．(2)求數(shù)列{an}的前n項和Sn.______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題26(人教A版選擇性必修第二冊P41習題4.3T8)若數(shù)列{an}的首項a1＝1，且滿足an＋1＝2an＋1，求數(shù)列{an}的通項公式及前10項的和．___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題27(人教A版選擇性必修第二冊P56復(fù)習參考題4T11)已知等差數(shù)列{an}的前n項和為Sn，且S4＝4S2，a2n＝2an＋1(n∈N*)．(1)求數(shù)列{an}的通項公式；(2)若bn＝3n－1，令cn＝anbn，求數(shù)列{cn}的前n項和Tn.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2023·全國甲卷)記Sn為數(shù)列{an}的前n項和，已知a2＝1，2Sn＝nan.(1)求{an}的通項公式；(2)求數(shù)列eq\b\lc\{\rc\}(\a\vs4\al\co1(\f(an＋1,2n)))的前n項和Tn.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________點評本題與教材習題相同，主要考查利用錯位相減法求數(shù)列的和．第七章立體幾何與空間向量[知識網(wǎng)絡(luò)][命題方向]1．空間向量與立體幾何是高考的熱點，難度以中檔或偏難為主，題型涵蓋選擇，填空和解答題，其中解答題必考，總分值大約22分．(1)從近幾年高考情況來看，本章內(nèi)容在客觀題中主要考查空間幾何體的表面積、體積以及與球相關(guān)的問題．(2)在解答題中主要考查線、面位置關(guān)系的證明及空間角的計算，如2023年新高考Ⅰ卷第18題第(1)問考查了證明線線平行的方法，第(2)問考查了面面角，建系是解題的關(guān)鍵．可以用試點法解決，坐標原點可以是頂點、底面邊上的一點或垂心、重心，要根據(jù)空間幾何體的結(jié)構(gòu)特征確定，并且建系后，要能求出有關(guān)點的坐標，難度不大，但需要計算細心，防止得分不全的情況．(3)本章重點考查的核心素養(yǎng)為邏輯推理、直觀想象和數(shù)學運算．2．空間向量與立體幾何考查的主要內(nèi)容有：(1)以空間幾何體或與球的切、接為背景考查幾何體的結(jié)構(gòu)特征、表面積、體積、球的性質(zhì)等；(2)以空間幾何體為載體考查空間線、面位置關(guān)系的證明以及空間角和距離的計算；(3)以空間向量為輔助工具解決空間角的計算與線、面位置關(guān)系的證明．探究1(人教A版必修第二冊P102)圓柱可以由矩形旋轉(zhuǎn)得到，圓錐可以由直角三角形旋轉(zhuǎn)得到．圓臺是否也可以由平面圖形旋轉(zhuǎn)得到？如果可以，由什么平面圖形旋轉(zhuǎn)得到？如何旋轉(zhuǎn)？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版必修第二冊P117)圓柱、圓錐、圓臺的體積公式之間有什么關(guān)系？結(jié)合棱柱、棱錐、棱臺的體積公式，你能將它們統(tǒng)一成柱體、錐體、臺體的體積公式嗎？柱體、錐體、臺體的體積公式之間又有怎樣的關(guān)系？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·新高考Ⅰ卷)南水北調(diào)工程緩解了北方一些地區(qū)水資源短缺問題，其中一部分水蓄入某水庫．已知該水庫水位為海拔148.5m時，相應(yīng)水面的面積為140.0km2；水位為海拔157.5m時，相應(yīng)水面的面積為180.0km2.將該水庫在這兩個水位間的形狀看作一個棱臺，則該水庫水位從海拔148.5m上升到157.5m時，增加的水量約為(eq\r(7)≈2.65)()A．1.0×109m3 B．1.2×109m3C．1.4×109m3 D．1.6×109m3點評本題與教材思考題都考查臺體的體積公式．探究3(人教A版必修第二冊P151)兩條相交直線可以確定一個平面，兩條平行直線也可以確定一個平面，那么定理“如果一條直線與一個平面內(nèi)的兩條相交直線垂直，那么該直線與此平面垂直”中的“兩條相交直線”可以改為“兩條平行直線”嗎？你能從向量的角度解釋原因嗎？如果改為“無數(shù)條直線”呢？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究4(人教A版必修第二冊P160)設(shè)平面α⊥平面β，點P在平面α內(nèi)，過點P作平面β的垂線a，直線a與平面α具有什么位置關(guān)系？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題1(人教A版必修第二冊P116練習T4)求證：直三棱柱的任意兩個側(cè)面的面積和大于第三個側(cè)面的面積．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題2(人教A版必修第二冊P120習題8.3T3)如圖，一個三棱柱形容器中盛有水，側(cè)棱AA1＝8.若側(cè)面AA1B1B水平放置時，水面恰好過AC，BC，A1C1，B1C1的中點．那么當?shù)酌鍭BC水平放置時，水面高為多少？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·北京卷)對24小時內(nèi)降水在平地上的積水厚度(mm)進行如下定義：0～1010～2525～5050～100小雨中雨大雨暴雨小明用一個圓錐形容器接了24小時的雨水，則這一天的雨水屬于哪個等級()A．小雨 B．中雨C．大雨 D．暴雨點評本題與教材習題的解題思路完全一樣，都是利用變換兩類幾何體的體積相等求水面的高度．典題3(人教A版必修第二冊P120習題8.3T4)如圖，圓錐PO的底面直徑和高均是a，過PO的中點O′作平行于底面的截面，以該截面為底面挖去一個圓柱，求剩下幾何體的表面積和體積．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·全國甲卷)甲、乙兩個圓錐的母線長相等，側(cè)面展開圖的圓心角之和為2π，側(cè)面積分別為S甲和S乙，體積分別為V甲和V乙．若eq\f(S甲,S乙)＝2，則eq\f(V甲,V乙)＝()A.eq\r(5) B．2eq\r(2)C.eq\r(10) D.eq\f(5\r(10),4)點評本題與教材習題都考查了空間幾何體體積的求解．典題4(人教A版必修第二冊P120習題8.3T9)如圖是一個獎杯的三視圖，試根據(jù)獎杯的三視圖計算它的表面積和體積．(可用計算工具，尺寸如圖，單位：cm，π取3.14，結(jié)果取整數(shù)．)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________點評千萬不要認為三視圖不在高考的范圍內(nèi)，此內(nèi)容在初中學過，在教材中也出現(xiàn)了，高考試題中出現(xiàn)了正常．典題5(人教A版必修第二冊P137例2)求證：空間四邊形相鄰兩邊中點的連線平行于經(jīng)過另外兩邊的平面．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題6(人教A版必修第二冊P138例3)如圖所示的一塊木料中，棱BC平行于面A′C′.(1)要經(jīng)過面A′C′內(nèi)的一點P和棱BC將木料鋸開，在木料表面應(yīng)該怎樣畫線？(2)所畫的線與平面AC是什么位置關(guān)系？______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題7(人教A版必修第二冊P144習題8.5T8)如圖，直線AA′，BB′，CC′相交于點O，AO＝A′O，BO＝B′O，CO＝C′O，求證：平面ABC∥平面A′B′C′.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題8(人教A版必修第二冊P144習題8.5T9)如圖，E，E′分別為長方體ABCD－A′B′C′D′的棱AD，A′D′的中點，求證：∠BEC＝∠B′E′C′.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題9(人教A版必修第二冊P145習題8.5T13)如圖，α∥β∥γ，直線a與b分別交α，β，γ于點A，B，C和點D，E，F(xiàn)，求證：eq\f(AB,BC)＝eq\f(DE,EF).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版必修第二冊P145習題8.5T15)如圖，透明塑料制成的長方體容器ABCD－A1B1C1D1內(nèi)灌進一些水，固定容器底面一邊BC于地面上，再將容器傾斜．隨著傾斜度的不同，有下面五個命題：(1)有水的部分始終呈棱柱形；(2)沒有水的部分始終呈棱柱形；(3)水面EFGH所在四邊形的面積為定值；(4)棱A1D1始終與水面所在平面平行；(5)當容器傾斜如圖(3)所示時，BE·BF是定值．其中所有正確命題的序號是________，為什么？_______________________________________________________________________________________________________________________________________________________________________________________________________________典題11(人教A版必修第二冊P147例1)如圖，已知正方體ABCD－A′B′C′D′.(1)哪些棱所在的直線與直線AA′垂直？(2)求直線BA′與CC′所成的角的大小．(3)求直線BA′與AC所成的角的大?。甠___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·全國乙卷)在正方體ABCD－A1B1C1D1中，P為B1D1的中點，則直線PB與AD1所成的角為()A.eq\f(π,2) B.eq\f(π,3)C.eq\f(π,4) D.eq\f(π,6)點評本題和教材例題類似，無論是問題的載體，還是考查角度、解題方法完全一樣．其實就是在教材例題的基礎(chǔ)上稍微換了一條直線而已．典題12(人教A版必修第二冊P147例2)如圖，在正方體ABCD－A1B1C1D1中，O1為底面A1B1C1D1的中心．求證：AO1⊥BD.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(多選)(2021·新高考Ⅱ卷)如圖，在正方體中，O為底面的中心，P為所在棱的中點，M，N為正方體的頂點，則滿足MN⊥OP的是()點評本題和教材例題類似，考查角度都是證明線線垂直，其方法都是通過線面垂直或利用空間向量解決問題．典題13(人教A版必修第二冊P151例3)求證：如果兩條平行直線中的一條直線垂直于一個平面，那么另一條直線也垂直于這個平面．已知：如圖，a∥b，a⊥α，求證：b⊥α._____________________________________________________________________________________________________________