人教A版高中數(shù)學(xué)必修第二冊(cè)同步講義第七章 復(fù)數(shù)單元測(cè)試（強(qiáng)化卷）（含解析）

第七章復(fù)數(shù)單元測(cè)試（強(qiáng)化卷）一、單選題1．已知SKIPIF1<0，則SKIPIF1<0（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【答案】C【詳解】因?yàn)镾KIPIF1<0，故SKIPIF1<0，故SKIPIF1<0故選：C.2．已知SKIPIF1<0，則SKIPIF1<0（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【答案】B【詳解】SKIPIF1<0，SKIPIF1<0.故選：B.3．已知復(fù)數(shù)SKIPIF1<0、SKIPIF1<0在復(fù)平面內(nèi)對(duì)應(yīng)的點(diǎn)關(guān)于虛軸對(duì)稱，SKIPIF1<0，則SKIPIF1<0=A．2 B．SKIPIF1<0 C．SKIPIF1<0 D．1【答案】D【詳解】由題意，復(fù)數(shù)SKIPIF1<0、SKIPIF1<0在復(fù)平面內(nèi)對(duì)應(yīng)的點(diǎn)關(guān)于虛軸對(duì)稱，SKIPIF1<0，則SKIPIF1<0，所以SKIPIF1<0，故選D.4．設(shè)SKIPIF1<0，則z的虛部為（）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．SKIPIF1<0【答案】C【詳解】SKIPIF1<0，則SKIPIF1<0的虛部為SKIPIF1<0.故選：C．5．已知SKIPIF1<0是虛數(shù)單位，則復(fù)數(shù)SKIPIF1<0對(duì)應(yīng)的點(diǎn)所在的象限是（）A．第一象限 B．第二象限 C．第三象限 D．第四象限【答案】D【詳解】SKIPIF1<0.所以復(fù)數(shù)對(duì)應(yīng)的點(diǎn)SKIPIF1<0在第四象限，故選：D6．若復(fù)數(shù)SKIPIF1<0滿足SKIPIF1<0，則下列說(shuō)法正確的是（）A．SKIPIF1<0的虛部為SKIPIF1<0 B．SKIPIF1<0的共軛復(fù)數(shù)為SKIPIF1<0C．SKIPIF1<0對(duì)應(yīng)的點(diǎn)在第二象限 D．SKIPIF1<0【答案】C【詳解】由SKIPIF1<0，得SKIPIF1<0，對(duì)于A，復(fù)數(shù)SKIPIF1<0的虛部為SKIPIF1<0，故A不正確；對(duì)于B，復(fù)數(shù)SKIPIF1<0的共軛復(fù)數(shù)為SKIPIF1<0，故B不正確；對(duì)于C，復(fù)數(shù)SKIPIF1<0對(duì)應(yīng)的點(diǎn)為SKIPIF1<0，所以復(fù)數(shù)SKIPIF1<0對(duì)應(yīng)的點(diǎn)在第二象限，故C正確；對(duì)于D，SKIPIF1<0，故D不正確.故選：C.7．已知SKIPIF1<0（SKIPIF1<0），則復(fù)數(shù)SKIPIF1<0（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【答案】C【詳解】∵SKIPIF1<0且SKIPIF1<0，則SKIPIF1<0，SKIPIF1<0，∴SKIPIF1<0.故選：C.8．已知復(fù)數(shù)SKIPIF1<0，則z的共軛復(fù)數(shù)在復(fù)平面內(nèi)對(duì)應(yīng)的點(diǎn)位于（）A．第一象限 B．第二象限 C．第三象限 D．第四象限【答案】A【詳解】復(fù)數(shù)SKIPIF1<0，則SKIPIF1<0所以復(fù)數(shù)SKIPIF1<0在復(fù)平面內(nèi)對(duì)應(yīng)的點(diǎn)的坐標(biāo)為SKIPIF1<0，位于復(fù)平面內(nèi)的第一象限.故選：A二、多選題9．下列命題中正確的是（）A．復(fù)數(shù)SKIPIF1<0的共軛復(fù)數(shù)是zB．若復(fù)數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0C．若復(fù)數(shù)SKIPIF1<0滿足SKIPIF1<0，則SKIPIF1<0D．在復(fù)平面中，虛軸上的點(diǎn)對(duì)應(yīng)的復(fù)數(shù)都是純虛數(shù)【答案】AC【詳解】選項(xiàng)A：復(fù)數(shù)SKIPIF1<0的共軛復(fù)數(shù)是z.判斷正確；選項(xiàng)B：令SKIPIF1<0，則有SKIPIF1<0，但SKIPIF1<0.判斷錯(cuò)誤；選項(xiàng)C：設(shè)復(fù)數(shù)SKIPIF1<0，則由SKIPIF1<0，可得SKIPIF1<0，則有SKIPIF1<0.判斷正確；選項(xiàng)D：在復(fù)平面中，虛軸上的點(diǎn)（除原點(diǎn)外）對(duì)應(yīng)的復(fù)數(shù)都是純虛數(shù).判斷錯(cuò)誤.故選：AC10．歐拉公式SKIPIF1<0(其中i為虛數(shù)單位，SKIPIF1<0)，是由瑞士著名數(shù)學(xué)家歐拉創(chuàng)立的，公式將指數(shù)函數(shù)的定義域擴(kuò)大到復(fù)數(shù)，建立了三角函數(shù)與指數(shù)的數(shù)的關(guān)聯(lián)，在復(fù)變函數(shù)論里面占有非常重要的地位，被譽(yù)為數(shù)學(xué)中的天橋，依據(jù)歐拉公式，下列選項(xiàng)能確的是（）A．復(fù)數(shù)SKIPIF1<0對(duì)應(yīng)的點(diǎn)位于第三象限 B．SKIPIF1<0為純虛數(shù)C．SKIPIF1<0的共軛復(fù)數(shù)為SKIPIF1<0； D．復(fù)數(shù)SKIPIF1<0的模長(zhǎng)等于SKIPIF1<0【答案】BCD【詳解】解：對(duì)于SKIPIF1<0，由于SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0表示的復(fù)數(shù)在復(fù)平面中位于第二象限，故SKIPIF1<0錯(cuò)誤；對(duì)于SKIPIF1<0，SKIPIF1<0，可得SKIPIF1<0為純虛數(shù)，故SKIPIF1<0正確；對(duì)于SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的共軛復(fù)數(shù)為SKIPIF1<0，故SKIPIF1<0正確．對(duì)于SKIPIF1<0，SKIPIF1<0，可得其模的長(zhǎng)為SKIPIF1<0SKIPIF1<0，故SKIPIF1<0正確；故選：SKIPIF1<0．11．任何一個(gè)復(fù)數(shù)SKIPIF1<0（其中SKIPIF1<0、SKIPIF1<0，SKIPIF1<0為虛數(shù)單位）都可以表示成：SKIPIF1<0的形式，通常稱之為復(fù)數(shù)SKIPIF1<0的三角形式.法國(guó)數(shù)學(xué)家棣莫弗發(fā)現(xiàn)：SKIPIF1<0，我們稱這個(gè)結(jié)論為棣莫弗定理.根據(jù)以上信息，下列說(shuō)法正確的是（）A．SKIPIF1<0B．當(dāng)SKIPIF1<0，SKIPIF1<0時(shí)，SKIPIF1<0C．當(dāng)SKIPIF1<0，SKIPIF1<0時(shí)，SKIPIF1<0D．當(dāng)SKIPIF1<0，SKIPIF1<0時(shí)，若SKIPIF1<0為偶數(shù)，則復(fù)數(shù)SKIPIF1<0為純虛數(shù)【答案】AC【詳解】對(duì)于A選項(xiàng)，SKIPIF1<0，則SKIPIF1<0，可得SKIPIF1<0，SKIPIF1<0，A選項(xiàng)正確；對(duì)于B選項(xiàng)，當(dāng)SKIPIF1<0，SKIPIF1<0時(shí)，SKIPIF1<0，B選項(xiàng)錯(cuò)誤；對(duì)于C選項(xiàng)，當(dāng)SKIPIF1<0，SKIPIF1<0時(shí)，SKIPIF1<0，則SKIPIF1<0，C選項(xiàng)正確；對(duì)于D選項(xiàng)，SKIPIF1<0，取SKIPIF1<0，則SKIPIF1<0為偶數(shù)，則SKIPIF1<0不是純虛數(shù)，D選項(xiàng)錯(cuò)誤.故選：AC.【點(diǎn)睛】本題考查復(fù)數(shù)的乘方運(yùn)算，考查了復(fù)數(shù)的模長(zhǎng)、共軛復(fù)數(shù)的運(yùn)算，考查計(jì)算能力，屬于中等題.12．關(guān)于復(fù)數(shù)SKIPIF1<0（i為虛數(shù)單位），下列說(shuō)法正確的是（）A．SKIPIF1<0 B．SKIPIF1<0在復(fù)平面上對(duì)應(yīng)的點(diǎn)位于第二象限C．SKIPIF1<0 D．SKIPIF1<0【答案】ACD【詳解】SKIPIF1<0所以SKIPIF1<0故A正確SKIPIF1<0，則SKIPIF1<0在復(fù)平面上對(duì)應(yīng)的點(diǎn)為SKIPIF1<0位于第三象限故B錯(cuò)誤SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0故C正確SKIPIF1<0故D正確故選：ACD三、填空題13．SKIPIF1<0是虛數(shù)單位，復(fù)數(shù)SKIPIF1<0_____________．【答案】SKIPIF1<0【分析】利用復(fù)數(shù)的除法化簡(jiǎn)可得結(jié)果.【詳解】SKIPIF1<0.故答案為：SKIPIF1<0.14．在復(fù)數(shù)范圍內(nèi)分解因式：SKIPIF1<0______．【答案】SKIPIF1<0【分析】將原式配成完全平方式，再根據(jù)SKIPIF1<0，即可得解；【詳解】解：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0故答案為：SKIPIF1<015．在復(fù)平面內(nèi)，復(fù)數(shù)SKIPIF1<0對(duì)應(yīng)點(diǎn)SKIPIF1<0，復(fù)數(shù)SKIPIF1<0對(duì)應(yīng)點(diǎn)SKIPIF1<0，把向量SKIPIF1<0繞點(diǎn)SKIPIF1<0順時(shí)針旋轉(zhuǎn)SKIPIF1<0得到向量SKIPIF1<0，則點(diǎn)P對(duì)應(yīng)的復(fù)數(shù)是______．【答案】SKIPIF1<0【分析】求出向量SKIPIF1<0對(duì)應(yīng)的復(fù)數(shù)，再由復(fù)數(shù)乘法的幾何意義求得向量SKIPIF1<0對(duì)應(yīng)的復(fù)數(shù)，最后由復(fù)數(shù)加法的幾何意義即可求得答案.【詳解】解：由題意知向量SKIPIF1<0對(duì)應(yīng)的復(fù)數(shù)是SKIPIF1<0，再由復(fù)數(shù)乘法的幾何意義得，向量SKIPIF1<0對(duì)應(yīng)的復(fù)數(shù)是SKIPIF1<0，最后由復(fù)數(shù)加法的幾何意義得，向量SKIPIF1<0，其對(duì)應(yīng)的復(fù)數(shù)是SKIPIF1<0，所以點(diǎn)P對(duì)應(yīng)的復(fù)數(shù)是SKIPIF1<0.故答案為：SKIPIF1<0.16．已知復(fù)數(shù)SKIPIF1<0滿足：SKIPIF1<0，則SKIPIF1<0____________，SKIPIF1<0____________．【答案】SKIPIF1<0SKIPIF1<0【詳解】因?yàn)镾KIPIF1<0，所以SKIPIF1<0，所以SKIPIF1<0SKIPIF1<0所以SKIPIF1<0.故答案為：（1）SKIPIF1<0；（2）SKIPIF1<0.四、解答題17．把下列復(fù)數(shù)的三角形式化成代數(shù)形式.（1）SKIPIF1<0；（2）SKIPIF1<0.【答案】（1）SKIPIF1<0（2）SKIPIF1<0【解析】（1）分別求出SKIPIF1<0再整理為SKIPIF1<0的形式.（2）分別求出SKIPIF1<0再整理為SKIPIF1<0的形式.【詳解】（1）SKIPIF1<0SKIPIF1<0.（2）SKIPIF1<0.18．已知復(fù)數(shù)SKIPIF1<0，其中SKIPIF1<0，i為虛數(shù)單位．(1)若z為實(shí)數(shù)，求m的值；(2)若z為純虛數(shù)，求SKIPIF1<0的虛部．【答案】(1)SKIPIF1<0(2)8(1)解：若z為實(shí)數(shù)，則SKIPIF1<0，解得SKIPIF1<0．(2)解：由題意得SKIPIF1<0解得SKIPIF1<0，∴SKIPIF1<0，故SKIPIF1<0，∴SKIPIF1<0的虛部為8．19．已知復(fù)數(shù)SKIPIF1<0，其中SKIPIF1<0為虛數(shù)單位.(1)當(dāng)SKIPIF1<0，且SKIPIF1<0是純虛數(shù)，求SKIPIF1<0的值；(2)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的取值范圍.【答案】(1)SKIPIF1<0(2)SKIPIF1<0(1)SKIPIF1<0是純虛數(shù)，故有SKIPIF1<0，經(jīng)計(jì)算有，SKIPIF1<0;(2)SKIPIF1<0,所以有SKIPIF1<0，如下圖，根據(jù)幾何意義，可知SKIPIF1<0.20．當(dāng)實(shí)數(shù)m分別為何值時(shí)，(1)復(fù)數(shù)SKIPIF1<0是：實(shí)數(shù)？虛數(shù)？(2)復(fù)數(shù)SKIPIF1<0純虛數(shù)？【答案】(1)當(dāng)SKIPIF1<0或SKIPIF1<0時(shí)復(fù)數(shù)SKIPIF1<0為實(shí)數(shù)，當(dāng)SKIPIF1<0且SKIPIF1<0時(shí)復(fù)數(shù)SKIPIF1<0為虛數(shù)(2)當(dāng)SKIPIF1<0時(shí)復(fù)數(shù)SKIPIF1<0為純虛數(shù)(1)若復(fù)數(shù)SKIPIF1<0為實(shí)數(shù)，則SKIPIF1<0∴SKIPIF1<0或SKIPIF1<0，若復(fù)數(shù)SKIPIF1<0為虛數(shù)，則SKIPIF1<0∴SKIPIF1<0且SKIPIF1<0，(2)若復(fù)數(shù)SKIPIF1<0純虛數(shù)，則SKIPIF1<0且SKIPIF1<0，由SKIPIF1<0可得SKIPIF1<0或SKIPIF1<0，又SKIPIF1<0時(shí)SKIPIF1<0不存在，SKIPIF1<0時(shí)SKIPIF1<0，所以SKIPIF1<0.21．（1）已知復(fù)數(shù)SKIPIF1<0，若SKIPIF1<0為實(shí)數(shù)，SKIPIF1<0為純虛數(shù)，求z；（2）已知關(guān)于x的方程SKIPIF1<0，若SKIPIF1<0是方程SKIPIF1<0的一個(gè)復(fù)數(shù)根，求出m，n的值【答案】（1）SKIPIF1<0；（2）SKIPIF1<0.【詳解】（1）已知SKIPIF1<0，∵SKIPIF1<0，SKIPIF1<0為實(shí)數(shù)，∴SKIPI