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录入空中课堂复数例题与习题

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wangweiye7840 2023-07-05 16:35:45 +08:00
parent 710e620f8f
commit a6f32bc1b2
2 changed files with 941 additions and 1 deletions
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2
工具v2/批量收录题目.py
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@ -1,5 +1,5 @@
#修改起始id,出处,文件名
starting_id = 18470 #起始id设置, 来自"寻找空闲题号"功能
starting_id = 18529 #起始id设置, 来自"寻找空闲题号"功能
raworigin = "" #题目来源的前缀(中缀在.tex文件中)
filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件
editor = "王伟叶" #编辑者姓名

940
题库0.3/Problems.json
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@ -476887,6 +476887,946 @@
"space": "4em",
"unrelated": []
},
"018529": {
"id": "018529",
"content": "计算:\\\\\n(1) $(1+3 \\mathrm{i})+(-4+2 \\mathrm{i})$;\\\\\n(2) $(3-2 \\mathrm{i})-(3+2 \\mathrm{i})$;\\\\\n(3) $(2-3 \\mathrm{i})(4+2 \\mathrm{i})$;\\\\\n(4) $(2+\\mathrm{i})(3+4 \\mathrm{i})(2-\\mathrm{i})$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018530": {
"id": "018530",
"content": "计算:\\\\\n(1) $\\dfrac{3+\\mathrm{i}}{2-\\mathrm{i}}$;\\\\\n(2) $\\dfrac{1+\\sqrt{2} \\mathrm{i}}{1-\\sqrt{2} \\mathrm{i}}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018531": {
"id": "018531",
"content": "计算虚数单位$\\mathrm{i}$的整数次幂, 并找出规律.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
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"edit": [
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],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018532": {
"id": "018532",
"content": "对任意整数$m$, 计算$\\mathrm{i}^m+\\mathrm{i}^{m+1}+\\mathrm{i}^{m+2}+\\mathrm{i}^{m+3}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018533": {
"id": "018533",
"content": "计算: $(a+b \\mathrm{i})^2-(a-b \\mathrm{i})^2$($a$、$b \\in \\mathbf{R}$).",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018534": {
"id": "018534",
"content": "对任意奇数$n$, 计算$(\\dfrac{1-\\mathrm{i}}{1+\\mathrm{i}})^{2 n}+(\\dfrac{1+\\mathrm{i}}{1-\\mathrm{i}})^{2 n}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018535": {
"id": "018535",
"content": "已知$m \\in \\mathbf{R}$, 设集合$M=\\{1,2,(m^2-3 m-1)+(m^2-5 m+6) \\mathrm{i}\\}$, $N=\\{-1,3\\}$. 若$M \\cap N \\neq \\varnothing$, 求$m$的值.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018536": {
"id": "018536",
"content": "填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline$z$ & 是否复数 & 是否实数 & 是否虚数 & 是否纯虚数 & $\\mathrm{Re} z$ & $\\mathrm{Im} z$ \\\\\n\\hline$-0.5 \\mathrm{i}$ & & & & & & \\\\\n\\hline $\\frac{1}{2}-\\sqrt{2} \\mathrm{i}$ & & & & & &\\\\\n\\hline $\\pi$ & & & & & &\\\\\n\\hline $0$ & & & & & &\\\\\n\\hline $\\sqrt{3}$ & & & & & &\\\\\n\\hline $2 \\mathrm{i}-5$ & & & & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018537": {
"id": "018537",
"content": "求实数$m$的值或取值范围, 使得复数$z=m^2+m-2+(m^2-1) \\mathrm{i}$分别是:\\\\\n(1) 实数;\\\\\n(2) 虚数;\\\\\n(3) 纯虚数;\\\\\n(4) $0$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
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],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018538": {
"id": "018538",
"content": "设$z$是复数, 求证: $\\overline {z}=z$是$z \\in \\mathbf{R}$的充要条件.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018539": {
"id": "018539",
"content": "设$z_1=1+3 \\mathrm{i}$, $z_2=1-\\mathrm{i}$. 求复数$z$, 使得$\\overline {z}=\\dfrac{z_1}{z_2}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018540": {
"id": "018540",
"content": "求角$\\theta$($\\theta \\in \\mathbf{R}$), 使得复数$z=(2 \\sin ^2 \\theta-\\sin \\theta)+(3 \\tan ^2 \\theta-1) \\mathrm{i}$分别是:\\\\\n(1) 实数;\\\\\n(2) 纯虚数;\\\\\n(3) 零.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018541": {
"id": "018541",
"content": "设$z$是复数, 你能说出$z$是纯虚数的一个充要条件吗? (通过$z$与$\\overline {z}$之间的关系描述)",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018542": {
"id": "018542",
"content": "在复平面上作出表示下列复数的向量: $z_1=2+2 \\mathrm{i}$, $z_2=-3-2 \\mathrm{i}$, $z_3=2 \\mathrm{i}$, $z_4=-4$, $z_5=2-2 \\mathrm{i}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018543": {
"id": "018543",
"content": "设复平面上的点$A$和点$B$所对应的复数分别为$z_A$和$z_B$, 试用$z_A$和$z_B$表示复平面上的向量$\\overrightarrow{AB}$所对应的复数$z$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018544": {
"id": "018544",
"content": "设$z \\in \\mathbf{C}$, 复平面上的点$Z$与$Z'$分别表示$z$与$z \\mathrm{i}$. 求证: $\\overrightarrow{OZ} \\perp \\overrightarrow{OZ'}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018545": {
"id": "018545",
"content": "如图, 在复平面上给定平行四边形$OABC$, 其中点$A$与点$C$分别对应于复数$z_A=-1+\\mathrm{i}$与$z_C=3+2 \\mathrm{i}$, 求点$B$所对应的复数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-2,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (-1,1) node [left] {$A$} coordinate (A);\n\\draw (3,2) node [right] {$C$} coordinate (C);\n\\draw ($(A)+(C)-(O)$) node [above] {$B$} coordinate (B);\n\\draw (O)--(A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018546": {
"id": "018546",
"content": "已知复数$z$、$2 \\overline {z}$在复平面上所对应的点分别为$Z$、$Z'$, 若向量$\\overrightarrow{ZZ'}=(1,3)$, 求$z+2 \\overline {z}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018547": {
"id": "018547",
"content": "如图, 已知复平面上点$A(-2,2)$, 点$B(1,3)$, 点$D$与点$B$关于$x$轴成轴对称. 若$ABCD$为平行四边形, 求点$C$所对应的复数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-3,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,2) node [left] {$A$} coordinate (A);\n\\draw (1,3) node [above] {$B$} coordinate (B);\n\\draw (1,-3) node [below] {$D$} coordinate (D);\n\\draw ($(B)+(D)-(A)$) node [right] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018548": {
"id": "018548",
"content": "已知复数$z$满足$|z|=1$, 求证: $z+\\dfrac{1}{z}$是实数.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018549": {
"id": "018549",
"content": "求下列复数的模:\\\\\n(1) $\\dfrac{(1-\\mathrm{i})(1+2 \\mathrm{i})}{4+3 \\mathrm{i}}$;\\\\\n(2) $\\dfrac{(7-3 \\mathrm{i})(5+4 \\mathrm{i})}{(7+3 \\mathrm{i})(-4-5 \\mathrm{i})}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018550": {
"id": "018550",
"content": "设复数$-\\sqrt{5}+2 \\mathrm{i}$和复数$2+\\sqrt{5} \\mathrm{i}$在复平面上分别对应点$A$和点$B$, 求$A$、$B$两点间的距离.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018551": {
"id": "018551",
"content": "若复数$z$满足$|z-3|+|z-4 \\mathrm{i}|=5$, 求$|z|$的取值范围.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018552": {
"id": "018552",
"content": "设复平面上三点$A$、$B$、$C$所对应的复数分别是$z_A$、$z_B$、$z_C$, 若$\\dfrac{z_B-z_A}{z_C-z_A}=2 \\mathrm{i}$, 求:\\\\\n(1) $\\angle BAC$的大小;\\\\\n(2) $\\angle ABC$的正切值.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018553": {
"id": "018553",
"content": "在复数范围求$25$与$-25$的平方根.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018554": {
"id": "018554",
"content": "在复数范围内解方程: $2 x^2-4 x+5=0$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018555": {
"id": "018555",
"content": "如果$p$、$q$都是实数, 而关于$x$的方程$2 x^2+p x+q=0$有一个根$-2+3 \\mathrm{i}$, 求$p$、$q$的值.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018556": {
"id": "018556",
"content": "已知实系数一元二次方程$x^2-2 x+m=0$的两根为$x_1$、$x_2$, 且$|x_1|+|x_2|=8$, 求$m$的值. 某同学的解答如下:\\\\\n{\\it 解: 因为$x_1$、$x_2$是$x^2-2 x+m=0$的两个根, 所以$\\begin{cases}x_1+x_2=2, \\\\ x_1 x_2=m.\\end{cases}$\n由 $|x_1|+|x_2|=8$, 即$x_1^2+2|x_1 x_2|+x_2^2=64$, 即$(x_1+x_2)^2-2 x_1 x_2+2|x_1 x_2|=64$, 即$4-2 m+2|m|=64$, 解得$m=-15$.}\\\\\n上述解法是否正确? 若有误, 请指出错误之处, 并写出正确的解答过程.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
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],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018557": {
"id": "018557",
"content": "分别写出下列复数的模$r$与辐角主值$\\theta$, 并把这些复数用三角形式表示:\\\\\n(1) $\\sqrt{3}+\\mathrm{i}$;\\\\\n(2) $-1+\\mathrm{i}$;\\\\\n(3) $-1$;\\\\\n(4) $-3-4 \\mathrm{i}$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
"usages": [],
"origin": "空中课堂必修第二册复数例题与习题",
"edit": [
"20230705\t王伟叶"
],
"same": [],
"related": [],
"remark": "",
"space": "4em",
"unrelated": []
},
"018558": {
"id": "018558",
"content": "把下列复数用三角形式表示:\\\\\n(1) $\\cos \\theta-\\mathrm{i} \\sin \\theta$;\\\\\n(2) $-2(\\cos \\alpha+\\mathrm{i} \\sin \\alpha)$.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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"018559": {
"id": "018559",
"content": "若复数$z$满足$|z|=2 \\sqrt{3}$, $\\arg z=\\dfrac{7 \\pi}{6}$, 则$z$的代数形式为\\blank{50}.",
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"018560": {
"id": "018560",
"content": "设$a \\in \\mathbf{R}$, 复数$z=(-2 a^2-1)+(a^2-a+1) \\mathrm{i}$的辐角主值是\\bracket{20}.\n\\fourch{第一象限的角}{第二象限的角}{第三象限的角}{第四象限的角}",
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"genre": "选择题",
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"018561": {
"id": "018561",
"content": "将复数$\\sin \\alpha+\\mathrm{i} \\cos \\alpha$用三角形式表示.",
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"genre": "解答题",
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"space": "4em",
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"018562": {
"id": "018562",
"content": "设$\\theta \\in(0, \\dfrac{\\pi}{2}) \\cup(\\dfrac{\\pi}{2}, \\pi)$, 将复数$1+\\mathrm{i}\\tan \\theta$用三角形式表示.",
"objs": [],
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"ans": "",
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"space": "4em",
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"018563": {
"id": "018563",
"content": "已知$z_1=\\dfrac{3}{2}(\\cos \\dfrac{\\pi}{12}+\\mathrm{i} \\sin \\dfrac{\\pi}{12})$, $z_2=2(\\cos \\dfrac{\\pi}{4}+\\mathrm{i} \\sin \\dfrac{\\pi}{4})$, 计算$z_1 z_2$, 结果用复数的代数形式表示.",
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"genre": "解答题",
"ans": "",
"solution": "",
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"018564": {
"id": "018564",
"content": "如图, 设复数$-2+2 \\mathrm{i}$在复平面上所对应的向量是$\\overrightarrow{OZ}$, 将$\\overrightarrow{OZ}$绕原点$O$逆时针旋转$120^{\\circ}$得到向量$\\overrightarrow{OZ'}$. 求向量$\\overrightarrow{OZ'}$所对应的复数. (结果用复数的代数形式表示)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3.5) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-2,-1,1}\n{\\draw [dashed, gray] (\\i,-3.5) -- (\\i,3);\n\\draw (\\i,0) node [below left] {$\\i$};};\n\\foreach \\i in {-3,-2,-1,1,2}\n{\\draw [dashed, gray] (-3,\\i) -- (3,\\i);\n\\draw (0,\\i) node [right] {$\\i$};};\n\\draw [->] (0,0) -- (-2,2) node [above] {$Z$} coordinate (Z);\n\\draw ($(O)!1!120:(Z)$) node [below] {$Z'$} coordinate (Z');\n\\draw [->] (O)--(Z');\n\\end{tikzpicture}\n\\end{center}",
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"genre": "解答题",
"ans": "",
"solution": "",
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"018565": {
"id": "018565",
"content": "计算$\\dfrac{4(\\cos \\dfrac{4 \\pi}{3}+\\mathrm{i} \\sin \\dfrac{4 \\pi}{3})}{2(\\cos \\dfrac{5 \\pi}{6}+\\mathrm{i} \\sin \\dfrac{5 \\pi}{6})}$, 并用复数的代数形式表示.",
"objs": [],
"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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"space": "4em",
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},
"018566": {
"id": "018566",
"content": "计算$(1-\\mathrm{i})^{20}$.",
"objs": [],
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"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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},
"018567": {
"id": "018567",
"content": "求$1$的三次方根.",
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"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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"018568": {
"id": "018568",
"content": "在复数范围内解方程: $(\\sqrt{3}-\\mathrm{i}) x^4=32 \\mathrm{i}$.",
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"tags": [],
"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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"018569": {
"id": "018569",
"content": "已知$n$为正整数, 若复数$z=(\\dfrac{3}{3+\\sqrt{3} \\mathrm{i}})^n$为实数, 求$n$的最小值及相应的$z$的值.",
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"genre": "解答题",
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"018570": {
"id": "018570",
"content": "求实数$m$的值或取值范围, 使得复数$z=(m+1)+(3 m-2) \\mathrm{i}$分别满足:\\\\\n(1) 复数$z$是虚数;\\\\\n(2) 复数$z$在复平面上所对应的点位于虚轴上;\\\\\n(3) 复数$z$在复平面上所对应的点位于第三象限;\\\\\n(4) 复数$z$在复平面上所对应的点到原点距离为$5$.",
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"genre": "解答题",
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"space": "4em",
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"018571": {
"id": "018571",
"content": "已知复平面上如图所示的平行四边形$ABCD$的顶点$A$、$B$、$D$三点所对应的复数分别是$2+3 \\mathrm{i}$、$5-\\mathrm{i}$、$6+\\mathrm{i}$, 求向量$\\overrightarrow{AC}$所对应的复数.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw [->] (-1,0) -- (10,0) node [below] {$x$};\n\\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (2,3) node [above] {$A$} coordinate (A);\n\\draw (5,-1) node [below] {$B$} coordinate (B);\n\\draw (6,1) node [above] {$D$} coordinate (D);\n\\draw (9,-3) node [right] {$C$} coordinate (C);\n\\draw (A)--(B)--(C)--(D)--cycle;\n\\end{tikzpicture}\n\\end{center}",
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"tags": [],
"genre": "解答题",
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"solution": "",
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"018572": {
"id": "018572",
"content": "已知复数$z_1=\\sqrt{3}+\\mathrm{i},|z_2|=1, z_1 \\overline{z_2}$是虚部为负数的纯虚数, 求复数$z_2$.",
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"018573": {
"id": "018573",
"content": "若虚数$z_1$、$z_2$满足$z_1^2=z_2$, 且$z_1$、$z_2$是一个实系数一元二次方程的两个根, 写出这样的一个实系数一元二次方程.",
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"genre": "解答题",
"ans": "",
"solution": "",
"duration": -1,
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"018574": {
"id": "018574",
"content": "是否存在虚数$z$, 使$z+\\dfrac{5}{z} \\in \\mathbf{R}$, 且$z+3$的实部与虚部互为相反数? 若存在, 求出虚数$z$; 若不存在, 说明理由.",
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"ans": "",
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"018575": {
"id": "018575",
"content": "已知复数$z$满足$z \\overline {z}=2$, $z^2$的虚部为$2$.\\\\\n(1) 求复数$z$;\\\\\n(2) 设$z, z^2, z-z^2$在复平面上所对应的点分别为$A, B, C$, 求$\\triangle ABC$的面积.",
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"020001": {
"id": "020001",
"content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",