diff --git a/工具/关键字筛选题号.py b/工具/关键字筛选题号.py
index 2903e1f4..30ea6f3d 100644
--- a/工具/关键字筛选题号.py
+++ b/工具/关键字筛选题号.py
@@ -2,7 +2,7 @@ import os,re,json
 
 """---设置关键字, 同一field下不同选项为or关系, 同一字典中不同字段间为and关系, 不同字典间为or关系, _not表示列表中的关键字都不含, 同一字典中的数字用来供应同一字段不同的条件之间的and---"""
 keywords_dict_table = [
-    {"origin":["双基百分百"],"origin2":["2022"],"usages":[r"0\.[3456]\d\d"],"usages_not":[r"0\.[9]",r"0\.8[56789]"]}
+    {"origin":["2013"]}
 ]
 """---关键字设置完毕---"""
 # 示例： keywords_dict_table = [
diff --git a/工具/批量收录题目.py b/工具/批量收录题目.py
index 0015b845..0b776a98 100644
--- a/工具/批量收录题目.py
+++ b/工具/批量收录题目.py
@@ -1,9 +1,9 @@
 #修改起始id,出处,文件名
-starting_id = 17400
+starting_id = 40799
 raworigin = ""
-filename = r"C:\Users\weiye\Documents\wwy sync\待整理word题目\2022全国各高考1.tex"
-editor = "20230531\t王伟叶"
-indexed = True
+filename = r"C:\Users\weiye\Documents\wwy sync\临时工作区\自拟题目12.tex"
+editor = "20230601\t王伟叶"
+indexed = False
 IndexDescription = "试题"
 
 import os,re,json
diff --git a/工具/文本文件/批量题目分类号记录.txt b/工具/文本文件/批量题目分类号记录.txt
index 5d450499..c704ab05 100644
--- a/工具/文本文件/批量题目分类号记录.txt
+++ b/工具/文本文件/批量题目分类号记录.txt
@@ -229,13 +229,17 @@ problems_dict =  {
 "2025届高一下学期周末卷10与11小测":"40707:40715",
 "2025届高一下学期周末卷13":"40716:40737",
 "2025届高一下学期周末卷14":"40753:40771",
+"2025届高一下学期周末卷14小测":"40799:40808",
 "2025届高一下学期周末卷15":"40772:40790",
 "2025届高一下学期周末卷15小测":"40791:40798",
+"2025届高一下学期周末卷16":"40809:40828",
+"2025届高一下学期周末卷16小测":"40829:40840",
 "2025届高一下学期期中复习一(集合逻辑不等式)":"40283:40298",
 "2025届高一下学期期中复习二(幂指对函数)":"40368:40386",
 "2025届高一下学期测验01":"40336:40349",
 "2025届高一下学期测验02":"40350:40367",
 "2025届高一下学期测验03":"40738:40752",
+"2025届高一下学期测验03":"40841:40861",
 "2025届高一下学期区期中统考试题":"15269:15289",
 "2024届高二下学期周末卷01":"40001:40017",
 "2024届高二下学期周末卷02":"40037:40056",
diff --git a/工具/文本文件/题号筛选.txt b/工具/文本文件/题号筛选.txt
index f38efdb4..fb1e28b3 100644
--- a/工具/文本文件/题号筛选.txt
+++ b/工具/文本文件/题号筛选.txt
@@ -1 +1 @@
-013310,013356,013358:013359,013368,013371,013457,013461,013466,013475:013476,013494,013505:013506,013535:013536
\ No newline at end of file
+011647:011669,012352:012382
\ No newline at end of file
diff --git a/工具/题号选题pdf生成.py b/工具/题号选题pdf生成.py
index 6506cf6b..348f755a 100644
--- a/工具/题号选题pdf生成.py
+++ b/工具/题号选题pdf生成.py
@@ -7,13 +7,13 @@ import os,re,time,json,sys
 """---设置题目列表---"""
 #留空为编译全题库, a为读取文本文件中的题号筛选.txt文件生成题库
 problems = r"""
-013310,013356,013358:013359,013368,013371,013457,013461,013466,013475:013476,013494,013505:013506,013535:013536
+a
 """
 """---设置题目列表结束---"""
 
 """---设置文件名---"""
 #目录和文件的分隔务必用/
-filename = "临时文件/双基冲刺卷较难题"
+filename = "临时文件/临时"
 """---设置文件名结束---"""
 
 """---设置是否需要解答题的空格---"""
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 10a9b002..6f7834a5 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -554597,5 +554597,1265 @@
         "remark": "",
         "space": "4em",
         "unrelated": []
+    },
+    "040799": {
+        "id": "040799",
+        "content": "若复数$z=\\dfrac{2 \\mathrm{i}}{1-\\mathrm{i}}$, 则$\\overline {z}=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040800": {
+        "id": "040800",
+        "content": "$1+\\mathrm{i}+\\mathrm{i}^2+\\mathrm{i}^3+\\cdots+\\mathrm{i}^{2025}=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040801": {
+        "id": "040801",
+        "content": "复数$z_1=1+2 \\mathrm{i}$, $z_2=-2+\\mathrm{i}$, $z_3=-1-2 \\mathrm{i}$, 它们在复平面上的对应点是一个正方形的三个顶点, 则这个正方形的第四个顶点对应的复数是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040802": {
+        "id": "040802",
+        "content": "设$a \\in \\mathbf{R}$, 复数$z_1=a^2-3+(a+5) \\mathrm{i}$, $z_2=a-1+(a^2+2 a-1) \\mathrm{i}$分别对应向量$\\overrightarrow{OZ_1}$、$\\overrightarrow{OZ_2}$($O$为原点), 若向量$\\overrightarrow{Z_1Z_2}$对应的复数为纯虚数, 则$a$的值是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040803": {
+        "id": "040803",
+        "content": "已知$z=\\dfrac{(1+\\mathrm{i})^3 \\cdot(a-\\mathrm{i})^2}{\\sqrt{2} \\cdot(a-3 \\mathrm{i})^2}$, 且$|z|=\\dfrac{2}{3}$, 则实数$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040804": {
+        "id": "040804",
+        "content": "设$z_1, z_2, z_3 \\in \\mathbf{C}$, 下列命题中假命题的是\\blank{50}.\\\\\n\\textcircled{1} $|-z_1|=|z_1|$;\\\\\n\\textcircled{2} 若$z_1^2=z_2^2$, 则$z_1 \\cdot \\overline{z_1}=z_2 \\cdot \\overline{z_2}$;\\\\\n\\textcircled{3} $|\\dfrac{z_1 \\cdot z_2}{z_3^2}|=\\dfrac{|z_1| \\cdot|z_2|}{|z_3|^2}$;\\\\\n\\textcircled{4} 若$(z_1-z_2)^2+(z_2-z_3)^2=0$, 则$z_1=z_2=z_3$.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040805": {
+        "id": "040805",
+        "content": "复数$z_1$、$z_2$、$z_3$所对应的复平面内的点$Z_1$、$Z_2$、$Z_3$不共线, 复数$z$满足$z=z_1+\\lambda(\\dfrac{z_2-z_1}{|z_2-z_1|}+\\dfrac{z_3-z_1}{|z_3-z_1|})=z_2+\\mu(\\dfrac{z_1-z_2}{|z_1-z_2|}+\\dfrac{z_3-z_2}{|z_3-z_2|})$($\\lambda$、$\\mu$为正数)则复数$z$对应的复平面内的点$Z$是$\\triangle Z_1Z_2Z_3$的\\blank{50}.(填内心、外心、重心、垂心等)",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040806": {
+        "id": "040806",
+        "content": "设$z_1, z_2$为复数, 下列等式错误的是\\bracket{20}.\n\\twoch{$|z_1 z_2|=|z_1||z_2|$}{$|\\dfrac{z_1}{z_2}|=\\dfrac{|z_1|}{|z_2|}$}{$|z_1+z_2|=|z_1|+|z_2|$}{$|z_1^n|=|z_1|^n$($n$为正整数)}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040807": {
+        "id": "040807",
+        "content": "设$z$是复数, 则下列命题中的假命题是\\bracket{20}.\n\\twoch{若$z^2 \\geq 0$, 则$z$是实数}{若$z^2<0$, 则$z$是虚数}{若$z$是虚数, 则$z^2 \\geq 0$}{若$z$是纯虚数, 则$z^2<0$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040808": {
+        "id": "040808",
+        "content": "在复平面上, 复数$z_1=3$、$z_2=4$i对应的点分别为$A$、$B$, 复数$z$对应的点为$P$. 若$P$在线段$AB$上, 求$|z|$的最小值, 以及取到最小值时的$z$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷14小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040809": {
+        "id": "040809",
+        "content": "直线$3 x+4 y-5=0$的倾斜角是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040810": {
+        "id": "040810",
+        "content": "已知$A(-3,2)$, $B(1,-4)$, 则线段$AB$的垂直平分线的斜截式方程是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040811": {
+        "id": "040811",
+        "content": "从原点引直线$l$, 使$l$与连结$A(1,2)$, $B(-1,3)$两点的线段有公共点, 则直线$l$的倾斜角的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040812": {
+        "id": "040812",
+        "content": "若直线$l$过点$(-2,3)$且与向量$\\overrightarrow {d}=(1,-4)$平行, 则直线$l$的点斜式方程是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040813": {
+        "id": "040813",
+        "content": "已知点$(1,2)$在直线$l$上的射影为$(-1,5)$, 则直线$l$的点法向式方程为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040814": {
+        "id": "040814",
+        "content": "已知直线$l$经过点$A(-4,3)$, 且在$x$轴上的截距是$-2$, 那么直线$l$在$y$轴上的截距是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040815": {
+        "id": "040815",
+        "content": "已知直线$l$的倾斜角满足$[0, \\dfrac{\\pi}{4}) \\cup[\\dfrac{2}{3} \\pi, \\pi)$, 则其斜率$k$的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040816": {
+        "id": "040816",
+        "content": "直线$l$过点$(1,2)$且与两坐标轴围成等腰直角三角形, 则$l$的一般式方程是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040817": {
+        "id": "040817",
+        "content": "如图, 已知四条直线$l_1, l_2, l_3, l_4$的斜率分别是$k_1, k_2, k_3, k_4$, 这四条直线的位置关系如右图所示, 请将$k_1,k_2,k_3,k_4\n$从小到大排序:\\blank{50}(用``$<$''号连接).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-2,-0.2) -- (2,0.5) node [right] {$l_1$};\n\\draw (-1.5,-2) -- (1,2) node [right] {$l_2$};\n\\draw (-1,-2) -- (-0.5,2) node [above] {$l_3$};\n\\draw (2,-1) -- (-2,2) node [above] {$l_4$};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040818": {
+        "id": "040818",
+        "content": "已知方程$(a^2+a) x+(a^2-1) y-1=0$表示一条直线, 则$a$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040819": {
+        "id": "040819",
+        "content": "直线$(m^2+1) x+y=1$, $m \\in \\mathbf{R}$的倾斜角的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040820": {
+        "id": "040820",
+        "content": "已知$P$是直线$l$上的一点, 将直线$l$绕点$P$沿逆时针方向旋转$\\alpha$($0<\\alpha<\\dfrac{\\pi}{2}$), 所得的直线方程是$x-y-2=0$, 若将它继续逆时针旋转$\\dfrac{\\pi}{2}-\\alpha$, 所得的直线方程是$2 x+y-1=0$, 则直线$l$的方程为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040821": {
+        "id": "040821",
+        "content": "一直线$l$被两直线$4 x+y+6=0$和$3 x-5 y-6=0$截得的线段中点恰好是坐标原点, 则直线$l$的方程是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040822": {
+        "id": "040822",
+        "content": "已知$m, n$是正的常数, $l$是经过原点的直线, 若直线$l$上的任意点先向左平移$m$个单位再向上平移$n$个单位后仍在直线$l$上, 则直线$l$的方程是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040823": {
+        "id": "040823",
+        "content": "已知$\\triangle ABC$的三个顶点为$A(1,6), B(-1,-2), C(6,3)$.\\\\\n(1) 求$BC$边上的高所在直线的方程;\\\\\n(2) 求与$AC$边平行的中位线所在直线的方程;\\\\\n(3) $AB$边上的垂直平分线所在的直线方程.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040824": {
+        "id": "040824",
+        "content": "在正方形$ABCO$中, $O$为原点, $\\overrightarrow{OA}=(3,4)$, $O, A, B, C$按逆时针排列, 求正方形$ABCO$各边所在的直线方程.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040825": {
+        "id": "040825",
+        "content": "已知直线$l: (3 a-1) x-(a-2) y-1=0$($a \\in \\mathbf{R}$).\\\\\n(1) 求证: 不论$a$取何值, $l$恒过定点;\\\\\n(2) 若$l$不通过第二象限, 求实数$a$的取值范围;\\\\\n(3) 记 (1) 中的定点为$P$, 若$l \\perp OP$($O$为原点), 求实数$a$的值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040826": {
+        "id": "040826",
+        "content": "过点$P(2,1)$作直线$l$分别交$x$轴、$y$轴的正半轴于$A, B$两点, $O$是坐标原点.\\\\\n(1) 当$\\triangle AOB$面积最小时, 求直线$l$的方程;\\\\\n(2) 当$|OA|+|OB|$最小时, 求直线$l$的方程;\\\\\n(3) 当$|PA| \\cdot|PB|$最大时, 求直线$l$的方程.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040827": {
+        "id": "040827",
+        "content": "已知$P$是直线$3 x-y=0$上位于第一象限的点, $M(3,2)$为一定点, 直线$PM$交$x$轴正半轴于点$Q$, 当$\\triangle POQ$的面积最小时, 求直线$PQ$的方程.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040828": {
+        "id": "040828",
+        "content": "设$u, v$是非零常数, $a$是正的常数, 在直角坐标平面内, 过点$P(u, v)$作直线$l$, 使其与两坐标轴相交所成的三角形的面积等于$a$, 当$u, v, a$取不同值时, 这样的直线$l$可能有\\blank{50}条.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040829": {
+        "id": "040829",
+        "content": "经过点$A(2,2)$、$B(3,-2)$的直线$l$的两点式方程为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040830": {
+        "id": "040830",
+        "content": "经过点$(3,2)$, 且倾斜角为$\\dfrac{5 \\pi}{6}$的直线$l$的点斜式方程为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040831": {
+        "id": "040831",
+        "content": "若直线$l_1: 2 x-3 y+2=0$与直线$l_2: x+t y+1=0$垂直, 则实数$t=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040832": {
+        "id": "040832",
+        "content": "若直线$(a-2) x+3 a y-4=0$在$x$轴上的截距为$2$, 则实数$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040833": {
+        "id": "040833",
+        "content": "若直线的倾斜角$\\theta \\in[0, \\dfrac{\\pi}{3}) \\cup[\\dfrac{3 \\pi}{4}, \\pi)$, 则其斜率$k$的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040834": {
+        "id": "040834",
+        "content": "若$\\overrightarrow {n}=(a, 1-a)$是直线$3 x+4 y-1=0$的一个法向量, 则实数$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040835": {
+        "id": "040835",
+        "content": "已知点$A(2,1), B(-1,3)$, 若过$(0,-2)$且斜率为$k$的直线$l$与线段$AB$有公共点, 则实数$k$的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040836": {
+        "id": "040836",
+        "content": "已知直线$(1-a) x+(2 a-3) y+1-2 a=0$不经过第一象限, 则实数$a$的取值范围为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040837": {
+        "id": "040837",
+        "content": "已知点$P$是直线$l: a x+y+2=0$上的一点, 将$l$绕点$P$沿逆时针方向旋转$\\dfrac{\\pi}{4}$后与直线$4 x-3 y+5=0$重合, 则实数$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040838": {
+        "id": "040838",
+        "content": "已知$\\triangle ABC$的顶点$B(2,3)$、$C(-2,1)$, 若其垂心$H$为$(-4,4)$, 则其顶点$A$的坐标为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040839": {
+        "id": "040839",
+        "content": "已知直线$l: (1-2 a) x+(a-1) y-a+2=0$, 点$A(-1,4), B(3,2)$, 是否存在实数$a$, 使得$l$与直线$AB$平行? 若存在, 求出$a$的值; 若不存在, 请说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040840": {
+        "id": "040840",
+        "content": "已知直线$l_1: m x-y-m+2=0$, 直线$l_2: (m+1) x+(m-1) y+1-3 m=0$, 其中$m>0$.\\\\\n(1) 求证: 不论$m$取何值时, 直线$l_1$与直线$l_2$均恒过一个定点;\\\\\n(2) 设$l_1$与$y$轴正半轴相交于点$M$, $l_2$与$x$轴正半轴相交于点$N$, $l_1$与$l_2$相交于点$A$, $O$为坐标原点, 求四边形$OMAN$的面积的最大值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期周末卷16小测",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040841": {
+        "id": "040841",
+        "content": "已知复数$z=1-2 \\mathrm{i}$, $\\mathrm{i}$为虚数单位, 则$\\mathrm{Im} z=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040842": {
+        "id": "040842",
+        "content": "复数$z$满足$(z-3)(2-\\mathrm{i})=5$, $\\mathrm{i}$为虚数单位, 则$\\overline {z}=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040843": {
+        "id": "040843",
+        "content": "若复数$z=-1-\\mathrm{i}$, $\\mathrm{i}$为虚数单位, 则$\\arg z=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040844": {
+        "id": "040844",
+        "content": "在$\\triangle ABC$中, $A$、$B$、$C$所对的边是$a$、$b$、$c$, $A=60^{\\circ}$, $b=1$, $\\triangle ABC$的面积为$\\sqrt{3}$, 则$a=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040845": {
+        "id": "040845",
+        "content": "已知向量$\\overrightarrow {a}=(\\sqrt{3}, 1)$, $\\overrightarrow {b}=(0,2)$, 则向量$\\overrightarrow {b}$在向量$\\overrightarrow {a}$方向上的数量投影是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040846": {
+        "id": "040846",
+        "content": "已知向量$\\overrightarrow {a}=(-1,2)$, $\\overrightarrow {b}=(1,2)$, $\\overrightarrow {c}=(k, 6)$, 若$(\\overrightarrow {a}-\\overrightarrow {c})$与$\\overrightarrow {b}$共线, 则$k=$\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040847": {
+        "id": "040847",
+        "content": "已知函数$f(x)=\\sin 2 x$的两条对称轴之间距离的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040848": {
+        "id": "040848",
+        "content": "设复数$z$满足$z^2-5|z|+6=0$, 则符合条件的$z$的解的个数为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040849": {
+        "id": "040849",
+        "content": "在集合$\\{1,2,3,4\\}$中, 任取一个偶数$a$和一个奇数$b$, 构成一个位置向量$\\overrightarrow {x}=(a, b)$, 以所有这些向量为邻边作的平行四边形中, 面积不超过$4$的平行四边形的个数为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040850": {
+        "id": "040850",
+        "content": "已知平面非零向量$\\overrightarrow {\\alpha}$、$\\overrightarrow {\\beta}$($|\\overrightarrow {\\alpha}| \\neq|\\overrightarrow {\\beta}|$), 满足$|\\overrightarrow {\\beta}|=2$, 且$\\overrightarrow {\\alpha}$与$\\overrightarrow {\\beta}-\\overrightarrow {\\alpha}$的夹角为$135^{\\circ}$, 则$|\\overrightarrow {\\alpha}|$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040851": {
+        "id": "040851",
+        "content": "已知点$P$在单位圆$O$的圆周上运动, 点$A(-3,0)$, 则$\\overrightarrow{AO} \\cdot \\overrightarrow{AP}$的取值范围是\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040852": {
+        "id": "040852",
+        "content": "已知$\\overrightarrow {a} \\overrightarrow {b}, \\overrightarrow {c}$为不共线向量, 且$|\\overrightarrow {a}|=3,|\\overrightarrow {b}|=2|\\overrightarrow {c}|$, 且$(\\overrightarrow {b}-\\overrightarrow{2 a}) \\perp(\\overrightarrow {c}-\\overrightarrow{2 a})$, 则$|\\overrightarrow {b}|$的最小值为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040853": {
+        "id": "040853",
+        "content": "函数$y=\\sqrt{\\cos (x-\\dfrac{\\pi}{3})}$的一个严格单调增区间为\\bracket{20}.\n\\fourch{$[0, \\dfrac{\\pi}{2}]$}{$[0, \\dfrac{\\pi}{4}]$}{$[-\\dfrac{\\pi}{2}, 0]$}{$[\\dfrac{\\pi}{3}, \\dfrac{\\pi}{2}]$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040854": {
+        "id": "040854",
+        "content": "设$O$为坐标原点, 复数$z_1$、$z_2$在复平面内对应的点分别为$P$、$Q$, 则下列结论中不一定正确的是\\bracket{20}.\n\\twoch{$|z_1+z_2|=|\\overrightarrow{OP}+\\overrightarrow{OQ}|$}{$|z_1-z_2|=| \\overrightarrow{OP}-\\overrightarrow{OQ}|$}{$|z_1|+|z_2|=|\\overrightarrow{OP}|+|\\overrightarrow{OQ}|$}{$|z_1 \\cdot z_2|=|\\overrightarrow{OP} \\cdot \\overrightarrow{OQ}|$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040855": {
+        "id": "040855",
+        "content": "下列所给的四个命题中, 不是真命题的是\\bracket{20}.\n\\twoch{$|z|=|\\overline {z}|$}{$z \\in \\mathbf{R} \\Leftrightarrow z=\\overline {z}$}{$|z|^2=z \\cdot \\overline {z}$}{$z^2=-|z|^2 \\Leftrightarrow z$为纯虚数}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040856": {
+        "id": "040856",
+        "content": "若向量$\\overrightarrow {a}, \\overrightarrow {b}, \\overrightarrow {c}$满足$\\overrightarrow {a}+\\overrightarrow {b}+\\overrightarrow {c}=\\overrightarrow{0}$, 且$\\overrightarrow {a}^2<\\overrightarrow {b}^2<\\overrightarrow {c}^2$, 则``$\\overrightarrow {a} \\cdot \\overrightarrow {b}$''、``$\\overrightarrow {b} \\cdot \\overrightarrow {c}$''、``$\\overrightarrow {c} \\cdot \\overrightarrow {a}$''中, 最大的是\\bracket{20}.\n\\fourch{$\\overrightarrow {a} \\cdot \\overrightarrow {b}$}{$\\overrightarrow {b} \\cdot \\overrightarrow {c}$}{$\\overrightarrow {c} \\cdot \\overrightarrow {a}$}{不能确定}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "040857": {
+        "id": "040857",
+        "content": "已知复数$z=1+m \\mathrm{i}$, 且$\\overline {z} \\cdot(1+\\mathrm{i})$是纯虚数($m \\in \\mathbf{R}$, $\\mathrm{i}$是虚数单位, $\\overline {z}$是$z$的共轭复数).\\\\\n(1) 求实数$m$的值和$|z|$;\\\\\n(2) 设$z_1=\\dfrac{a-\\mathrm{i}^{2023}}{z}$, 且复数$z_1$对应的点在第二象限, 求实数$a$的取值范围.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040858": {
+        "id": "040858",
+        "content": "已知关于$x$的实系数一元二次方程$x^2+m x+9=0$.\\\\\n(1) 若复数$z$是该方程的一个虚根, 且$|z|+\\overline {z}=4-2 \\sqrt{2} \\mathrm{i}$, 求实数$m$的值;\\\\\n(2) 若该方程的两根为$x_1$、$x_2$, 若$|x_1-x_2|=2 \\sqrt{3}$, 求实数$m$的值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040859": {
+        "id": "040859",
+        "content": "复数$z_1=2 \\sin \\theta-\\sqrt{3} \\cdot \\mathrm{i}$, $z_2=1+2 \\cos \\theta \\cdot \\mathrm{i}$, $i$是虚数单位.\\\\\n(1) 若$\\theta \\in[0, \\dfrac{\\pi}{2}]$, 且$z_1 \\cdot z_2$为实数, 求$\\theta$的值;\\\\\n(2) 若$\\theta \\in[\\dfrac{\\pi}{2}, \\dfrac{5 \\pi}{6}]$, 复数$z_1$、$z_2$对应的向量分别是$\\overrightarrow {a}$、$\\overrightarrow {b}$, 且存在$\\theta$使等式$(\\lambda \\overrightarrow {a}-\\overrightarrow {b}) \\cdot(\\overrightarrow {a}-\\lambda \\overrightarrow {b})=0$, 求实数$\\lambda$的取值范围.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040860": {
+        "id": "040860",
+        "content": "如图是函数$f(x)=A \\sin (\\omega x+\\varphi),(A>0, \\omega>0,0<\\omega<\\dfrac{\\pi}{2})$图像的一部分, $M$、$N$是它与$x$轴的两个交点, $C$、$D$分别是它的最高点和最低点, $E(0,1)$是线段$MC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-pi/4}:{7*pi/4}, samples = 100] plot (\\x,{2*sin(\\x/pi*180+45)});\n\\draw ({pi/4},2) node [above] {$C$} coordinate (C);\n\\draw ({-pi/4},0) node [below] {$M$} coordinate (M);\n\\draw ($(C)!0.5!(M)$) node [right] {$E$} coordinate (E);\n\\draw ({3*pi/4},0) node [below left] {$N$} coordinate (N);\n\\draw ({5*pi/4},-2) node [below] {$D$} coordinate (D);\n\\draw (M)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 若点$M$的坐标为$(-1,0)$, 求$C$、$N$、$D$的坐标;\\\\\n(2) 若点$M$的坐标为$(-m, 0)$($m>0$), 且$\\overrightarrow{MC} \\cdot \\overrightarrow{MD}=\\dfrac{3 \\pi^2}{4}-4$, 试确定函数$f(x)$的解析式, 并求$f(x)$在$[0, \\dfrac{\\pi}{2}]$上的值域.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "040861": {
+        "id": "040861",
+        "content": "设$\\triangle ABC$是边长为$2$的正三角形, 点$P_1$、$P_2$、$P_3$依次四等分线段$BC$(如图所示).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0) node [below] {$C$} coordinate (C);\n\\draw (60:2) node [above] {$A$} coordinate (A);\n\\draw (0.5,0) node [below] {$P_1$} coordinate (P_1);\n\\draw (1,0) node [below] {$P_2$} coordinate (P_2);\n\\draw (1.5,0) node [below] {$P_3$} coordinate (P_3);\n\\draw (A)--(B)--(C)--cycle;\n\\foreach \\i in {1,2,3}\n{\\draw [->] (A) -- (P_\\i);};\n\\end{tikzpicture}\n\\end{center}\n(1) 求$\\overrightarrow{AB} \\cdot \\overrightarrow{AP_1}+\\overrightarrow{AP_1} \\cdot \\overrightarrow{AP_2}+\\overrightarrow{AP_2} \\cdot \\overrightarrow{AP_3}+\\overrightarrow{AP_3} \\cdot \\overrightarrow{AC}$的值;\\\\\n(2) 若$Q$为线段$AP_1$上的一点, 且$\\overrightarrow{AQ}=m \\overrightarrow{AB}+\\dfrac{1}{16} \\overrightarrow{AC}$, 求实数$m$的值;\\\\\n(3) $P$在边$BC$的何处时, $\\overrightarrow{PA} \\cdot \\overrightarrow{PC}$取得最小值, 并求出此最小值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "2025届高一下学期测验04",
+        "edit": [
+            "20230601\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
     }
 }
\ No newline at end of file