修改两个答案和一个题目

题库0.3/Problems.json

@@ -588912,7 +588912,7 @@
     },
     "021669": {
         "id": "021669",
-        "content": "函数$y=\\dfrac{\\cos x+1}{\\cos x-1}$的定义域是值域是\\blank{50}.",
+        "content": "函数$y=\\dfrac{\\cos x+1}{\\cos x-1}$的定义域是\\blank{50}, 值域是\\blank{50}.",
         "objs": [],
         "tags": [
             "第三单元"
@@ -588933,7 +588933,8 @@
         ],
         "origin": "2025届高一下校本作业",
         "edit": [
-            "20230209\t王伟叶"
+            "20230209\t王伟叶",
+            "20240210\t王伟叶"
         ],
         "same": [],
         "related": [],
@@ -589292,7 +589293,7 @@
             "第三单元"
         ],
         "genre": "填空题",
-        "ans": "$0<\\beta<\\alpha<\\dfrac{\\pi}{2}$",
+        "ans": "$\\beta<\\alpha$",
         "solution": "",
         "duration": -1,
         "usages": [
@@ -589307,7 +589308,8 @@
         ],
         "origin": "2025届高一下校本作业",
         "edit": [
-            "20230209\t王伟叶"
+            "20230209\t王伟叶",
+            "20240210\t王伟叶"
         ],
         "same": [],
         "related": [],
@@ -644327,14 +644329,15 @@
             "第三单元"
         ],
         "genre": "解答题",
-        "ans": "当$a \\in (-\\infty,-1)\\cup (1,+\\infty)$ 时, 方程实数解个数为$0$个; \\\\\n当$a \\in (0,1]$时，方程实数解个数为$1$个;\\\\\n当$a \\in \\{0,-1\\}$时，方程实数解个数为$2$个;\\\\\n当$a \\in (-1,0)$时, 方程实数解个数为$3$个",
+        "ans": "图像如下:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-pi,-1) cos (-pi/2,0) sin (0,1);\n\\draw (0,0) sin (pi/2,-1) cos (pi,0);\n\\draw [dashed] (-pi,0) node [above] {$\\pi$} -- (-pi,-1) -- (0,-1) node [right] {$-1$};\n\\draw (0,1) node [right] {$1$} (pi,0) node [above] {$\\pi$};\n\\foreach \\i/\\j in {-pi/-1,0/1,pi/0}\n{\\filldraw (\\i,\\j) circle (0.03);};\n\\filldraw [fill = white] (0,0) circle (0.03);\n\\end{tikzpicture}\n\\end{center}\n当$a \\in (-\\infty,-1)\\cup (1,+\\infty)$ 时, 方程实数解个数为$0$个; \\\\\n当$a \\in (0,1]$时，方程实数解个数为$1$个;\\\\\n当$a \\in \\{0,-1\\}$时，方程实数解个数为$2$个;\\\\\n当$a \\in (-1,0)$时, 方程实数解个数为$3$个",
         "solution": "",
         "duration": -1,
         "usages": [],
         "origin": "2025届高一下校本作业-20240116修改",
         "edit": [
             "20230209\t王伟叶",
-            "20240116\t朱敏慧"
+            "20240116\t朱敏慧",
+            "20240210\t王伟叶"
         ],
         "same": [],
         "related": [