添加三道自拟题目

题库0.3/Problems.json

@@ -638855,6 +638855,66 @@
         "space": "4em",
         "unrelated": []
     },
+    "023635": {
+        "id": "023635",
+        "content": "如果 $\\overrightarrow{e_1}$ 与 $\\overrightarrow{e_2}$ 是平面上两个不平行的向量, 那么该平面上的任意向量 $\\overrightarrow{a}$, 都可唯一地表示为 $\\overrightarrow{e_1}$ 与 $\\overrightarrow{e_2}$ 的线性组合, 即存在唯一的一对实数 $\\lambda$ 与 $\\mu$, 使得\n$\\overrightarrow{a}=\\lambda \\overrightarrow{e_1}+\\mu \\overrightarrow{e_2}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240122\t赵琍琍"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "023636": {
+        "id": "023636",
+        "content": "给定平面上不共线的三点 $O$、$A$、$B$, 根据平面向量基本定理, 对平面上任定一点 $P$, 都有唯一的一对实数 $\\lambda, \\mu$, 使得 $\\overrightarrow{OP}=\\lambda \\overrightarrow{OA}+\\mu \\overrightarrow{OB}$. 求证: $A$、$B$、$P$ 三点共线的一个充要条件是 $\\lambda+\\mu=1$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240122\t赵琍琍"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "023637": {
+        "id": "023637",
+        "content": "矩形 $ABCD$ 中, $AB=1$, $AD=2$, $CE \\perp BD$ 于 $E$, 点 $P$ 在以 $C$ 为圆心, $CE$为半径的圆周上运动, 若 $\\overrightarrow{AP}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{AD}$ ($\\lambda$、$\\mu$ 为实数), 求 $\\lambda+\\mu$ 的范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0) node [below] {$B$} coordinate (B);\n\\draw (2,1) node [above] {$C$} coordinate (C);\n\\draw (0,1) node [above] {$D$} coordinate (D);\n\\draw ($(B)!0.2!(D)$) node [left] {$E$} coordinate (E);\n\\draw (C) circle ({2/sqrt(5)});\n\\draw (C) ++ (30:{2/sqrt(5)}) node [above right] {$P$} coordinate (P);\n\\draw (A) rectangle (C) (B)--(D)(C)--(E);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240122\t赵琍琍"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "030001": {
         "id": "030001",
         "content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",