收录高三寒假作业36新题

工具v2/文本文件/新题收录列表.txt

@@ -106,3 +106,6 @@
 20240125-115551
 023897:023905
 
+20240125-115808
+023906:023910,023151
+

题库0.3/Problems.json

@@ -64463,7 +64463,9 @@
             "20220625\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "023907"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -64520,7 +64522,9 @@
             "20220625\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "023906"
+        ],
         "remark": "",
         "space": "",
         "unrelated": []
@@ -596129,7 +596133,9 @@
             "20230319\t王伟叶"
         ],
         "same": [],
-        "related": [],
+        "related": [
+            "023906"
+        ],
         "remark": "",
         "space": "4em",
         "unrelated": []
@@ -644730,6 +644736,111 @@
         "space": "4em",
         "unrelated": []
     },
+    "023906": {
+        "id": "023906",
+        "content": "若 $A(1,0,0)$、$B(0,2,0)$、$C(0,0,3)$ 为空间三点, 则平面 $ABC$ 的一个法向量为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "022074",
+            "001975"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "023907": {
+        "id": "023907",
+        "content": "设平面 $\\alpha$ 的一个法向量为 $\\overrightarrow{n}=(-1,2,4)$, 直线 $l$ 的一个方向向量为 $\\overrightarrow{d}=(x, y, z)$.\\\\\n(1) 当 $l \\parallel $ 平面 $\\alpha$ 时, $x, y, z$ 满足\\blank{50};\\\\\n(2) 当 $l \\perp$ 平面 $\\alpha$ 时, $x, y, z$ 满足\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [
+            "001973"
+        ],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "023908": {
+        "id": "023908",
+        "content": "``直线的方向向量与平面的法向量垂直''是``直线与平面平行''的\\blank{50}条件.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "023909": {
+        "id": "023909",
+        "content": "异面直线所成的角 $\\theta$ 与它们方向向量的夹角 $\\varphi$ 之间的关系为\\blank{50}; 直线与平面所成的角 $\\theta$ 与直线方向向量和平面法向量的夹角 $\\varphi$ 之间的关系为\\blank{50}; 二面角的大小 $\\theta$ 与两平面法向量的夹角 $\\varphi$ 之间的关系为\\blank{50}.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "023910": {
+        "id": "023910",
+        "content": "如图所示是一个直三棱柱 (以 $A_1B_1C_1$ 为底面) 被一平面所截得到的几何体, 截面为 $ABC$. 已知 $A_1B_1=B_1C_1=1$, $\\angle A_1B_1C_1=90^{\\circ}$, $AA_1=4$, $BB_1=2$, $CC_1=3$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw ({-sqrt(2)/2},0,0) node [left] {$A_1$} coordinate (A_1);\n\\draw ({sqrt(2)/2},0,0) node [right] {$C_1$} coordinate (C_1);\n\\draw (0,0,{sqrt(2)/2}) node [below] {$B_1$} coordinate (B_1);\n\\draw (A_1) ++ (0,4,0) node [left] {$A$} coordinate (A);\n\\draw (B_1) ++ (0,2,0) node [right] {$B$} coordinate (B);\n\\draw (C_1) ++ (0,3,0) node [right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(B)$) node [below right] {$O$} coordinate (O);\n\\draw (A_1)--(B_1)--(C_1)--(C)--(A)--cycle(A)--(B)--(C)(O)--(C)(B)--(B_1);\n\\draw [dashed] (A_1)--(C_1);\n\\end{tikzpicture}\n\\end{center}\n(1) 设点 $O$ 是 $AB$ 的中点, 证明: $OC \\parallel $ 平面 $A_1B_1C_1$;\\\\\n(2) 求二面角 $B-AC-A_1$ 的大小.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "自拟题目",
+        "edit": [
+            "20240125\t毛培菁"
+        ],
+        "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.",