diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb
index 4f64c059..49cf995d 100644
--- a/工具/批量添加题库字段数据.ipynb
+++ b/工具/批量添加题库字段数据.ipynb
@@ -9,220 +9,50 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "题号: 000270 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000271 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000271 , 字段: objs 中已有该数据: K0714005X\n",
- "题号: 000284 , 字段: objs 中已有该数据: K0715003X\n",
- "题号: 000285 , 字段: objs 中已添加数据: K0714004X\n",
- "题号: 000288 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000379 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000485 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000485 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 000510 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 000560 , 字段: objs 中已添加数据: K0714003X\n",
- "题号: 000625 , 字段: objs 中已添加数据: K0714004X\n",
- "题号: 000625 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000705 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 000723 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 000755 , 字段: objs 中已添加数据: K0714002X\n",
- "题号: 000801 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 000813 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 000895 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 000908 , 字段: objs 中已添加数据: K0714003X\n",
- "题号: 002306 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002307 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002307 , 字段: objs 中已添加数据: K0714005X\n",
- "题号: 002308 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002308 , 字段: objs 中已添加数据: K0714005X\n",
- "题号: 002309 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002310 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002311 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002313 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002314 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002315 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002316 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002317 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002318 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002319 , 字段: objs 中已添加数据: KNONE\n",
- "题号: 002321 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002322 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002323 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002324 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002325 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002326 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002327 , 字段: objs 中已添加数据: K0714003X\n",
- "题号: 002327 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002328 , 字段: objs 中已添加数据: KNONE\n",
- "题号: 002329 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 002330 , 字段: objs 中已添加数据: K0714003X\n",
- "题号: 002330 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002331 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002332 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002333 , 字段: objs 中已添加数据: K0714002X\n",
- "题号: 002334 , 字段: objs 中已添加数据: K0713001X\n",
- "题号: 002335 , 字段: objs 中已添加数据: K0714005X\n",
- "题号: 002335 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002337 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002338 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002339 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002342 , 字段: objs 中已添加数据: K0714005X\n",
- "题号: 002342 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002343 , 字段: objs 中已添加数据: K0714005X\n",
- "题号: 002343 , 字段: objs 中已添加数据: K0714002X\n",
- "题号: 002344 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002345 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002346 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002347 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002356 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002363 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002367 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002380 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002389 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002428 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002428 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002443 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 002447 , 字段: objs 中已添加数据: KNONE\n",
- "题号: 002469 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 002480 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002481 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002482 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 002683 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 002686 , 字段: objs 中已添加数据: KNONE\n",
- "题号: 002688 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003399 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003400 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003401 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 003401 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 003402 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 003403 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003404 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003406 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 003407 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003408 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003409 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003409 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003410 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 003411 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003411 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 003413 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003414 , 字段: objs 中已添加数据: KNONE\n",
- "题号: 003415 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 003416 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003428 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 003608 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003619 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003650 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003664 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003704 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003733 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 003748 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003826 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003858 , 字段: objs 中已添加数据: K0714004X\n",
- "题号: 003895 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 003909 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 003909 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 003918 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004078 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004099 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004120 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004129 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004152 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004162 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 004162 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004182 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004204 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004204 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004221 , 字段: objs 中已添加数据: K0713001X\n",
- "题号: 004221 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004242 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004242 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004288 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004288 , 字段: objs 中已添加数据: K0715002X\n",
- "题号: 004288 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004303 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004309 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004309 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004361 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004487 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004524 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004529 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004529 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004561 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004633 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 004701 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004717 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004722 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 004722 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 004743 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 004743 , 字段: objs 中已添加数据: K0715003X\n",
- "题号: 008873 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 008873 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 008875 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 008876 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 008877 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 008878 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 008879 , 字段: objs 中已添加数据: K0713003X\n",
- "题号: 008879 , 字段: objs 中已添加数据: K0713004X\n",
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- "题号: 008881 , 字段: objs 中已添加数据: K0714003X\n",
- "题号: 008882 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 008883 , 字段: objs 中已添加数据: K0714003X\n",
- "题号: 008884 , 字段: objs 中已添加数据: K0713002X\n",
- "题号: 008884 , 字段: objs 中已添加数据: K0713004X\n",
- "题号: 008885 , 字段: objs 中已添加数据: K0715003X\n",
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- "题号: 009820 , 字段: objs 中已添加数据: K0714003X\n",
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- "题号: 010704 , 字段: objs 中已添加数据: K0715003X\n"
+ "题号: 012033 , 字段: tags 中已添加数据: 第一单元\n",
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+ "题号: 012036 , 字段: tags 中已添加数据: 第八单元\n",
+ "题号: 012037 , 字段: tags 中已添加数据: 第六单元\n",
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+ "题号: 012039 , 字段: tags 中已添加数据: 第五单元\n",
+ "题号: 012040 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012041 , 字段: tags 中已添加数据: 第九单元\n",
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+ "题号: 012043 , 字段: tags 中已添加数据: 第一单元\n",
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+ "题号: 012047 , 字段: tags 中已添加数据: 第九单元\n",
+ "题号: 012048 , 字段: tags 中已添加数据: 第六单元\n",
+ "题号: 012049 , 字段: tags 中已添加数据: 第五单元\n",
+ "题号: 012050 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012051 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012051 , 字段: tags 中已添加数据: 第六单元\n",
+ "题号: 012052 , 字段: tags 中已添加数据: 第六单元\n",
+ "题号: 012053 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012053 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012054 , 字段: tags 中已添加数据: 第一单元\n",
+ "题号: 012055 , 字段: tags 中已添加数据: 第五单元\n",
+ "题号: 012056 , 字段: tags 中已添加数据: 第一单元\n",
+ "题号: 012057 , 字段: tags 中已添加数据: 第四单元\n",
+ "题号: 012058 , 字段: tags 中已添加数据: 第五单元\n",
+ "题号: 012059 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012060 , 字段: tags 中已添加数据: 第四单元\n",
+ "题号: 012061 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012062 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012063 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012064 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012065 , 字段: tags 中已添加数据: 第四单元\n",
+ "题号: 012066 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012067 , 字段: tags 中已添加数据: 第四单元\n",
+ "题号: 012068 , 字段: tags 中已添加数据: 第一单元\n",
+ "题号: 012069 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012070 , 字段: tags 中已添加数据: 第一单元\n",
+ "题号: 012071 , 字段: tags 中已添加数据: 第三单元\n",
+ "题号: 012072 , 字段: tags 中已添加数据: 第二单元\n",
+ "题号: 012073 , 字段: tags 中已添加数据: 第四单元\n",
+ "题号: 012074 , 字段: tags 中已添加数据: 第二单元\n"
]
}
],
diff --git a/工具/批量题号选题pdf生成.ipynb b/工具/批量题号选题pdf生成.ipynb
index fbe5247f..733f5467 100644
--- a/工具/批量题号选题pdf生成.ipynb
+++ b/工具/批量题号选题pdf生成.ipynb
@@ -2,16 +2,16 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "开始编译教师版本pdf文件: 临时文件/赋能批量_教师用_20221105.tex\n",
+ "开始编译教师版本pdf文件: 临时文件/抛物线预选_教师用_20221119.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/赋能批量_学生用_20221105.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/抛物线预选_学生用_20221119.tex\n",
"0\n"
]
}
@@ -26,10 +26,18 @@
"\"\"\"---设置题目列表---\"\"\"\n",
"#字典字段为文件名, 之后为内容的题号\n",
"problems_dict = {\n",
- "\"赋能6\":\"376:385\",\n",
- "\"赋能7\":\"386:395\",\n",
- "\"赋能8\":\"396:405\",\n",
- "\"赋能9\":\"406:415\"\n",
+ "\"K0713001X\":\"002334,008886,008894,010671\",\n",
+ "\"K0719001X\":\"004654\",\n",
+ "\"K0719002X\":\"002393,009099\",\n",
+ "\"K0719003X\":\"000369,000467,000669,000728,000806,000833,000878,000909,000957,000968,002395,002397,002405,002406,002407,003448,003639,003837,003930,003945,004197,004221,004495,004514,004524,004550,004570,008921,008924,008926,008928,008935,008960,009077,009105,009837,009840,010682\",\n",
+ "\"K0719004X\":\"000337,000707,000739,000783,000864,002408,002409,002410,002439,003437,003449,008920,008927,008953,008956,009081,009836,010684\",\n",
+ "\"K0719005X\":\"000290,002440,002441,002450,004078,010703\",\n",
+ "\"K0719006X\":\"000275,002402,002421,002689,003440,003599,003781,008923,010683\",\n",
+ "\"K0720002X\":\"002426\",\n",
+ "\"K0720003X\":\"000282,000464,002398,002400,002401,002403,002404,002411,002412,002413,002417,002418,002419,002420,002422,002423,002424,002425,002427,002429,002430,002434,002436,002437,002445,002690,003438,003439,003441,003443,003444,003446,003447,003450,003451,004141,004225,004351,004372,004626,004639,004713,004764,008922,008925,008930,008931,008932,008933,008934,008936,008954,008955,008968,009098,009106,009110,009838,009839,010685,010688,010689\",\n",
+ "\"K0720004X\":\"000804,003795,004065,008929,008937,008957,010686\",\n",
+ "\"K0720005X\":\"002438\"\n",
+ "\n",
"}\n",
"\n",
"\"\"\"---设置题目列表结束---\"\"\"\n",
@@ -37,7 +45,7 @@
"\"\"\"---设置文件保存路径---\"\"\"\n",
"#目录和文件的分隔务必用/\n",
"directory = \"临时文件/\"\n",
- "filename = \"赋能批量\"\n",
+ "filename = \"抛物线预选\"\n",
"\"\"\"---设置文件名结束---\"\"\"\n",
"if directory[-1] != \"/\":\n",
" directory += \"/\"\n",
diff --git a/工具/文本文件/metadata.txt b/工具/文本文件/metadata.txt
index d9fcc11d..2a2f3ad5 100644
--- a/工具/文本文件/metadata.txt
+++ b/工具/文本文件/metadata.txt
@@ -1,941 +1,214 @@
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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 3dbd0a8e..ee09b4b4 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
@@ -1,7 +1 @@
-000020,000033,000048,000049,000054,000061,000101,000119,000153,000163,000165,000168,000170,000189,000197,000236,000326,000339,000347,000348,000349,000357,000367,000368,000371,000387,000411,000431,000436,000481,000622,000637,000738,000757,000788,000792,000858,000871,000883,000894,000924,001073,001086,001120,001226,001238,001245,001286,001292,001300,001305,001314,001326,001371,001480,001506,001510,001597,001623,001649,001659,001676,001847,001848,001852,001857,001858,001871,001877,001992,002011,002027,002081,002697,002710,002772,002784,002793,002798,002800,002833,002847,002874,002909,002925,002955,003041,003092,003095,003097,003130,003140,003142,003179,003328,003332,003345,003346,003355,003454,003455,003466,003501,003519,003520,003544,003632,003636,003658,003662,003798,003815,003841,003875,003891,003936,003982,003986,003996,004080,004081,004084,004092,004122,004124,004125,004144,004167,004240,004248,004283,004312,004313,004328,004365,004400,004414,004420,004439,004446,004447,004449,004451,004454,004455,004458,004460,004492,004557,004768,004794,004878,004928,004929,005150,005199,005225,005230,005300,005464,005568,005610,005723,005899,006062,006115,006129,006166,006177,006264,006346,006414,007042,007314,007684,007709,007959,008106,008168,008215,009137,009186,009227,009468,009518,009530,009531,009532,009547,009698,009720,009998,010110,010125,010137,010454,010497,010498,010500,010501,010508,010531,010923,010924,010925,010926,010928,010930,010932,010937,010946,010948,010952,010956,011332,011989,011990,011991,011992,011996,011997,012000,012001,012002,012009,012010,012013,012016,012017,012021,020065,020079,030013,030017,030021,030031,030076,030078,030087,030088,030089,030090,030091,030093,030095,030097,030116,030119,030120,030122,030128,030137,030145,030149,030151,030152,030169,030200,030211,030212,030243,030246,030247,030263,030264,030273,030278,030432,030434,030436,030440,030450
-
-未使用题号:
-000054,000153,000163,000189,000197,000236,000788,000792,000858,000871,000883,001120,001226,001305,001371,001597,001623,001649,001659,001848,001877,003179,003328,003332,003454,003466,003501,003544,003875,003982,003996,004084,004167,004283,004768,004794,004928,006062,007314,007684,009137,009186,009227,009698,009720,010110,010454,010498,010500,010531,011332,011989,011990,011991,011992,011996,011997,012000,012001,012002,012009,012010,012013,012016,012017,012021,020065,030078,030087,030088,030089,030090,030091,030093,030095,030097,030116,030119,030120,030122,030128,030137,030145,030149,030151,030200,030211,030212,030243,030246,030247,030263,030264
-
-已使用题号:
-000020,000033,000048,000049,000061,000101,000119,000165,000168,000170,000326,000339,000347,000348,000349,000357,000367,000368,000371,000387,000411,000431,000436,000481,000622,000637,000738,000757,000894,000924,001073,001086,001238,001245,001286,001292,001300,001314,001326,001480,001506,001510,001676,001847,001852,001857,001858,001871,001992,002011,002027,002081,002697,002710,002772,002784,002793,002798,002800,002833,002847,002874,002909,002925,002955,003041,003092,003095,003097,003130,003140,003142,003345,003346,003355,003455,003519,003520,003632,003636,003658,003662,003798,003815,003841,003891,003936,003986,004080,004081,004092,004122,004124,004125,004144,004240,004248,004312,004313,004328,004365,004400,004414,004420,004439,004446,004447,004449,004451,004454,004455,004458,004460,004492,004557,004878,004929,005150,005199,005225,005230,005300,005464,005568,005610,005723,005899,006115,006129,006166,006177,006264,006346,006414,007042,007709,007959,008106,008168,008215,009468,009518,009530,009531,009532,009547,009998,010125,010137,010497,010501,010508,010923,010924,010925,010926,010928,010930,010932,010937,010946,010948,010952,010956,020079,030013,030017,030021,030031,030076,030152,030169,030273,030278,030432,030434,030436,030440,030450
\ No newline at end of file
+000263,000267,000268,000273,000278,000282,000283,000286,000287,000290,000395,000421,000484,000523,000544,000825,000839,002095,002096,002097,002098,002099,002100,002101,002102,002103,002104,002105,002106,002108,002109,002110,002111,002113,002114,002115,002116,002117,002119,002120,002121,002122,002123,002124,002135,002153,002181,002192,002202,002203,002211,002220,002237,002238,002246,002248,002249,002253,002292,002293,002294,002296,002297,002298,002299,002300,002301,002302,002303,002304,002305,002312,002320,002336,002340,002341,002355,002357,002358,002359,002383,002388,002394,002396,002399,002414,002415,002416,002444,002446,002448,002449,002453,002454,002462,002463,002464,002465,002473,002477,002691,003371,003379,003387,003405,003425,003433,003442,003445,003659,003663,003671,003744,003804,003819,003831,003933,003937,004158,004237,004246,004267,004343,004504,004508,004566,004718,004734,004742,004749,008764,008782,008789,008836,008837,008838,008839,008840,008841,008842,008843,008844,008845,008846,008847,008848,008849,008850,008851,008852,008853,008857,008867,008874,008888,008896,008911,008918,008938,008944,008963,008967,009075,009079,009082,009103,009108,009810,009811,009818,009826,009835,009841,009842,009843,009999,010611,010625,010632,010633,010634,010641,010663,010665,010681,010687,010691,010692,010693,010694,010695,010696,010699,010951,010956,010963,010969,010976,010979,010984,011000,011058,011061,011068,011075,011089,011092,011096,011099,011110,011223,011227,011232,011240,011246,011247,011261,011267,011272,011282,011290,011291,011298,011303,011309,011311,011315,011324,011329,011330,011337,011339,011345,011348,011355,011366,011373,011374,011387,011394,011395,011399,011406,011425,011429,011436,011440,011441,011443,011450,011456,011464,011467,011471,011487,011491,011495,011499,011506,011513,011516,011522,011534,011542,011544,011546,011551,011555,011562,011568,011572,011576,011580,011582,011588,011590,011593,011600,011603,011623,011627,011644,011645,011655,011659,011668,011672,011686,011691,011697,011701,011713,011718,011736,012030,012031,012032
\ No newline at end of file
diff --git a/工具/根据目标列表批量生成对应题目的字典.ipynb b/工具/根据目标列表批量生成对应题目的字典.ipynb
index 6085d94b..500863b5 100644
--- a/工具/根据目标列表批量生成对应题目的字典.ipynb
+++ b/工具/根据目标列表批量生成对应题目的字典.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
@@ -14,38 +14,44 @@
},
{
"cell_type": "code",
- "execution_count": 4,
+ "execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"# 输入目标列表\n",
- "t = \"\"\"K0629001X\n",
- "K0629002X\n",
- "K0629003X\n",
- "K0630001X\n",
- "K0630002X\n",
- "K0630003X\n",
- "K0630004X\n",
- "K0631001X\n",
- "K0631002X\n",
- "K0631003X\n",
+ "t = \"\"\"K0719001X\n",
+ "K0719002X\n",
+ "K0719003X\n",
+ "K0719004X\n",
+ "K0719005X\n",
+ "K0719006X\n",
+ "K0720001X\n",
+ "K0720002X\n",
+ "K0720003X\n",
+ "K0720004X\n",
+ "K0720005X\n",
"\"\"\""
]
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "\"K0629001X\":\"000293,000297,009868,010740,030465\",\n",
- "\"K0630002X\":\"000298,000302,000304,001948,004348,009870,010721,010730\",\n",
- "\"K0630004X\":\"000294,000300,003647,004656,004698,004740,009871,010732,010735,030462,030472\",\n",
- "\"K0631002X\":\"000303,000305,001981,004243,009873,010739\",\n",
- "\"K0631003X\":\"000295,000296,000299,004096,009872,010737,030461,030468\",\n"
+ "\"K0719001X\":\"004654\",\n",
+ "\"K0719002X\":\"002393,009099\",\n",
+ "\"K0719003X\":\"000369,000467,000669,000728,000806,000833,000878,000909,000957,000968,002395,002397,002405,002406,002407,003448,003639,003837,003930,003945,004197,004221,004495,004514,004524,004550,004570,008921,008924,008926,008928,008935,008960,009077,009105,009837,009840,010682\",\n",
+ "\"K0719004X\":\"000337,000707,000739,000783,000864,002408,002409,002410,002439,003437,003449,008920,008927,008953,008956,009081,009836,010684\",\n",
+ "\"K0719005X\":\"000290,002440,002441,002450,004078,010703\",\n",
+ "\"K0719006X\":\"000275,002402,002421,002689,003440,003599,003781,008923,010683\",\n",
+ "\"K0720002X\":\"002426\",\n",
+ "\"K0720003X\":\"000282,000464,002398,002400,002401,002403,002404,002411,002412,002413,002417,002418,002419,002420,002422,002423,002424,002425,002427,002429,002430,002434,002436,002437,002445,002690,003438,003439,003441,003443,003444,003446,003447,003450,003451,004141,004225,004351,004372,004626,004639,004713,004764,008922,008925,008930,008931,008932,008933,008934,008936,008954,008955,008968,009098,009106,009110,009838,009839,010685,010688,010689\",\n",
+ "\"K0720004X\":\"000804,003795,004065,008929,008937,008957,010686\",\n",
+ "\"K0720005X\":\"002438\",\n"
]
}
],
diff --git a/工具/目标挂钩简要清点.ipynb b/工具/目标挂钩简要清点.ipynb
index 04ba1a49..e58c0688 100644
--- a/工具/目标挂钩简要清点.ipynb
+++ b/工具/目标挂钩简要清点.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
@@ -24,30 +24,22 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "第一单元 . 总题数: 1758 , 完成对应题数: 1400\n",
- "第二单元 . 总题数: 2485 , 完成对应题数: 2127\n",
- "第三单元 . 总题数: 2071 , 完成对应题数: 394\n",
- "第四单元 . 总题数: 1116 , 完成对应题数: 1027\n",
- "第五单元 . 总题数: 1324 , 完成对应题数: 304\n",
- "第六单元 . 总题数: 1069 , 完成对应题数: 429\n",
- "第七单元 . 总题数: 1338 , 完成对应题数: 59\n",
- "第八单元 . 总题数: 1219 , 完成对应题数: 195\n",
- "第九单元 . 总题数: 250 , 完成对应题数: 101\n"
- ]
- },
- {
- "ename": "",
- "evalue": "",
- "output_type": "error",
- "traceback": [
- "\u001b[1;31mThe Kernel crashed while executing code in the the current cell or a previous cell. Please review the code in the cell(s) to identify a possible cause of the failure. Click here for more info. View Jupyter log for further details."
+ "第一单元 . 总题数: 1766 , 完成对应题数: 1400\n",
+ "第二单元 . 总题数: 2495 , 完成对应题数: 2127\n",
+ "第三单元 . 总题数: 2079 , 完成对应题数: 395\n",
+ "第四单元 . 总题数: 1126 , 完成对应题数: 1032\n",
+ "第五单元 . 总题数: 1329 , 完成对应题数: 305\n",
+ "第六单元 . 总题数: 1074 , 完成对应题数: 431\n",
+ "第七单元 . 总题数: 1338 , 完成对应题数: 953\n",
+ "第八单元 . 总题数: 1221 , 完成对应题数: 195\n",
+ "第九单元 . 总题数: 252 , 完成对应题数: 101\n"
]
}
],
@@ -58,42 +50,22 @@
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
- "11357"
+ "(12680, 6939)"
]
},
- "execution_count": 6,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "sum(count1)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "12682"
- ]
- },
- "execution_count": 7,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "len(pro_dict)"
+ "(sum(count1),sum(count2))"
]
},
{
diff --git a/工具/讲义生成.ipynb b/工具/讲义生成.ipynb
index d6372089..0b2e5c2e 100644
--- a/工具/讲义生成.ipynb
+++ b/工具/讲义生成.ipynb
@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 8,
+ "execution_count": 5,
"metadata": {},
"outputs": [
{
@@ -13,11 +13,9 @@
"题块 1 处理完毕.\n",
"正在处理题块 2 .\n",
"题块 2 处理完毕.\n",
- "正在处理题块 3 .\n",
- "题块 3 处理完毕.\n",
- "开始编译教师版本pdf文件: 临时文件/周末卷09_教师_20221115.tex\n",
+ "开始编译教师版本pdf文件: 临时文件/37_抛物线的概念及性质_教师_20221119.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/周末卷09_学生_20221115.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/37_抛物线的概念及性质_学生_20221119.tex\n",
"0\n"
]
}
@@ -30,19 +28,19 @@
"\"\"\"---设置模式结束---\"\"\"\n",
"\n",
"\"\"\"---设置模板文件名---\"\"\"\n",
- "# template_file = \"模板文件/第一轮复习讲义模板.tex\"\n",
- "template_file = \"模板文件/测验周末卷模板.tex\"\n",
+ "template_file = \"模板文件/第一轮复习讲义模板.tex\"\n",
+ "# template_file = \"模板文件/测验周末卷模板.tex\"\n",
"# template_file = \"模板文件/日常选题讲义模板.tex\"\n",
"\"\"\"---设置模板文件名结束---\"\"\"\n",
"\n",
"\"\"\"---设置其他预处理替换命令---\"\"\"\n",
"#2023届第一轮讲义更换标题\n",
- "# exec_list = [(\"标题数字待处理\",\"32\"),(\"标题文字待处理\",\"空间向量的概念与性质及立体几何中的证明问题\")] \n",
- "# enumi_mode = 0\n",
+ "exec_list = [(\"标题数字待处理\",\"37\"),(\"标题文字待处理\",\"抛物线的概念及性质\")] \n",
+ "enumi_mode = 0\n",
"\n",
"#2023届测验卷与周末卷\n",
- "exec_list = [(\"标题替换\",\"基础考\")]\n",
- "enumi_mode = 1\n",
+ "# exec_list = [(\"标题替换\",\"基础考\")]\n",
+ "# enumi_mode = 1\n",
"\n",
"# 日常选题讲义\n",
"# exec_list = [(\"标题文字待处理\",\"三角向量复数立几易错题\")] \n",
@@ -51,15 +49,14 @@
"\"\"\"---其他预处理替换命令结束---\"\"\"\n",
"\n",
"\"\"\"---设置目标文件名---\"\"\"\n",
- "destination_file = \"临时文件/周末卷09\"\n",
+ "destination_file = \"临时文件/37_抛物线的概念及性质\"\n",
"\"\"\"---设置目标文件名结束---\"\"\"\n",
"\n",
"\n",
"\"\"\"---设置题号数据---\"\"\"\n",
"problems = [\n",
- "\"11049:11060\",\n",
- "\"11061:11064\",\n",
- "\"11065:11069\"\n",
+ "\"4654,2393,669,3837,4495,4550,864,3437,2450,2441,2402,2689,2426,282,2403,2417,2418,804\",\n",
+ "\"9099,8926,9077,9840,8956,9081,2440,8923,3781,2424,2429,2434,3438,8937\"\n",
"\n",
"]\n",
"\"\"\"---设置题号数据结束---\"\"\"\n",
diff --git a/工具/题号选题pdf生成.ipynb b/工具/题号选题pdf生成.ipynb
index abcc734c..b9084049 100644
--- a/工具/题号选题pdf生成.ipynb
+++ b/工具/题号选题pdf生成.ipynb
@@ -2,16 +2,16 @@
"cells": [
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "开始编译教师版本pdf文件: 临时文件/必修前八单元简单题_教师用_20221115.tex\n",
+ "开始编译教师版本pdf文件: 临时文件/题库_教师用_20221119.tex\n",
"0\n",
- "开始编译学生版本pdf文件: 临时文件/必修前八单元简单题_学生用_20221115.tex\n",
+ "开始编译学生版本pdf文件: 临时文件/题库_学生用_20221119.tex\n",
"0\n"
]
}
@@ -26,7 +26,8 @@
"\"\"\"---设置题目列表---\"\"\"\n",
"#留空为编译全题库, a为读取临时文件中的题号筛选.txt文件生成题库\n",
"problems = r\"\"\"\n",
- "000054,000153,000163,000189,000197,000788,000792,000858,000871,000883,001120,001226,001305,001371,001597,001623,001649,001659,001848,001877,003179,003328,003332,003454,003466,003544,003875,003982,003996,004084,004167,004283,004768,004794,004928,006062,007314,007684,009137,009186,009227,009698,009720,010110,010454,010498,010500,010531,011332,011989,011990,011991,011992,011996,011997,012000,012001,012002,012009,012010,012013,012016,012017,012021,020065,030078,030087,030088,030089,030090,030091,030093,030095,030097,030116,030119,030120,030122,030128,030137,030145,030149,030151\n",
+ "\n",
+ "\n",
"\n",
"\n",
"\"\"\"\n",
@@ -34,7 +35,7 @@
"\n",
"\"\"\"---设置文件名---\"\"\"\n",
"#目录和文件的分隔务必用/\n",
- "filename = \"临时文件/必修前八单元简单题\"\n",
+ "filename = \"临时文件/题库\"\n",
"\"\"\"---设置文件名结束---\"\"\"\n",
"\n",
"\n",
@@ -175,7 +176,7 @@
],
"metadata": {
"kernelspec": {
- "display_name": "Python 3.9.7 ('base')",
+ "display_name": "Python 3.8.8 ('base')",
"language": "python",
"name": "python3"
},
@@ -189,12 +190,12 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.9.7"
+ "version": "3.8.8"
},
"orig_nbformat": 4,
"vscode": {
"interpreter": {
- "hash": "e4cce46d6be9934fbd27f9ca0432556941ea5bdf741d4f4d64c6cd7f8dfa8fba"
+ "hash": "d311ffef239beb3b8f3764271728f3972d7b090c974f8e972fcdeedf230299ac"
}
}
},
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 68531b76..3c2f7542 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -7001,7 +7001,8 @@
"content": "若椭圆$\\dfrac{x^2}{4}+\\dfrac{y^2}{a^2}=1$与双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}2=1$有相同的焦点, 求实数$a$的值.",
"objs": [
"K0713004X",
- "K0713002X"
+ "K0713002X",
+ "K0716002X"
],
"tags": [
"第七单元",
@@ -7456,7 +7457,8 @@
"objs": [
"K0713003X",
"K0716003X",
- "K0713002X"
+ "K0713002X",
+ "K0716002X"
],
"tags": [
"第七单元",
@@ -7482,7 +7484,8 @@
"content": "求双曲线$y=\\dfrac 1x$的焦点坐标与准线方程.",
"objs": [
"K0716002X",
- "K0717002X"
+ "K0717002X",
+ "KNONE"
],
"tags": [
"第七单元",
@@ -8711,7 +8714,8 @@
],
"tags": [
"第四单元",
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\dfrac 12$",
@@ -8762,7 +8766,9 @@
"000335": {
"id": "000335",
"content": "若双曲线的一条渐近线为$x+2y=0$, 且双曲线与抛物线$y=x^2$的准线仅有一个公共点, 则此双曲线的标准方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -8837,7 +8843,10 @@
"000337": {
"id": "000337",
"content": "已知抛物线$C$的顶点在平面直角坐标系原点, 焦点在$x$轴上, 若$C$经过点$M(1,3)$, 则其焦点到准线的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X",
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -8941,7 +8950,8 @@
"content": "在$(x+\\dfrac{2}{x^2})^6$的二项展开式中第四项的系数是\\blank{50}(结果用数值表示).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$160$",
@@ -9314,7 +9324,8 @@
"content": "已知$(a+3b)^n$的展开式中, 各项系数的和与各项二项式系数的和之比为$64$, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$6$",
@@ -9644,7 +9655,9 @@
"000360": {
"id": "000360",
"content": "设$k\\in \\mathbf{R}$, $\\dfrac{y^2}{k}-\\dfrac{x^2}{k-2}=1$表示焦点在$y$轴上的双曲线, 则半焦距的取值范围是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -9978,7 +9991,9 @@
"000369": {
"id": "000369",
"content": "抛物线$y=x^2$上一点$M$到焦点的距离为$1$, 则点$M$的纵坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -10124,7 +10139,8 @@
"content": "若$(2x^2+\\dfrac1x)^n$($n\\in \\mathbf{N}$, $n\\ge 1$)的二项展开式中的第$9$项是常数项, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$12$",
@@ -10500,7 +10516,8 @@
"content": "设常数$a>0$, 若$(x+\\dfrac ax)^9$的二项展开式中$x^5$的系数为$144$, 则$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$2$",
@@ -10843,7 +10860,8 @@
"content": "在二项式$(x+\\dfrac2x)^6$的展开式中, 常数项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$160$",
@@ -10873,7 +10891,9 @@
"000399": {
"id": "000399",
"content": "等轴双曲线$x^2-y^2=a^2$与抛物线$y^2=16x$的准线交于$A$、$B$两点, 且$|AB|=4\\sqrt3$, 则该双曲线的实轴长等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -11162,7 +11182,8 @@
"content": "$(1+2x)^6$展开式中$x^3$项的系数为\\blank{50}(用数字作答).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$160$",
@@ -11381,7 +11402,8 @@
"content": "$(1-\\dfrac x2)^8$的二项展开式中含$x^2$项的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$7$",
@@ -11533,7 +11555,9 @@
"000424": {
"id": "000424",
"content": "过双曲线$C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}4=1$的右焦点$F$作一条垂直于$x$轴的垂线交双曲线$C$的两条渐近线于$A$、$B$两点, $O$为坐标原点, 则$\\triangle OAB$的面积的最小值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -11755,7 +11779,10 @@
"000432": {
"id": "000432",
"content": "若双曲线$x^2-\\dfrac{y^2}{b^2}=1$的一个焦点到其渐近线距离为$2\\sqrt2$, 则该双曲线焦距等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -11835,7 +11862,8 @@
"content": "设函数$f(x)=\\begin{cases} x^6, & x\\ge 1, \\\\ -2x-1, & x\\le -1, \\end{cases}$ 则当$x\\le -1$时, $f[f(x)]$表达式的展开式中含${x^2}$项的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$60$",
@@ -11936,7 +11964,8 @@
"content": "二项式$(x^2+\\dfrac 1x)^5$的展开式中, $x$的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$10$",
@@ -12238,7 +12267,9 @@
"000451": {
"id": "000451",
"content": "点$(1,0)$到双曲线$\\dfrac{x^2}4-y^2=1$的渐近线的距离是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -12344,7 +12375,8 @@
],
"tags": [
"第四单元",
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$2$",
@@ -12540,7 +12572,8 @@
"content": "在$(\\dfrac3{x^2}+\\sqrt{x})^n$的二项展开式中, 所有项的二项式系数之和为$1024$, 则常数项的值等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$405$",
@@ -12586,7 +12619,10 @@
"000464": {
"id": "000464",
"content": "已知抛物线$C$的顶点为坐标原点, 双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}{144}=1$的右焦点是$C$的焦点$F$. 若斜率为$-1$, 且过$F$的直线与$C$交于$A,B$两点, 则$|AB|=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -12659,7 +12695,9 @@
"000467": {
"id": "000467",
"content": "抛物线$y^2=4x$的焦点坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -12756,7 +12794,8 @@
"content": "在代数式$(x+\\dfrac 1{x^2})^7$的展开式中, 一次项的系数是\\blank{50}(用数字作答).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$21$",
@@ -13106,7 +13145,8 @@
"content": "$(x+\\dfrac{1}{x^2})^9$的二项展开式中, 常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$84$",
@@ -13365,7 +13405,9 @@
"000493": {
"id": "000493",
"content": "在平面直角坐标系中, 双曲线$\\dfrac{x^2}{a^2}-y^2=1 $的一个顶点与抛物线$y^2=12x$的焦点重合, 则双曲线的两条渐近线的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -13605,7 +13647,8 @@
"objs": [],
"tags": [
"第五单元",
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$5\\sqrt 2$",
@@ -14060,7 +14103,10 @@
"000518": {
"id": "000518",
"content": "与双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$的渐近线相同, 且经过点$A(-3,2 \\sqrt3)$的双曲线的方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -14139,7 +14185,8 @@
"content": "在$(x-a)^{10}$的展开式中, $x^7$的系数是$15$, 则实数$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$-\\frac 12$",
@@ -14418,7 +14465,8 @@
"content": "在$(1+2x)^5$的展开式中, $x^2$项系数为\\blank{50}(用数字作答).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$40$",
@@ -14593,7 +14641,8 @@
"content": "$(\\sqrt x-\\dfrac1x)^9$的二项展开式中的常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$-84$",
@@ -15152,7 +15201,8 @@
"id": "000560",
"content": "在平面直角坐标系$xOy$中, 以直线$y=\\pm 2x$为渐近线, 且经过椭圆$x^2+\\dfrac{y^2}4=1$右顶点的双曲线的方程是\\blank{50}.",
"objs": [
- "K0714003X"
+ "K0714003X",
+ "K0717006X"
],
"tags": [
"第七单元",
@@ -15231,7 +15281,8 @@
"content": "已知$(1+2x)^6$展开式的二项式系数的最大值为$a$, 系数的最大值为$b$, 则$\\dfrac ba=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$12$",
@@ -15363,7 +15414,8 @@
"content": "二项式$(x-\\dfrac1{2x})^4$的展开式中的常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\frac 32$",
@@ -15679,7 +15731,8 @@
"content": "在$(x-\\dfrac2x)^6$的二项展开式中, 常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\dfrac 1{12}$",
@@ -15787,7 +15840,10 @@
"000584": {
"id": "000584",
"content": "抛物线$y^2=-8x$的焦点与双曲线$\\dfrac{x^2}{a^2}-y^2=1$的左焦点重合, 则这条双曲线的两条渐近线的夹角为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -16030,7 +16086,8 @@
"content": "若$(2x+\\dfrac 1x)^n$的二项展开式中的所有二项式系数之和等于$256$, 则该展开式中常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$1120$",
@@ -16208,7 +16265,8 @@
],
"tags": [
"第四单元",
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$8$",
@@ -16969,7 +17027,9 @@
"000629": {
"id": "000629",
"content": "设$m$为常数, 若点$F(0,5)$是双曲线$\\dfrac{y^2}m-\\dfrac{x^2}9=1$的一个焦点, 则$m=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -17049,7 +17109,8 @@
"content": "若$(\\sqrt x-\\dfrac1x)^n$的二项展开式中各项的二项式系数的和是$64$, 则展开式中的常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$15$",
@@ -17277,7 +17338,8 @@
"content": "若$(x+a)^7$的二项展开式中, 含$x^6$项的系数为$7$, 则实数$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$1$",
@@ -17298,7 +17360,9 @@
"000641": {
"id": "000641",
"content": "已知双曲线$x^2-\\dfrac{y^2}{a^2}=1 \\ (a>0)$,它的渐近线方程是$y=\\pm 2x$,则$a$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -17751,7 +17815,8 @@
"content": "已知$(x-\\dfrac ax)^7$展开式中$x^3$的系数为$84$, 则正实数$a$的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$2$",
@@ -18027,7 +18092,9 @@
"000669": {
"id": "000669",
"content": "抛物线$y=\\dfrac14{x^2}$的焦点到准线的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -18410,7 +18477,10 @@
"000683": {
"id": "000683",
"content": "已知双曲线$C_1$与双曲线$C_2$的焦点重合,$C_1$的方程为$\\dfrac{x^2}3-{y^2}=1$, 若$C_2$的一条渐近线的倾斜角是$C_1$的一条渐近线的倾斜角的$2$倍, 则$C_2$的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -18580,7 +18650,9 @@
"000689": {
"id": "000689",
"content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{(a+3)^2}=1 \\ (a>0)$的一条渐近线方程为$y=\\pm 2x$, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -18720,7 +18792,8 @@
"content": "设多项式$1+x+(1+x)^2+(1+x)^3+\\cdots+(1+x)^n\\ (x\\ne 0, \\ n\\in \\mathbf{N}^*)$的展开式中$x$项的系数为$T_n$, 则$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{T_n}{n^2}=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\frac 12$",
@@ -19064,7 +19137,9 @@
"000707": {
"id": "000707",
"content": "设抛物线的焦点坐标为$(1,0)$, 则此抛物线的标准方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -19470,7 +19545,8 @@
"content": "若二项式$(2x+\\dfrac ax)^7$的展开式中一次项的系数是$-70$, 则$\\displaystyle\\lim_{n\\to\\infty}(a+a^2+a^3+\\cdots+a^n)=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$-\\frac 13$",
@@ -19626,7 +19702,9 @@
"000728": {
"id": "000728",
"content": "抛物线$y=x^2$的焦点坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -19816,7 +19894,8 @@
"content": "代数式$(x^2+2)(\\dfrac1{x^2}-1)^5$的展开式的常数项是\\blank{50}(用数字作答).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$3$",
@@ -19865,7 +19944,8 @@
"content": "在$(x+\\dfrac1x)^6$的二项展开式中, 常数项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$20$",
@@ -19922,7 +20002,9 @@
"000739": {
"id": "000739",
"content": "已知抛物线$x^2=ay$的准线方程是$y=-\\dfrac14$, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -20228,7 +20310,9 @@
"000751": {
"id": "000751",
"content": "从集合$\\{-1,1,2,3\\}$随机取一个为$m$, 从集合$\\{-2,-1,1,2\\}$随机取一个为$n$, 则方程$\\dfrac{x^2}m+\\dfrac{y^2}n=1$表示双曲线的概率为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -20852,7 +20936,8 @@
"content": "$(1+2x)^n$的二项展开式中, 含$x^3$项的系数等于含$x$项的系数的$8$倍, 则正整数$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$5$",
@@ -21094,7 +21179,9 @@
"000783": {
"id": "000783",
"content": "已知抛物线的顶点在坐标原点, 焦点在$y$轴上, 抛物线上一点$M(a,-4) \\ (a>0)$到焦点F的距离为$5$. 则该抛物线的标准方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -21169,7 +21256,9 @@
"000786": {
"id": "000786",
"content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}9=1 \\ (a>0)$的渐近线方程为$3x\\pm 2y=0$, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -21577,7 +21666,8 @@
"content": "$(\\sqrt x+\\dfrac1x)^9$二项展开式中的常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$84$",
@@ -21677,7 +21767,9 @@
"000804": {
"id": "000804",
"content": "已知抛物线型拱桥的顶点距水面$2$米时, 量得水面宽为$8$米. 当水面下降$1$米后, 水面的宽为\\blank{50}米.",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -21726,7 +21818,9 @@
"000806": {
"id": "000806",
"content": "抛物线$x^2=12y$的准线方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -21858,7 +21952,8 @@
"content": "若$(x^3-\\dfrac1{x^2})^n$的展开式中含有非零常数项, 则正整数$n$的最小值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$5$",
@@ -22181,7 +22276,8 @@
"content": "$(1+\\dfrac1{x^2})(1+x)^6$展开式中$x^2$的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$30$",
@@ -22329,7 +22425,8 @@
"content": "若$(1+3x)^n$的二项展开式中$x^2$项的系数是$54$, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$4$",
@@ -22451,7 +22548,10 @@
"000833": {
"id": "000833",
"content": "若双曲线$\\dfrac{x^2}3-\\dfrac{16y^2}{p^2}=1\\ (p>0)$的左焦点在抛物线$y^2=2px$的准线上, 则$p=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"双曲线",
@@ -22556,7 +22656,8 @@
"content": "$(x+\\dfrac1x)^n$的展开式中的第$3$项为常数项, 则正整数$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$4$",
@@ -22846,7 +22947,9 @@
"000848": {
"id": "000848",
"content": "双曲线$2 x^2-y^2=6$的焦距为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -22872,7 +22975,8 @@
"content": "已知$(ax+\\dfrac1x)^6$二项展开式中的第五项系数为$\\dfrac{15}2$, 则正实数$a$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\frac{\\sqrt 2}2$",
@@ -23258,7 +23362,9 @@
"000864": {
"id": "000864",
"content": "在平面直角坐标系$xOy$中, 有一定点$A(1,1)$, 若线段$OA$的垂直平分线过抛物线$C:y^2=2px \\ (p>0)$的焦点, 则抛物线$C$的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -23309,7 +23415,8 @@
"content": "记$(2x+\\dfrac1x)^n \\ (n\\in \\mathbf{N}^*$)的展开式中第$m$项的系数为$b_m$, 若$b_3=2b_4$, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$5$",
@@ -23506,7 +23613,8 @@
"content": "$(\\dfrac1x-\\sqrt x)^6$的展开式中, 常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$15$",
@@ -23580,7 +23688,9 @@
"000876": {
"id": "000876",
"content": "已知双曲线$x^2-\\dfrac{y^2}4=1$的右焦点为$F$, 过点$F$且平行于双曲线的一条渐近线的直线与双曲线交于点$P$, $M$在直线$PF$上, 且满足$\\overrightarrow{OM}\\cdot \\overrightarrow{PF}=0$, 则$\\dfrac{|\\overrightarrow{PM}|}{|\\overrightarrow{PF}|}=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -23627,7 +23737,9 @@
"000878": {
"id": "000878",
"content": "抛物线$y^2=4x$的焦点坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -23832,7 +23944,8 @@
"content": "试写出$(x-\\dfrac1x)^7$展开式中系数最大的项\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\frac{35}x$",
@@ -24387,7 +24500,8 @@
"content": "若二项式$(2x-\\dfrac1{\\sqrt x})^n$展开式中的第$5$项为常数项, 则展开式中各项的二项式系数之和为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$64$",
@@ -24457,7 +24571,9 @@
"000909": {
"id": "000909",
"content": "若经过抛物线 $y^2=4x$焦点的直线$l$与圆$(x-4)^2+y^2=4$相切, 则直线$l$的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -24587,7 +24703,8 @@
"content": "在$(x^3-\\dfrac1x)^8$的展开式中, 其常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$28$",
@@ -24862,7 +24979,9 @@
"000925": {
"id": "000925",
"content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,\\ b>0)$的一条渐近线方程是$y=\\sqrt3x$, 它的一个焦点与抛物线$y^2=16x$的焦点相同, 则双曲线的标准方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -24962,7 +25081,8 @@
"content": "设$a\\ne 0$, $n$是大于$1$的自然数, $(1+\\dfrac xa)^n$的展开式为$a_0+a_1x+a_2x^2+\\cdots+a_nx^n$. 若$a_1=3$, $a_2=4$, 则$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$3$",
@@ -25546,7 +25666,8 @@
"content": "在代数式$(4x^2-2x-5)(1+\\dfrac1{x^2})^5$的展开式中, 常数等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$15$",
@@ -25673,7 +25794,9 @@
"000956": {
"id": "000956",
"content": "双曲线$4 x^2-y^2=1$的一条渐近线与直线$tx+y+1=0$垂直, 则$t=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -25699,7 +25822,9 @@
"000957": {
"id": "000957",
"content": "已知抛物线$y^2=4x$上一点$M(x_0,2 \\sqrt3)$, 则点$M$到抛物线焦点的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -25970,7 +26095,9 @@
"000968": {
"id": "000968",
"content": "抛物线$y^2=x$上一点$M$到焦点的距离为$1$, 则点$M$的横坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -26069,7 +26196,10 @@
"000972": {
"id": "000972",
"content": "已知双曲线$x^2-\\dfrac{y^2}{m^2}=1 \\ (m>0)$的渐近线与圆$x^2+(y+2)^2=1$没有公共点, 则该双曲线的焦距的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63187,7 +63317,9 @@
"002348": {
"id": "002348",
"content": "双曲线$\\dfrac{x^2}{4}-y^2=1$的两个焦点的坐标为\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63211,7 +63343,9 @@
"002349": {
"id": "002349",
"content": "双曲线$y^2-\\dfrac{x^2}{5}=1$的两个焦点的坐标为\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63235,7 +63369,9 @@
"002350": {
"id": "002350",
"content": "已知双曲线的焦点为$(6,0)$和$(-6,0)$, 且过$(-5,2)$, 则此双曲线方程为\\blank{100}.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63257,7 +63393,9 @@
"002351": {
"id": "002351",
"content": "若方程$x^2\\sin\\theta+y^2\\cos\\theta=1$表示双曲线, 则$\\theta$的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63279,7 +63417,9 @@
"002352": {
"id": "002352",
"content": "已知双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$上的点$P$到右焦点的距离为$14$, 则$P$到左准线的距离为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63301,7 +63441,9 @@
"002353": {
"id": "002353",
"content": "已知双曲线$kx^2-2ky^2+1=0$的一个焦点为$(-4,0)$, 则实数$k=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63323,7 +63465,9 @@
"002354": {
"id": "002354",
"content": "已知双曲线的半焦距为$c$, 两准线间的距离为$d$, 且$c=d$, 则双曲线的离心率等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63454,7 +63598,9 @@
"002360": {
"id": "002360",
"content": "[选做]\n已知坐标平面内的定点$P(a,b)$. 根据点$P$的不同位置, 讨论是否存在以${{F}_{1}}(3,0),{{F}_{2}}(-3,0)$为焦点的双曲线过点$P$.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63476,7 +63622,9 @@
"002361": {
"id": "002361",
"content": "双曲线$kx^2-2ky^2=4$的一条准线是$y=1$, 则实数$k$的值等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63498,7 +63646,9 @@
"002362": {
"id": "002362",
"content": "若方程$\\dfrac{x^2}{2-m}+\\dfrac{y^2}{|m|-3}=1$表示双曲线, 则实数$m$的取值范围为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63521,7 +63671,8 @@
"id": "002363",
"content": "与两圆$x^2+y^2=1$和$x^2+y^2-8x+7=0$都相切的圆的圆心轨迹是\\bracket{20}.\n\\twoch{两个椭圆}{两条双曲线}{一条双曲线和一条直线}{一个椭圆和一条双曲线}",
"objs": [
- "K0713002X"
+ "K0713002X",
+ "K0716002X"
],
"tags": [
"第七单元",
@@ -63545,7 +63696,10 @@
"002364": {
"id": "002364",
"content": "已知$E,F$分别是离心率为$\\dfrac{\\sqrt{5}+1}{2}$的双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,b>0)$的左顶点与右焦点, 再记$M(0,b)$, 则$\\angle EMF$等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717002X",
+ "K0717007X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63567,7 +63721,9 @@
"002365": {
"id": "002365",
"content": "已知$P$为双曲线$3x^2-5y^2=15$上的一点, $F_1,F_2$为其两个焦点, 且$S_{\\triangle F_1PF_2}=3\\sqrt{3}$, 求$\\angle F_1PF_2$的大小.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63589,7 +63745,10 @@
"002366": {
"id": "002366",
"content": "已知双曲线$\\dfrac{{{x}^{2}}}{16}-\\dfrac{{{y}^{2}}}{20}=1$的左右焦点分别为${{F}_{1}},{{F}_{2}}$, 设双曲线上有一动点$P$.\\\\ \n(1) 若$|P{{F}_{1}}|=9$, 求$|P{{F}_{2}}|$;\\\\ \n(2) 若$|P{{F}_{1}}|=19$, 求$|P{{F}_{2}}|$;\\\\ \n(3) 设$a$是实常数, 若$P$到定点$A(a,0)$的距离最小值为$10$, 求$a$的值.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0715002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63612,7 +63771,8 @@
"id": "002367",
"content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1\\ (a>b>0)$和双曲线$\\dfrac{x^2}{m^2}-\\dfrac{y^2}{n^2}=1 \\ (m,n>0)$有公共的焦点$F_1,F_2$, $P$是两曲线的一个交点.\\\\ \n(1) 证明: $\\angle F_1PF_2=2\\arctan \\dfrac{n}{b}$;\\\\ \n(2) 证明: $\\triangle F_1PF_2$的面积为$bn$.",
"objs": [
- "K0715002X"
+ "K0715002X",
+ "K0716002X"
],
"tags": [
"第七单元",
@@ -63636,7 +63796,9 @@
"002368": {
"id": "002368",
"content": "双曲线与其共轭双曲线有共同的\\bracket{20}.\n\\fourch{焦点}{准线}{离心率}{渐近线}",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63658,7 +63820,9 @@
"002369": {
"id": "002369",
"content": "中心在原点, 一个焦点为$(3,0)$的等轴双曲线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63680,7 +63844,9 @@
"002370": {
"id": "002370",
"content": "中心在原点, 一个焦点为$(3,0)$, 一条渐近线方程为$2x-3y=0$的双曲线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63702,7 +63868,9 @@
"002371": {
"id": "002371",
"content": "中心在原点, 焦距为$6$, 一条渐近线方程为$2x-3y=0$的双曲线的标准方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63724,7 +63892,10 @@
"002372": {
"id": "002372",
"content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的离心率$e=\\dfrac{5}{4}$, 半虚轴长为$2$, 则该双曲线的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716003X",
+ "K0717007X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63746,7 +63917,9 @@
"002373": {
"id": "002373",
"content": "若双曲线的离心率为$2$, 则它的共轭双曲线的离心率为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63768,7 +63941,9 @@
"002374": {
"id": "002374",
"content": "双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的两条渐近线夹角的大小为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63793,7 +63968,9 @@
"002375": {
"id": "002375",
"content": "[选做]\n已知函数$y=x+\\dfrac{1}{x}$的图像是双曲线, 则该双曲线的离心率为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63815,7 +63992,10 @@
"002376": {
"id": "002376",
"content": "求过点$(2,-2)$, 且与双曲线$x^2-2y^2=2$有公共渐近线的双曲线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63837,7 +64017,10 @@
"002377": {
"id": "002377",
"content": "若双曲线的虚轴长为$6$, 一条渐近线的方程为$3x-y=0$, 则此双曲线的标准方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0717002X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63859,7 +64042,9 @@
"002378": {
"id": "002378",
"content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,b>0)$的一条准线$l$与一条渐近线交于$P$点, $F$是与$l$相应的焦点.\\\\ \n(1) 求证: 直线$PF$与这条渐近线垂直;\\\\ \n(2) 求$|PF|$.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63881,7 +64066,9 @@
"002379": {
"id": "002379",
"content": "已知双曲线$\\dfrac{x^2}{9}-\\dfrac{y^2}{16}=1$的焦点分别为$F_1,F_2$, $P$为双曲线上一点, 满足\n$|PF_1|\\cdot|PF_2|=32$. 求证: $PF_1\\perp PF_2$.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63906,7 +64093,8 @@
"id": "002380",
"content": "与椭圆$x^2+4y^2=64$有共同焦点, 且一条渐近线的方程为$x+\\sqrt{3}y=0$的双曲线的标准方程为\\blank{80}.",
"objs": [
- "K0713002X"
+ "K0713002X",
+ "K0717006X"
],
"tags": [
"第七单元",
@@ -63930,7 +64118,9 @@
"002381": {
"id": "002381",
"content": "已知双曲线的中心在原点, 且一条渐近线方程为$12x-5y=0$, 一条准线方程为$y=\\dfrac{144}{13}$, 则该双曲线的标准方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63952,7 +64142,9 @@
"002382": {
"id": "002382",
"content": "已知双曲线以两条坐标轴为对称轴, 点$M(\\dfrac{16}{5},\\dfrac{12}{5})$是其准线和渐近线的交点, 则此双曲线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -63995,7 +64187,10 @@
"002384": {
"id": "002384",
"content": "过双曲线$\\dfrac{{{x}^{2}}}{3}-{{y}^{2}}=1$的左焦点${{F}_{1}}$作倾斜角为$\\dfrac{\\pi}{3}$的\n弦$AB$, 求三角形${{F}_{2}}AB$的周长及面积.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64017,7 +64212,10 @@
"002385": {
"id": "002385",
"content": "设$a$是实常数, 已知直线$y=ax+1$与双曲线$3x^2-y^2=1$.\\\\ \n(1) 若直线与双曲线只有一个公共点, 求$a$的值;\\\\ \n(2) 若直线和双曲线的右支有两个公共点, 求$a$的取值范围;\\\\ \n(3) 在(2)的条件下, 设公共点为$A,B$, 是否存在实数$a$使得以线段$AB$为直径的圆经过坐标原点? 若存在, 求$a$ 的值; 若不存在, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0718001X",
+ "K0716004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64039,7 +64237,9 @@
"002386": {
"id": "002386",
"content": "已知双曲线$2x^2-y^2=2$, 试问过点$N(1,1)$能否作一直线与双曲线交于$C,D$两点, 且使$N$为$CD$的中点, 这样的直线如果存在, 求出它的方程, 如果不存在, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64061,7 +64261,9 @@
"002387": {
"id": "002387",
"content": "[选做]\n已知双曲线$2x^2-y^2=2$, 若过点$N(m,n)$能作一直线与双曲线交于$C,D$两点, 且使$N$为$CD$的中点,\n求$(m,n)$所满足的条件.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64128,7 +64330,9 @@
"002390": {
"id": "002390",
"content": "已知$AB$是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$过左焦点$F_1$的任意一条弦, 以$AB$为直径的圆被左准线截得圆弧$\\overset\\frown{CD}$, 求证: $\\overset\\frown{CD}$所对的圆心角的度数为定值.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64150,7 +64354,9 @@
"002391": {
"id": "002391",
"content": "已知双曲线$2x^2-y^2=2$.\\\\ \n(1) 求斜率为$2$的平行弦的中点轨迹方程;\\\\ \n(2) 过$A(2,1)$的直线与双曲线交于$P,Q$两点, 线段$PQ$中点$M$落在一条二次曲线上, 求这个二次曲线的方程, 并回答(不需要理由)中点是否能取到该二次曲线上的每一点.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64172,7 +64378,9 @@
"002392": {
"id": "002392",
"content": "已知直线$l$和双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0,b>0)$有两个交点$A,B$. 与该双曲线的渐近线也有两个交点$C,D$. 证明: $|AC|=|BD|$.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -64194,7 +64402,9 @@
"002393": {
"id": "002393",
"content": "若$A$是直线$l$外的一定点, 则过$A$且与$l$相切的圆的圆心轨迹是\\bracket{20}.\n\\fourch{圆}{双曲线一支}{抛物线}{以上都不是}",
- "objs": [],
+ "objs": [
+ "K0719002X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64241,7 +64451,9 @@
"002395": {
"id": "002395",
"content": "抛物线$y^2=10x$的焦点到准线的距离是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64287,7 +64499,9 @@
"002397": {
"id": "002397",
"content": "在抛物线$y^2=8x$上有一点$P$, 它到焦点的距离为$20$, 则$P$点的坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64309,7 +64523,9 @@
"002398": {
"id": "002398",
"content": "抛物线$y^2=2x$的焦点弦(过焦点的弦)的端点为$A(x_1,y_1)$与$B(x_2,y_2)$, 且$x_1+x_2=3$, 则$|AB|=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64352,7 +64568,9 @@
"002400": {
"id": "002400",
"content": "过抛物线$y^2=2px$的对称轴上一点$C(p,0)$作一条直线与抛物线交于$A,B$两点,\n若$A$点的纵坐标为$-\\dfrac{p}{2}$, 则$B$点的纵坐标为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64374,7 +64592,9 @@
"002401": {
"id": "002401",
"content": "若正三角形的一个顶点在原点, 另两个顶点在抛物线$y^2=2px \\ (p>0)$上, 则这个三角形的面积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64396,7 +64616,9 @@
"002402": {
"id": "002402",
"content": "已知$F$是抛物线$y^2=4x$的焦点, $A(3,2)$是一个定点, $P$是抛物线上的动点, 当$|PA|+|PF|$取到最小值时, 点$P$的坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64418,7 +64640,9 @@
"002403": {
"id": "002403",
"content": "过抛物线$y^2=2px(p>0)$的焦点的一条直线与抛物线相交于两个不同的点, 若两个交点的纵坐标分别为$y_1,y_2$,\n则$y_1y_2$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64442,7 +64666,9 @@
"002404": {
"id": "002404",
"content": "设过抛物线$y^2=2px(p>0)$的焦点的直线交抛物线于$A,B$两点, 若直线$AB$的倾斜角为$\\theta$,\n求$|AB|$.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64464,7 +64690,9 @@
"002405": {
"id": "002405",
"content": "抛物线$x^2=-32y$的焦点坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64495,7 +64723,9 @@
"002406": {
"id": "002406",
"content": "抛物线$y=ax^2 \\ (a\\ne 0)$的准线方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64517,7 +64747,9 @@
"002407": {
"id": "002407",
"content": "抛物线$y^2=16x$上的一点$P$到$x$轴的距离为$12$, 则$P$与焦点$F$间的距离$|PF|=$\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64539,7 +64771,9 @@
"002408": {
"id": "002408",
"content": "已知抛物线顶点在原点, 焦点在$y$轴上, 又抛物线上一点$(m,-3)$到焦点的距离为$5$, 则此抛物线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64561,7 +64795,9 @@
"002409": {
"id": "002409",
"content": "已知抛物线的顶点在原点, 对称轴与坐标轴重合, 且过点$(-2,3)$, 则抛物线的标准方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64583,7 +64819,9 @@
"002410": {
"id": "002410",
"content": "抛物线的顶点在原点, 焦点在$x$轴上, 其通径(过焦点, 且与轴垂直的弦)的两端点与顶点连成的三角形面积为$4$, 则此抛物线方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64605,7 +64843,9 @@
"002411": {
"id": "002411",
"content": "若顶点在原点, 焦点在$x$轴上的抛物线截直线$y=2x+1$所得的弦长为$\\sqrt{15}$, 则此抛物线的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64627,7 +64867,9 @@
"002412": {
"id": "002412",
"content": "若直线$y=kx-2$交抛物线$y^2=8x$于$A,B$两点, 且$AB$中点的横坐标是$2$, 则$|AB|=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64649,7 +64891,9 @@
"002413": {
"id": "002413",
"content": "已知$AB$是抛物线$y^2=4x$的焦点弦, 其坐标$A(x_1,y_1),B(x_2,y_2)$, 满足$x_1+x_2=6$, 则直线$AB$的斜率是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64734,7 +64978,9 @@
"002417": {
"id": "002417",
"content": "抛物线$y^2=-8x$被点$(-1,1)$平分的弦所在直线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64756,7 +65002,9 @@
"002418": {
"id": "002418",
"content": "抛物线$y=2x^2$的一组斜率为$2$的平行弦的中点的轨迹方程是\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64778,7 +65026,9 @@
"002419": {
"id": "002419",
"content": "已知抛物线$y^2=2x$的弦$AB$过定点$(-2,0)$, 则$AB$中点的轨迹方程是\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64800,7 +65050,9 @@
"002420": {
"id": "002420",
"content": "过抛物线焦点$F$的直线交此抛物线于$A,B$两点, 弦$AB$的垂直平分线交此曲线的对称轴于$R$. 证明: $|FR|=\\dfrac{1}{2}|AB|$.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64822,7 +65074,9 @@
"002421": {
"id": "002421",
"content": "已知过抛物线焦点$F$的直线与抛物线相交于$A,B$两点, 点$A,B$在此抛物线准线上的射影分别为$A_1,B_1$. 证明: $\\angle A_1FB_1=90^\\circ$.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64844,7 +65098,9 @@
"002422": {
"id": "002422",
"content": "已知长度固定的线段$AB$的端点$A,B$在抛物线$y={{x}^{2}}$上移动.\\\\ \n(1) 若$|AB|=2$, 求$AB$的中点$M$到$x$轴的距离的最小值;\\\\ \n(2) (选做)若$|AB|=l(l>0)$, 求$AB$的中点$M$到$x$轴的距离的最小值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64866,7 +65122,9 @@
"002423": {
"id": "002423",
"content": "抛物线$y^2=2px \\ (p>0)$的弦$PQ$的中点为$(x_0,y_0)$, 则弦$PQ$所在直线的一个法向量为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64888,7 +65146,9 @@
"002424": {
"id": "002424",
"content": "设抛物线$y=ax^2 \\ (a>0)$与直线$y=kx+b$相交于两点, 它们的横坐标为$x_1,x_2$, 而$x_3$是直线与$x$轴交点的横坐标, 那么$x_1,x_2,x_3$的关系是\\bracket{20}.\n\\twoch{$x_3=x_1+x_2$}{$x_3=\\dfrac{1}{x_1}+\\dfrac{1}{x_2}$}{$x_1x_2=x_2x_3+x_3x_1$}{$x_3=\\sqrt{x_1x_2}$}",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64910,7 +65170,9 @@
"002425": {
"id": "002425",
"content": "抛物线$y^2=x$上的点到直线$x-2y+4=0$的距离最小的点是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64932,7 +65194,9 @@
"002426": {
"id": "002426",
"content": "若点$P$在抛物线$y^2=x$上, 点$Q$在圆$(x-3)^2+y^2=1$上, 则$|PQ|$的最小值等于\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0720002X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -64954,7 +65218,9 @@
"002427": {
"id": "002427",
"content": "设$A,B$是抛物线$y^2=2px$上的点, 且满足$\\angle AOB=90^\\circ$($O$是坐标原点). 证明, 直线$AB$过定点, 并求此定点的坐标. (注: 此题可改编成``证明在抛物线的轴上存在一点$P$, 使得过$P$的弦的两端点$AB$总是满足$\\angle AOB=90^\\circ$, 其中$O$是抛物线的顶点'')",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65001,7 +65267,9 @@
"002429": {
"id": "002429",
"content": "已知正方形的一条边$AB$在直线$y=x+4$上, 顶点$C,D$在抛物线$y^2=x$上, 求此正方形的边长.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65023,7 +65291,9 @@
"002430": {
"id": "002430",
"content": "[选做]\n已知$PQ$是圆$x^2+y^2=1$中的一条垂直于$x$轴的, 不同于直径的定弦. 求证: 所有被$PQ$平分的弦所在的直线都与同一条抛物线有且仅有一个公共点.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65099,7 +65369,9 @@
"002433": {
"id": "002433",
"content": "已知双曲线$x^2-\\dfrac{y^2}{3}=1$.\\\\ \n(1) 若双曲线上存在两点关于直线$y=-\\dfrac{1}{3}x+b$对称, 求实数$b$的取值范围;\\\\ \n(2) (选做)若双曲线上存在两点关于直线$y=kx+4$对称, 求实数$k$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -65124,7 +65396,9 @@
"002434": {
"id": "002434",
"content": "已知抛物线$x^2=3y$.\\\\ \n(1) 求该抛物线过点$A(3,3)$的切线的方程;\\\\ \n(2) 求该抛物线过点$B(1,-1)$的切线的方程.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65149,7 +65423,9 @@
"002435": {
"id": "002435",
"content": "已知双曲线$\\dfrac{y^2}{4}-x^2=1$.\\\\ \n(1) 求该双曲线过点$A(\\sqrt{3},4)$的切线的方程;\\\\ \n(2) 求该双曲线过点$B(1,1)$的切线的方程.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -65174,7 +65450,10 @@
"002436": {
"id": "002436",
"content": "关于过圆锥曲线上一点的切线.\\\\ \n(1) 求过双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$上一点$P(x_0,y_0)$的切线方程;\\\\ \n(2) 求过抛物线$x^2=2py(p>0)$上一点$P(x_0,y_0)$的切线方程.",
- "objs": [],
+ "objs": [
+ "K0718001X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"双曲线",
@@ -65200,7 +65479,10 @@
"002437": {
"id": "002437",
"content": "关于``切点弦''.\\\\ \n(1) 过点$P(x_0,y_0)$引双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$的切线, 若有两条切线, 设切点分别为\n$A,B$, 求直线$AB$的方程;\\\\ \n(2) 过点$P(x_0,y_0)$引抛物线$x^2=2py(p>0)$的切线, 若有两条切线, 设切点分别为\n$A,B$, 求直线$AB$的方程.",
- "objs": [],
+ "objs": [
+ "K0718001X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"双曲线",
@@ -65226,7 +65508,10 @@
"002438": {
"id": "002438",
"content": "关于光学性质.\\\\ \n(1) 已知双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$的上下焦点分别为$F_1,F_2$,\n设$P(x_0,y_0)$在双曲线上, 直线$l$为过点$P$的双曲线的切线. 求证: $\\angle F_1PF_2$被直线$l$平分;\\\\ \n(2) 已知抛物线$x^2=2py(p>0)$的焦点为$F$,\n设$P(x_0,y_0)$在抛物线上, 直线$l$为过点$P$的抛物线的切线. 求证: 射线$FP$经过直线$l$反射后, 反射光线与$x$轴垂直.",
- "objs": [],
+ "objs": [
+ "K0718001X",
+ "K0720005X"
+ ],
"tags": [
"第七单元",
"双曲线",
@@ -65252,7 +65537,9 @@
"002439": {
"id": "002439",
"content": "焦点为$(-3,5)$, 准线为$y=7$的抛物线的方程是\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65277,7 +65564,9 @@
"002440": {
"id": "002440",
"content": "抛物线$(x+2)^2=-4(y-1)$的准线方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719005X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65307,7 +65596,10 @@
"002441": {
"id": "002441",
"content": "已知抛物线$y^2=a(x+1)$的准线方程是$x=-3$, 则其焦点坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719005X",
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65332,7 +65624,9 @@
"002442": {
"id": "002442",
"content": "双曲线$9y^2-x^2-2x-10=0$的渐近线方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -65408,7 +65702,9 @@
"002445": {
"id": "002445",
"content": "将直线$x-2y+b=0$左移一个单位, 再下移两个单位后, 它与抛物线$y^2=4x$有且仅有一个公共点, 则实数$b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65532,7 +65828,9 @@
"002450": {
"id": "002450",
"content": "顶点在$(1,2)$, 对称轴平行于坐标轴, 且过点$(4,5)$的抛物线的方程为\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0719005X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -65557,7 +65855,9 @@
"002451": {
"id": "002451",
"content": "双曲线$x^2-y^2+8x+2y+24=0$的焦点坐标是\\blank{80}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -65582,7 +65882,9 @@
"002452": {
"id": "002452",
"content": "[选做]\n求证:\\\\ \n(1) 曲线$y=\\dfrac{1}{x}$是双曲线;\\\\ \n(2) (选做)曲线$y=2x+\\dfrac{1}{x}$是双曲线.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -66037,7 +66339,9 @@
"002470": {
"id": "002470",
"content": "双曲线$\\left\\{\\begin{array}{l}x=1+\\sqrt{3}\\tan\\theta,\\\\y=1+3\\sec\\theta,\\end{array}\\right.$的两条渐近线的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -69573,7 +69877,8 @@
"content": "$(x-2y)^{10}$的展开式中, 第$4$项的二项式系数为\\blank{80}, 第$4$项的系数为\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69597,7 +69902,8 @@
"content": "$\\left(\\dfrac{1}{\\sqrt{x}}-\\sqrt[3]{x}\\right)^{20}$的展开式中, 不含$x$的项是\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69621,7 +69927,8 @@
"content": "$\\left(x+2x^{-2}\\right)^n$的展开式中, 第三项为常数, 则中间项为\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69645,7 +69952,8 @@
"content": "若$\\left(\\sqrt[3]{x}-\\dfrac{2}{x}\\right)^{n}$的展开式中, 第$8$项含$x^{\\frac{1}{3}}$,\n则含$x^{-1}$的是第\\blank{80}项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69669,7 +69977,8 @@
"content": "在$\\left(\\sqrt{2}+\\sqrt[3]{3}\\right)^{100}$的展开式中, 有\\blank{80}项是有理项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69717,7 +70026,8 @@
"content": "在$(ax+1)^7$的展开式中, 已知$x^3$的系数是$x^2$的系数与$x^4$的系数的等差中项, 且实数$a>1$, 则$a=$\\blank{30}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69743,7 +70053,8 @@
"content": "$(1+x)+(1+x)^2+(1+x)^3+\\cdots+(1+x)^{2n} \\ (n\\in \\mathbf{Z}^+)$的展开式中$x^n$项的系数为\\blank{30}(用单个组合数表示).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69769,7 +70080,8 @@
"content": "利用$(1+t)^5$的展开式化简: $(2x+1)^5-5(2x+1)^4+10(2x+1)^3-10(2x+1)^2+5(2x+1)-1=$\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69793,7 +70105,8 @@
"content": "$(x+2y+z)^9$的展开式中含$x^2y^3z^4$项的系数为\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69819,7 +70132,8 @@
"content": "[选做]\n设$n$是正整数, 将$(1+x+x^2)^n$的展开式中, $x^k$的系数记作$\\mathrm{D}_n^k$, 称为三项式系数.\n参照二项式系数杨辉三角形中的规律, 探究三项式系数的规律. 要求写出相应的三角数阵, 证明并描述所得数阵的规律.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -69843,7 +70157,8 @@
"content": "$(1+\\mathrm{i})^{10}$的展开式中, 第八项为\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69867,7 +70182,8 @@
"content": "在$(1+x)^n$的展开式中, 若第三项和第六项的系数相等, 则$n=$\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69891,7 +70207,8 @@
"content": "二项式$(x-y)^{99}$中, 系数最小的项是第\\blank{40}项, 系数最大的项是第\\blank{40}项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69915,7 +70232,8 @@
"content": "$(2-3x)^n$展开式中各项系数之和为\\blank{40}, 各项二项式系数之和为\\blank{40},\n$x$的奇次幂的项的系数之和为\\blank{40}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -69939,7 +70257,8 @@
"content": "$(2x-1)^5$的展开式中, 各项系数的绝对值之和为\\blank{80}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -70013,7 +70332,8 @@
"content": "求$(2x+3y)^{11}$的展开式中系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -70037,7 +70357,8 @@
"content": "在$(1+x)^n$的展开式中, 有连续三项的二项式系数之比为$3:8:14$, 试在展开式的所有项中, 求系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -70133,7 +70454,8 @@
"content": "$(1+2x+x^2)^{10}(1-x)^5$的展开式中, 各项系数的和为\\blank{30}, 常数项为\\blank{30},\n$x$的奇次项系数的和为\\blank{30}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -70183,7 +70505,8 @@
"content": "$\\left(1+\\sqrt{2}\\right)^{50}$的展开式中最大的项为\\blank{100}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -70255,7 +70578,8 @@
"content": "[选做]\n利用复数的三角形式的有关性质及二项式定理证明:\\\\ \n(1) $1+\\mathrm{C}_n^1\\cos\\alpha+\\mathrm{C}_n^2\\cos 2\\alpha+\\mathrm{C}_n^3\\cos 3\\alpha+\\cdots+\\mathrm{C}_n^n\\cos n\\alpha=2^n\\cos^n\\dfrac{\\alpha}{2}\\cos\\dfrac{n\\alpha}{2}$;\\\\ \n(2) $\\mathrm{C}_n^1\\sin\\alpha+\\mathrm{C}_n^2\\sin 2\\alpha+\\mathrm{C}_n^3\\sin 3\\alpha+\\cdots+\\mathrm{C}_n^n\\sin n\\alpha=2^n\\cos^n\\dfrac{\\alpha}{2}\\sin\\dfrac{n\\alpha}{2}$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -71404,7 +71728,9 @@
"002684": {
"id": "002684",
"content": "已知方程$xy=1$表示双曲线, 求它的任意一组焦点和准线.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -71426,7 +71752,9 @@
"002685": {
"id": "002685",
"content": "已知方程$x^2-2xy+y^2-8x-8y=0$表示一条抛物线, 求它的焦点和准线.",
- "objs": [],
+ "objs": [
+ "KNONE"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -71472,7 +71800,9 @@
"002687": {
"id": "002687",
"content": "在双曲线$x^2-3y^2=1$上有两个不同的点$A$与$B$.\\\\ \n(1) 若$A,B$同在右支上, 求直线$AB$倾斜角的范围;\\\\ \n(2) 若$A,B$分别在两支上, 求直线$AB$倾斜角的范围.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -71518,7 +71848,9 @@
"002689": {
"id": "002689",
"content": "设$AB$是抛物线的一条过焦点$F$的弦, $A'$及$B'$分别是$A$和$B$在准线上的射影. 证明: $\\angle A'FB'=90^\\circ$.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -71540,7 +71872,9 @@
"002690": {
"id": "002690",
"content": "设$AB$是抛物线的一条过焦点的弦.\\\\ \n(1) 证明: 以$AB$为直径的圆与该抛物线的准线相切;\\\\ \n(2) 证明: 从(1)中的圆和准线的切点$T$出发作已知抛物线的两切线, 切点恰为$A$和$B$.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -89921,7 +90255,9 @@
"003418": {
"id": "003418",
"content": "已知双曲线的中心在原点, 焦点在坐标轴上. 分别求满足下列条件的双曲线的标准方程:\\\\\n(1) 以椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的长轴顶点为焦点, 且过$P(4\\sqrt 2,3)$的双曲线方程为\\blank{50};\\\\\n(2) 点$P(\\sqrt 2,1)$在双曲线上, 且它到双曲线的右焦点的距离是$1$, 该双曲线方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -89943,7 +90279,9 @@
"003419": {
"id": "003419",
"content": "双曲线顶点间距离为$6$, 渐近线方程为$y=\\pm \\dfrac 32x$, 该双曲线方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -89965,7 +90303,9 @@
"003420": {
"id": "003420",
"content": "双曲线$4x^2+ky^2-4k=0$的虚轴长为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -89987,7 +90327,9 @@
"003421": {
"id": "003421",
"content": "双曲线$\\dfrac{x^2}4-\\dfrac{y^2}8=1$的两条渐近线所夹的锐角的大小为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90013,7 +90355,9 @@
"003422": {
"id": "003422",
"content": "已知$F_1$、$F_2$是双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}{20}=1$的焦点, 点$P$在双曲线上. 若$|PF_1|=9$, 则$|PF_2|=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90035,7 +90379,10 @@
"003423": {
"id": "003423",
"content": "直线$y=kx+2$与双曲线$x^2-y^2=6$的右支交于不同的两点, 则$k$的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0718001X",
+ "K0716004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90100,7 +90447,10 @@
"003426": {
"id": "003426",
"content": "已知双曲线$C$的一个顶点$A(0,\\sqrt 2)$, 其渐近线经过原点且与圆$M:(x-\\sqrt 2)^2+y^2=1$相切.\\\\\n(1) 求双曲线$C$的方程;\\\\\n(2) 已知直线$l:y=x-\\sqrt 2$, 在双曲线的上支求点$P$, 使点$P$与直线$l$的距离等于$\\sqrt 2$.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90122,7 +90472,9 @@
"003427": {
"id": "003427",
"content": "在双曲线$\\dfrac{y^2}{12}-\\dfrac{x^2}{13}=1$上支上有不同三点$A(x_1,y_1)$、$B(\\sqrt{26},6)$、$C(x_2,y_2)$到焦点$F(0,5)$的距离成等差数列.\\\\\n(1) 求$y_1+y_2$的值;\\\\\n(2) 证明: 线段$AC$的垂直平分线经过一个定点$T$并且求出这个点$T$的坐标.",
- "objs": [],
+ "objs": [
+ "K0715002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90169,7 +90521,9 @@
"003429": {
"id": "003429",
"content": "若双曲线$8mx^2-my^2=8$的一个焦点是$(0,3)$, 则$m=$\\blank{50};",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90191,7 +90545,11 @@
"003430": {
"id": "003430",
"content": "和双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$有共同的渐近线, 并且实轴长为$12$的双曲线方程是\\blank{50};",
- "objs": [],
+ "objs": [
+ "K0717002X",
+ "K0716003X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90213,7 +90571,9 @@
"003431": {
"id": "003431",
"content": "过$P(1,0)$作直线$l$与双曲线${x^2}-\\dfrac{y^2}4=1$只有一个公共点, 则这样的直线共有\\blank{50}条.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90235,7 +90595,10 @@
"003432": {
"id": "003432",
"content": "已知双曲线$\\dfrac{x^2}{12}-\\dfrac{y^2}4=1$的右焦点为$F$, 若过点$F$的直线$l$与双曲线的右支有且只有一个公共点, 则直线$l$的斜率的取值范围为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0718001X",
+ "K0716004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90280,7 +90643,10 @@
"003434": {
"id": "003434",
"content": "求渐近线为$3x\\pm 4y=0$, 焦点为椭圆$\\dfrac{x^2}{10}+\\dfrac{y^2}5=1$的一对顶点的双曲线方程.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90304,7 +90670,11 @@
"003435": {
"id": "003435",
"content": "已知$F_1F_2$是双曲线${x^2}-\\dfrac{y^2}{b^2}=1$($b>0$)的左、右焦点, 直线$l$过$F_2$且与双曲线交于$AB$两点.\\\\\n(1) 若$l$的倾斜角为$\\dfrac{\\pi}2$, $\\triangle F_1AB$是等边三角形, 求双曲线的渐近线方程;\\\\\n(2) 设$b=\\sqrt 3$, 若$l$的斜率存在, 且$(\\overrightarrow{F_1A}+\\overrightarrow{F_1B})\\cdot \\overrightarrow{AB}=0$, 求$l$的斜率.",
- "objs": [],
+ "objs": [
+ "K0716003X",
+ "K0717004X",
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90328,7 +90698,9 @@
"003436": {
"id": "003436",
"content": "设双曲线${x^2}-\\dfrac{y^2}4=1$的右顶点为$A$, 定点$B$的坐标为$(\\dfrac 12,1)$.\\\\\n(1) 是否存在过$B(\\dfrac 12,1)$点且被点$B$平分的双曲线的弦$PQ$, 若存在求出弦$PQ$所在直线方程, 若不存在说明理由;\\\\\n(2) 过点$B$的动直线$l$交双曲线于$P,Q$两点, $M$为线段$PQ$的中点, 求直线$AM$的斜率的取值范围.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -90350,7 +90722,9 @@
"003437": {
"id": "003437",
"content": "已知抛物线的顶点在原点, 焦点在坐标轴上. 分别求适合下列条件的抛物线的标准方程:\\\\\n(1) 过点$(-2,3)$的抛物线为\\blank{50};\\\\\n(2) 准线过点$(2,3)$的抛物线为\\blank{50};\\\\\n(3) 焦点在直线$3x-4y-12=0$上的抛物线为\\blank{50};\\\\\n(4) 焦点在$y$轴上, 抛物线上一点$M(m,-3)$到焦点的距离等于$5$的抛物线为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90372,7 +90746,9 @@
"003438": {
"id": "003438",
"content": "过点$(2,1)$与抛物线$y=x^2$恰有一个公共点的直线有\\blank{50}条.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90394,7 +90770,9 @@
"003439": {
"id": "003439",
"content": "抛物线$y=x^2$上到直线$2x-y=4$距离最短的点的坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90416,7 +90794,9 @@
"003440": {
"id": "003440",
"content": "已知点$A(3,4)$, $F$是抛物线$y^2=8x$的焦点, $M$是抛物线上的动点.当$|MA|+|MF|$最小时, $M$的坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90438,7 +90818,9 @@
"003441": {
"id": "003441",
"content": "若$AB$是抛物线$y=x^2$的一条过焦点的弦, 且$|AB|=4$, 则$AB$的中点到直线$y+1=0$的距离是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90481,7 +90863,9 @@
"003443": {
"id": "003443",
"content": "已知$F$是抛物线$C:y^2=4x$的焦点, $AB$是抛物线$C$上的两个点, 线段$AB$的中点为$M(2,2)$, 则$\\triangle ABF$的面积等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90503,7 +90887,9 @@
"003444": {
"id": "003444",
"content": "已知$A,B$是抛物线$y^2=2px(p>0)$上的两个点, $O$为坐标原点, 若$|OA|=|OB|$, 且抛物线的焦点恰为$\\triangle AOB$的垂心, 则直线$AB$的方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90546,7 +90932,9 @@
"003446": {
"id": "003446",
"content": "如图, $M$是抛物线上$y^2=x$上的一点(异于原点), 动弦$ME$、$MF$分别交$x$轴于$AB$两点, 且$MA=MB$.\\\\\n(1) 若$M$为定点, 证明: 直线$EF$的斜率为定值;\\\\\n(2) 若$M$为动点, 且$\\angle EMF={90}^{\\circ}$, 求$\\triangle EMF$的重心$G$的轨迹.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 1.5]\n \\draw [->] (-0.5,0) -- (4,0) node [below] {$x$};\n \\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$}; \n \\draw [domain = -1.8:1.8, samples = 1000] plot (\\x*\\x, \\x);\n \\draw (1.44,1.2) node [above] {$M$};\n \\draw (0.49,-0.7) node [below] {$E$} -- (1.44,1.2) -- (2.89,-1.7) node [below] {$F$} -- cycle;\n \\draw (0.84,0) node [below right] {$A$} (2.04,0) node [above right] {$B$};\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90568,7 +90956,9 @@
"003447": {
"id": "003447",
"content": "求证: 抛物线的准线上任意一点引抛物线的两切线互相垂直并且切点弦过定点.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90590,7 +90980,9 @@
"003448": {
"id": "003448",
"content": "抛物线$y=-4x^2$的焦点坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90618,7 +91010,9 @@
"003449": {
"id": "003449",
"content": "抛物线的焦点在双曲线$\\dfrac{x^2}{25}-\\dfrac{y^2}4=1$上, 则抛物线的标准方程\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90640,7 +91034,10 @@
"003450": {
"id": "003450",
"content": "设直线$a$与抛物线$\\Omega:y^2=4x$相交于不同的两点$AB$, $O$为坐标原点.\\\\\n(1) 求抛物线$\\Omega$的焦点到准线的距离;\\\\\n(2) 若$\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=0$, 点$Q$在线段$AB$上, 满足$OQ\\perp AB$, 求点$Q$的轨迹.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -90662,7 +91059,9 @@
"003451": {
"id": "003451",
"content": "如图, 已知点$P$是$y$轴左侧(不含$y$轴)一点, 抛物线$C:y^2=4x$上存在不同的两点$A$、$B$满足$PA$、$PB$的中点均在$C$上.\\\\\n(1) 设$AB$中点为$M$, 证明: $PM\\perp y$轴;\\\\\n(2) 若$P$是半椭圆${x^2}+\\dfrac{y^2}4=1x<0$上的动点, 求$\\triangle PAB$面积的取值范围.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, line cap = round, line join = round, scale = 0.7]\n \\draw [->] (-1.5,0) -- (5,0) node [below] {$x$};\n \\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n \\draw (0,0) node [below right] {$O$}; \n \\draw [domain = -4.5:4.5, samples = 1000] plot (\\x*\\x/4, \\x);\n \\draw ({11/4+sqrt(10)/2},{1+sqrt(10)}) node [above] {$A$} coordinate (A);\n \\draw ({11/4-sqrt(10)/2},{1-sqrt(10)}) node [below] {$B$} coordinate (B);\n \\draw ($(A)!0.5!0:(B)$) node [right] {$M$} coordinate (M);\n \\draw (-1,1) node [left] {$P$} coordinate (P);\n \\draw (P) -- (M) (A) -- (P) -- (B) -- cycle;\n\\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -93722,7 +94121,8 @@
"content": "$(2x-\\dfrac 1{\\sqrt x})^6$的展开式中第三项的二项式系数为\\blank{50}, 第三项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -93850,7 +94250,8 @@
"content": "求二项式$(2+x^2)^{32}$展开式中,\\\\\n(1) 二项式系数最大的项;\\\\\n(2) 系数最大的项;\\\\\n(3) 所有项的系数之和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -93958,7 +94359,8 @@
"content": "在$(\\sqrt[3]x-\\dfrac 1{2\\cdot \\sqrt[3]x})^{10}$的二项展开式中, 常数项是\\blank{50}; 含$x^2$项的二项式系数是\\blank{50}; 含$x^2$项的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -94209,7 +94611,8 @@
"content": "已知二项式$(x+a)^5$展开式中, $x^2$项的系数为$80$, 则$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -94327,7 +94730,9 @@
"003599": {
"id": "003599",
"content": "已知抛物线:$y^2=2px \\ (p>0)$, 若第一象限的$A,B$两点在抛物线上, 焦点为$F$, $|AF|=2$, $|BF|=4$, $|AB|=3$, 则直线$AB$的斜率为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -95091,7 +95496,10 @@
"003629": {
"id": "003629",
"content": "双曲线$C_1:\\dfrac{x^2}4-\\dfrac{y^2}{b^2}=1$与圆$C_2:x^2+y^2=4+b^2 \\ (b>0)$交于点$A(x_A,y_A)$(第一象限), 曲线$\\Gamma$由所有在$C_1$或$C_2$上, 且满足$|x|>x_A$的点组成, $C_2$与$x$轴的左、右交点分别记作$F_1,F_2$.\\\\\n(1) 若$x_A=\\sqrt6$, 求$b$的值;\\\\\n(2) 若$b=\\sqrt5$, 点$P$在曲线$\\Gamma$上, 且在第一象限, $|PF_1|=8$, 求$\\angle F_1PF_2$;\\\\\n(3) 点$D(0,\\dfrac{b^2}2+2)$, 过该点的直线斜率为$-\\dfrac b2$的$l$和$\\Gamma$有且只有两个交点, 记作$M,N$, 用$b$表示$\\overrightarrow{OM}\\cdot \\overrightarrow{ON}$, 并求$\\overrightarrow{OM}\\cdot \\overrightarrow{ON}$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -95218,7 +95626,8 @@
"content": "在二项式$(2x+1)^5$的展开式中, $x^2$的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -95347,7 +95756,9 @@
"003639": {
"id": "003639",
"content": "过曲线$y^2=4x$的焦点$F$并垂直于$x$轴的直线分别与曲线$y^2=4x$交于$A$、$B$, $A$在$B$的上方, $M$为抛物线上一点, $\\overrightarrow{OM}=\\lambda \\overrightarrow{OA}+(\\lambda-2) \\overrightarrow{OB}$, 则$\\lambda=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -95392,7 +95803,10 @@
"003641": {
"id": "003641",
"content": "已知数列$\\{a_n\\}$满足$a_n0)$的焦点为$F_1$、$F_2$, $P$为该双曲线上的一点, 若$|PF_1|=5$, 则$|PF_2|=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -97701,7 +98120,8 @@
"content": "若二项式$\\left(ax-\\dfrac{\\sqrt{3}}{6}\\right)^3$的展开式的第二项系数为$-\\dfrac{\\sqrt{3}}{2}$, 则实数$a$的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -97785,7 +98205,9 @@
"003739": {
"id": "003739",
"content": "点$P$在双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0, \\ b>0)$上, $F_1,F_2$是该双曲线的两个焦点, $\\angle F_1PF_2=90^\\circ$, 且$\\triangle F_1PF_2$的三条边长成等差数列, 则$a:b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -98047,7 +98469,8 @@
"content": "已知$\\left(\\sqrt{x}+\\dfrac 3{\\sqrt[3]{x}}\\right)^n$展开式中, 各项系数的和与各项二项式系数的和之差为$56$, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -98381,7 +98804,8 @@
"content": "设常数$a\\in \\mathbf{R}$, 若$\\left(x^2+\\dfrac ax\\right)^5$的二项展开式中$x^7$的系数为$-10$, 则$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -98402,7 +98826,9 @@
"003765": {
"id": "003765",
"content": "已知双曲线$C_1: \\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1 \\ (a>0,\\ b>0)$与双曲线$C_2: \\dfrac{x^2}{4}-\\dfrac{y^2}{16}=1$有相同的渐近线, 且$C_1$的右焦点为$F(\\sqrt{5},0)$, 则$a=$\\blank{50}, $b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -98766,7 +99192,9 @@
"003781": {
"id": "003781",
"content": "已知直线$l_1:4x-3y+6=0$和直线$l_2:x+1=0$, 抛物线$y^2=4x$上的动点$P$到直线$l_1$和$l_2$的距离之和的最小值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -99090,7 +99518,9 @@
"003795": {
"id": "003795",
"content": "某抛物线形拱桥的跨度为$20$米, 拱高是$4$米, 在建桥时, 每隔$4$米需用一根支柱支撑, 其中最高支柱的高度是\\blank{50}米.(答案保留两位小数)",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -100041,7 +100471,9 @@
"003837": {
"id": "003837",
"content": "若抛物线$y^2=2mx$的焦点与双曲线$\\dfrac{x^2}{2}-\\dfrac{y^2}{2}=1$的右焦点重合, 则$m=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -100109,7 +100541,8 @@
"content": "若$(1+x)^n+\\left(1+x^\\frac 12\\right)^n+\\left(1+x^\\frac 13\\right)^n+\\cdots+\\left(1+x^\\frac 1n\\right)^n \\ (n\\in \\mathbf{N}^*)$的展开式中$x$的系数是$a_n$, 展开式中所有项的系数和为$b_n$, 则$\\displaystyle\\lim_{n\\to \\infty}\\dfrac{na_n}{b_n}=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -100352,7 +100785,8 @@
"content": "在$(1+x)^n$的展开式中, 第十项是使得二项式系数最大的唯一的项, 则$n$的值是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -100717,7 +101151,8 @@
"content": "如果$\\left(\\dfrac{1}{\\sqrt{x}}-\\dfrac{\\sqrt{2}}{2}\\right)^n$展开式第三项的二项式系数为$66$, 那么展开式第六项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -101076,7 +101511,8 @@
"content": "二项式$\\left(x+\\dfrac 1{2x}\\right)^8$展开式中的二项式系数最大的项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -101121,7 +101557,9 @@
"003885": {
"id": "003885",
"content": "点$P$是双曲线$\\dfrac{x^2}{4}-y^2=1$的右支上一点, $M,N$分别是圆$(x+\\sqrt{5})^2+y^2=1$和圆$(x-\\sqrt{5})^2+y^2=1$上的点, 则$|PM|-|PN|$的最大值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -101451,7 +101889,8 @@
"content": "设$A_n$为$(1+x)^{n+1}$的展开式中含$x^{n-1}$项的系数, $B_n$为$(1+x)^{n-1}$的展开式中二项式系数的和($n\\in \\mathbf{N}^*$), 则能使$A_n\\ge B_n$成立的$n$的最大值是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -102151,7 +102590,9 @@
"003930": {
"id": "003930",
"content": "以抛物线$C:y^2=8x$上的一点$A$为圆心作圆, 若该圆经过抛物线$C$的顶点和焦点, 那么该圆的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -102493,7 +102934,9 @@
"003945": {
"id": "003945",
"content": "从抛物线$y^2=4x$上一点$P$引抛物线的垂线, 垂足为$M$, 且$|PM|=5$, 设抛物线的焦点为$F$, 则$\\triangle MPF$的面积为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -102624,7 +103067,9 @@
"003951": {
"id": "003951",
"content": "双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$的左焦点为$F_1$, 顶点为$A_1,A_2$, $P$是该双曲线右支上任意一点, 则分别以线段$PF_1,A_1A_2$为直径的两圆一定\\bracket{20}.\n\\fourch{相交}{内切}{外切}{相离}",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -102930,7 +103375,9 @@
"003964": {
"id": "003964",
"content": "$F$为双曲线$C: \\dfrac{x^2}{64}-\\dfrac{y^2}{16}=1$的左焦点, 双曲线$C$上的点$P_i$与$P_{7-i} \\ (i=1,2,3)$关于$y$轴对称, 且$P_1,P_2,P_3$在双曲线的右支上, 则$|P_1F|+|P_2F|+|P_3F|-|P_4F|-|P_5F|-|P_6F|$的值是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -103196,7 +103643,8 @@
"content": "设$\\left(3x^{\\frac 13}+x^\\frac 12\\right)^n$展开式的各项系数之和为$t$, 其二项式系数之和为$h$, 且$t+h=272$, 则展开式中$x^2$的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -103703,7 +104151,8 @@
"content": "如果$\\left(3x^2-\\dfrac{2}{x^3}\\right)^n$的展开式中含有非零常数项, 那么正整数$n$的最小值为\\bracket{20}.\n\\fourch{$10$}{$6$}{$5$}{$3$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -104224,7 +104673,8 @@
"K0819005X"
],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -104271,7 +104721,8 @@
"K0819005X"
],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -104437,7 +104888,8 @@
"K0819006X"
],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -104483,7 +104935,8 @@
"K0819005X"
],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -105314,7 +105767,9 @@
"004064": {
"id": "004064",
"content": "焦点在$y$轴上, 焦距为$6$, 且经过点$(0,\\sqrt 5)$的双曲线的标准方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -105338,7 +105793,9 @@
"004065": {
"id": "004065",
"content": "已知抛物线型拱桥的顶点距水面$2$米时, 量得水面宽为$8$米, 当水面下降$1$米后, 水面的宽为\\blank{50}米.",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -105660,7 +106117,9 @@
"id": "004078",
"content": "(1) 设椭圆$C_1:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$与双曲线$C_2:9{x^2}-\\dfrac{9y^2}8=1$有相同的焦点$F_1$、$F_2$, $M$是椭圆$C_1$与双曲线$C_2$的公共点, 且$\\triangle MF_1F_2$的周长为$6$, 求椭圆$C_1$的方程;\\\\\n我们把具有公共焦点、公共对称轴的两段圆锥曲线弧合成的封闭曲线称为``盾圆''.\\\\\n(2) 如图, 已知``盾圆$D$''的方程为$y^2=\\begin{cases}\n4x, & 0\\le x\\le 3,\\\\ -12(x-4), & 3=latex]\n \\draw [->] (-0.5,0) -- (6,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\draw [domain = {-2*sqrt(3)}:{2*sqrt(3)}, samples = 200] plot ({\\x*\\x/4},\\x);\n \\draw [domain = {-2*sqrt(3)}:{2*sqrt(3)}, samples = 200] plot ({\\x*\\x/(-12)+4},\\x);\n \\draw [dashed] (3,-4) -- (3,4);\n \\draw (3,0) node [below left] {$3$};\n \\end{tikzpicture}\n\\end{center}",
"objs": [
- "K0713003X"
+ "K0713003X",
+ "K0716002X",
+ "K0719005X"
],
"tags": [
"第七单元",
@@ -105821,7 +106280,9 @@
"004083": {
"id": "004083",
"content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{81}=1$($a>0$)的一条渐近线方程为$y=3x$, 则$a=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -106399,7 +106860,8 @@
"content": "在$(x-\\sqrt 2)^8$的二项展开式中, $x^5$项的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -107057,7 +107519,8 @@
"content": "在$(x^2+\\dfrac 2x)^7$的二项展开式中, $x^2$的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -107423,7 +107886,10 @@
"004141": {
"id": "004141",
"content": "已知常数$p>0$, 抛物线$\\Gamma:y^2=2px$的焦点为$F$.\\\\\n(1) 若直线$x=2$被$\\Gamma$截得的弦长为$4$, 求$p$的值;\\\\\n(2) 设$E$为点$F$关于原点$O$的对称点, $P$为$\\Gamma$上的动点, 求$\\dfrac{|PE|}{|PF|}$的取值范围;\\\\\n(3) 设$p=2$. 两条互相垂直的直线$l_1,l_2$均过点$F$, $l_1$与$\\Gamma$相交于$A,B$两点, $l_2$与$\\Gamma$相交于$C,D$两点. 若$AC\\perp BC$, 求四边形$ACBD$的面积.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -108206,7 +108672,9 @@
"004169": {
"id": "004169",
"content": "双曲线$\\dfrac{x^2}4-\\dfrac{y^2}9=1$的焦点到渐近线的距离等于\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -108235,7 +108703,8 @@
"content": "在二项式$(1+ax)^7(a\\in \\mathbf{R})$的展开式中, $x$的系数为$\\dfrac 73$, 则$\\displaystyle\\lim_{n\\to \\infty}(a+a^2+a^3+\\cdots +a^n)$的值是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -108304,7 +108773,9 @@
"004173": {
"id": "004173",
"content": "已知双曲线$C:\\dfrac{x^2}9-\\dfrac{y^2}8=1$, 左、右焦点分别为$F_1$、$F_2$, 过点$F_2$作一直线与双曲线$C$的右支交于$P$、$Q$两点, 使得$\\angle {F_1}PQ=90^\\circ$, 则$\\triangle {F_1}PQ$的内切圆的半径$r=$\\blank{50}",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -108697,7 +109168,10 @@
"004188": {
"id": "004188",
"content": "已知双曲线$C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的实轴与虚轴长度相等, 则$C$的渐近线方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717002X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -108796,7 +109270,8 @@
"content": "已知$(x-\\dfrac 1{2x})^n$的展开式的常数项为第$6$项, 则常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -108920,7 +109395,9 @@
"004197": {
"id": "004197",
"content": "抛物线$y=4x^2$的准线方程是\\bracket{20}.\n\\fourch{$x=-2$}{$x=-1$}{$y=-\\dfrac 18$}{$y=-\\dfrac 1{16}$}",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -109259,7 +109736,8 @@
"content": "在$(\\sqrt[3]x+\\dfrac 2x)^n$的二项展开式中, 所有项的系数之和为81, 则其常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -109506,7 +109984,9 @@
"content": "在圆锥$PO$中, 已知高$PO=2$, 底面圆的直径$AB=8$, $M$为母线$PB$的中点. 根据圆锥曲线的定义, 下列四个图中的截面边界曲线分别为圆(截面平行于底面)、椭圆(椭圆长轴为线段$AM$)、双曲线的一部分(双曲线所在平面垂直于$AB$)及抛物线的一部分(抛物线对称轴为$MO$所在直线), 下面四个命题:\\\\\n\\textcircled{1} 圆的面积为$4\\pi$; \\textcircled{2} 椭圆的长轴为$\\sqrt{37}$; \\textcircled{3} 双曲线两渐近线的夹角为$\\arcsin \\dfrac 35$; \\textcircled{4} 抛物线中焦点到准线的距离为$\\dfrac{8\\sqrt 5}5$中, 正确的个数为\\bracket{20}.\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 90:270,samples = 200,dashed] plot ({1/2*sin(\\x)},{-1/6*(cos(\\x))-sqrt(3)/2});\n \\draw [domain = 90:-90,samples = 200] plot ({1/2*sin(\\x)},{-1/6*(cos(\\x))-sqrt(3)/2});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [right] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 90:270,samples = 200,dashed] plot ({2*sin(\\x)/(3+sin(\\x))},{-1/3*2*cos(\\x)/(3+sin(\\x))-2*sqrt(3)/(3+sin(\\x))});\n \\draw [domain = 90:-90,samples = 200] plot ({2*sin(\\x)/(3+sin(\\x))},{-1/3*2*cos(\\x)/(3+sin(\\x))-2*sqrt(3)/(3+sin(\\x))});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [right] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 30:90,samples = 200,dashed] plot ({1/2},{cos(\\x)/6/sin(\\x)-sqrt(3)/2/sin(\\x)});\n \\draw [domain = 150:90,samples = 200] plot ({1/2},{cos(\\x)/6/sin(\\x)-sqrt(3)/2/sin(\\x)});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [left] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n \\begin{tikzpicture}\n \\draw (1,{-sqrt(3)}) node [right] {$B$} -- (0,0) node [above] {$P$} -- (-1,{-sqrt(3)}) node [left] {$A$};\n \\draw [domain = 0:90,samples = 200,dashed] plot ({sin(\\x)/(1+sin(\\x))},{1/3*cos(\\x)/(1+sin(\\x))-sqrt(3)/(1+sin(\\x))});\n \\draw [domain = 180:90,samples = 200] plot ({sin(\\x)/(1+sin(\\x))},{1/3*cos(\\x)/(1+sin(\\x))-sqrt(3)/(1+sin(\\x))});\n \\draw [domain = 90:270,samples = 200,dashed] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\draw [domain = 90:-90,samples = 200] plot ({sin(\\x)},{-1/3*(cos(\\x))-sqrt(3)});\n \\filldraw (0,{-sqrt(3)}) circle (0.03) node [left] {$O$} ({1/2},{-sqrt(3)/2}) node [right] {$M$};\n \\end{tikzpicture}\n\\end{center}\n\\fourch{$1$个}{$2$个}{$3$个}{$4$个}",
"objs": [
"K0713001X",
- "K0713002X"
+ "K0713002X",
+ "K0717004X",
+ "K0719003X"
],
"tags": [
"第六单元",
@@ -109619,7 +110099,9 @@
"004225": {
"id": "004225",
"content": "设抛物线$y^2=4px\\ (p>0)$的准线与$x$轴的交点为$M$, 过$M$作直线$l$交抛物线于$A$、$B$两点.\\\\\n(1) 求线段$AB$中点的轨迹方程;\\\\\n(2) 若线段$AB$的垂直平分线交对称轴于$N(x_0,0)$, 求$x_0$的取值范围;\\\\\n(3) 若直线$l$的斜率依次取$p, p^2,p^3,\\cdots ,p^n,\\cdots$时, 线段$AB$的垂直平分线与对称轴的交点依次为$N_1, N_2, N_3,\\cdots ,N_n, \\cdots$, 当$00$).\\\\\n(1) 若$\\Gamma$的一条渐近线方程为$y=2x$, 求$\\Gamma$的方程;\\\\\n(2) 设$F_1$、$F_2$是$\\Gamma$的两个焦点, $P$为$\\Gamma$上一点, 且$PF_1\\perp PF_2$, $\\triangle PF_1F_2$的面积为$9$, 求$b$的值;\\\\\n(3) 已知斜率为$2$的直线与$\\Gamma$交于$A$、$B$两点, 点$M$是线段$AB$的中点, 设点$M$的横坐标的集合为$\\Omega$. 若$\\{x|x=2n,\\ n\\in \\mathbf{N}^* \\}\\subseteq \\Omega$, 求正数$b$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0716002X",
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -112721,7 +113212,8 @@
],
"tags": [
"第四单元",
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -112939,7 +113431,9 @@
"004351": {
"id": "004351",
"content": "设抛物线$\\Gamma$的方程为$y^2=2px$, 其中常数$p>0$. $F$是抛物线$\\Gamma$的焦点.\\\\\n(1) 若直线$x=3$被抛物线$\\Gamma$所截得的弦长为$6$, 求$p$的值;\\\\\n(2) 设$A$是点$F$关于顶点$O$的对称点. $P$是抛物线$\\Gamma$上的动点, 求$\\dfrac{|PA|}{|PF|}$的最大值;\\\\\n(3) 设$p=2$, $l_1,l_2$是两条互相垂直, 且均经过点$F$的直线. $l_1$与抛物线$\\Gamma$交于点$A$、$B$, $l_2$与抛物线交于点$C$、$D$. 若点$G$满足$4\\overrightarrow{FG}=\\overrightarrow{FA}+\\overrightarrow{FB}+\\overrightarrow{FC}+\\overrightarrow{FD}$, 求点$G$的轨迹方程.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -113526,7 +114020,9 @@
"004372": {
"id": "004372",
"content": "在平面直角坐标系$xOy$中, 抛物线$\\Gamma:y^2=4x$, 点$C(1,0)$. $A,B$为$\\Gamma$上的两点, $A$在第一象限, 满足$\\overrightarrow{OA}\\cdot \\overrightarrow{OB}=-4$.\\\\\n(1) 求证: 直线$AB$过定点, 并求定点坐标;\\\\\n(2) 设$P$为$\\Gamma$上的动点, 求$\\dfrac{|OP|}{|CP|}$的取值范围;\\\\\n(3) 记$\\triangle AOB$的面积为$S_1$, $\\triangle BOC$的面积为$S_2$, 求$S_1+S_2$的最小值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -114099,7 +114595,8 @@
"content": "已知$(x^2+\\dfrac 2x)^n$的二项展开式中, 所有二项式系数的和为$512$, 则展开式中的常数项为\\blank{50}(结果用数值表示).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -115064,7 +115561,8 @@
"content": "在$(x-a)^{10}$的展开式中, $x^7$的系数是$15$, 则实数$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -115611,7 +116109,8 @@
"content": "在$(x+\\dfrac 1x)^{10}$的展开式中, 常数项等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$252$",
@@ -116433,7 +116932,8 @@
"content": "设函数$f(x)=\\begin{cases} x^6, & x\\ge 1, \\\\ -2x-1, & x\\le -1, \\end{cases}$ 则当$x\\le -1$时, 则$f[f(x)]$表达式的展开式中含$x^2$项的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -117078,7 +117578,9 @@
"004495": {
"id": "004495",
"content": "在平面直角坐标系$xOy$中, 已知抛物线$y^2=4x$上一点$P$到焦点的距离为$3$, 则点$P$的横坐标是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -117587,7 +118089,9 @@
"004514": {
"id": "004514",
"content": "抛物线$x^2=-4y$的准线方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -117670,7 +118174,8 @@
"content": "在$(x+\\sqrt 2)^7$的二项展开式中任取一项, 则该项系数为有理数的概率为\\blank{50}(用数字作答).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -117841,7 +118346,8 @@
"id": "004524",
"content": "以抛物线$y^2=4x$的焦点为右焦点, 且长轴为$4$的椭圆的标准方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}+\\dfrac{y^2}{15}=1$}{$\\dfrac{x^2}{16}+\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4+\\dfrac{y^2}3=1$}{$\\dfrac{x^2}4+{y^2}=1$}",
"objs": [
- "K0713003X"
+ "K0713003X",
+ "K0719003X"
],
"tags": [
"第七单元",
@@ -118087,7 +118593,9 @@
"004534": {
"id": "004534",
"content": "双曲线$\\dfrac{x^2}4-\\dfrac{y^2}9=1$的两渐近线的夹角的大小为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -118142,7 +118650,8 @@
"content": "已知$(1+ax)^6$的展开式中, 含有$x^3$项的系数为$160$, 则实数$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -118506,7 +119015,9 @@
"004550": {
"id": "004550",
"content": "已知圆$C$过定点$A(0,1)$, 圆心$C$在抛物线$x^2=2y$上, $M,N$为圆$C$与$x$轴的交点.\\\\\n(1) 当圆心$C$是抛物线的顶点时, 求抛物线的准线被该圆截得的弦长;\\\\\n(2) 当圆心$C$在抛物线上运动时, $|MN|$是否为一定值? 证明你的结论;\\\\\n(3) 当圆心$C$在抛物线上运动时, 记$|AM|=m$, $|AN|=n$, 求$\\dfrac mn+\\dfrac nm$的最大值, 并求出此时圆$C$的方程.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -118746,7 +119257,8 @@
"content": "在$(x+\\dfrac{1}{\\sqrt{x}})^6$的二项展开式中, 常数项的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -119046,7 +119558,10 @@
"004570": {
"id": "004570",
"content": "已知抛物线$y^2=4x$, $F$为焦点, $P$为准线$l$上一动点, 线段$PF$与抛物线交于点$Q$, 定义$d(P)=\\dfrac{|FP|}{|FQ|}$.\\\\\n(1) 若点$P$坐标为$(-1,-\\dfrac 83)$, 求$d(P)$;\\\\\n(2) 求证: 存在常数$a$, 使得$2d(P)=|FP|+a$恒成立;\\\\\n(3) 设$P_1,P_2,P_3$为准线$l$上的三点, 且$|P_1P_2|=|P_2P_3|$, 试比较$d(P_1)+d(P_3)$与$2d(P_2)$的大小.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0719002X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -120427,7 +120942,8 @@
"content": "$(2+x)^6$的二项展开式中, 系数最大的项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -120449,7 +120965,9 @@
"004626": {
"id": "004626",
"content": "设$A,B$是一条斜率为$4$的直线与抛物线$y^2=x$的两个交点, 则线段$AB$的中点的坐标可能是\\blank{50}(写出一个可能的点的坐标).",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -120805,7 +121323,9 @@
"004639": {
"id": "004639",
"content": "已知抛物线$C$的方程为$y^2=x$, 圆$M$的方程为$(x-2)^2+y^2=1$.\\\\\n(1) 设$P$是抛物线$C$上的动点, 证明: $P$在圆$M$外;\\\\\n(2) 设斜率为$1$的直线$l$与圆$M$相切, 且与抛物线$C$交于$Q_1,Q_2$两点, 求$|Q_1Q_2|$的值;\\\\\n(3) 设$A_1,A_2,A_3$是抛物线$C$上的三点, 直线$A_1A_2$, 直线$A_1A_3$均与圆$M$相切, 判断直线$A_2A_3$与圆$M$的位置关系, 说明理由.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -121216,7 +121736,9 @@
"004654": {
"id": "004654",
"content": "已知点$O$是坐标原点, 点$A(0,2)$点$P$是抛物线$y=4x^2$上的点, 则使得$OPA$是等腰三角形的点$P$为\\bracket{20}.\n\\fourch{$2$}{$4$}{$6$}{$8$}",
- "objs": [],
+ "objs": [
+ "K0719001X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -121347,7 +121869,10 @@
"004659": {
"id": "004659",
"content": "已知双曲线$\\Gamma:\\dfrac{x^2}2-\\dfrac{y^2}4=1$的右顶点为$A$, 点$B$的坐标为$(1,\\sqrt 2)$.\\\\\n(1) 设双曲线$\\Gamma$的两条渐近线的夹角为$\\theta$, 求$\\cos\\theta$;\\\\\n(2) 设点$D$是双曲线$\\Gamma$上的动点, 若点$N$满足$\\overrightarrow{BN}=\\overrightarrow{ND}$, 求点$N$的轨迹方程;\\\\\n(3) 过点$B$的动直线$l$交双曲线$\\Gamma$于$PQ$两个不同的点, $M$为线段$PQ$的中点, 求直线$AM$的斜率的取值范围.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -121844,7 +122369,8 @@
"content": "已知代数式$(\\dfrac 2m+\\dfrac mx)^n$($m>0$, $x>0$).\\\\\n(1) 当$m=2$, $n=6$时, 求二项展开式中二项式系数最大的项;\\\\\n(2) 若$(\\dfrac 2m+\\dfrac mx)^{10}=a_0+\\dfrac{a_1}x+\\dfrac{a_2}{x^2}+\\cdots +\\dfrac{a_{10}}{x^{10}}$, 且$a_2=180$, 求$a_i$($0\\le i \\le 10$, $i\\in \\mathbf{N}$)的最大值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -122022,7 +122548,9 @@
"004685": {
"id": "004685",
"content": "若双曲线$x^2-\\dfrac{y^2}m=1$的渐近线方程为$y=\\pm 2x$, 则实数$m=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -122048,7 +122576,8 @@
"content": "在$(1+2x)^6$的二项展开式中, $x^2$项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -122701,7 +123230,8 @@
"content": "若$(x+\\dfrac 1{\\sqrt{x}})^n$的二项展开式中各项系数的和等于$64$, 则其中$x^3$的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -122747,7 +123277,9 @@
"004713": {
"id": "004713",
"content": "已知抛物线$y^2=4x$, 斜率为$k$的直线$l$经过抛物线的焦点$F$, 与抛物线交于$P$、$Q$两点, 点$Q$关于$x$轴的对称点为$Q'$, 点$P$关于直线$x=1$的对称点为$P'$, 且满足$P'Q'\\perp PQ$, 则直线$l$的方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -123106,7 +123638,8 @@
"content": "$(1+2x)^{10}$ 的二项展开式中, $x^2$ 项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -123578,7 +124111,9 @@
"004746": {
"id": "004746",
"content": "若双曲线方程为${x^2}-\\dfrac{y^2}{16}=1$, 则该双曲线的渐近线方程为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -123604,7 +124139,8 @@
"content": "在$(1+2x)^6$的二项展开式中, $x^5$项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -124054,7 +124590,9 @@
"004764": {
"id": "004764",
"content": "已知抛物线$y^2=4x$的焦点为$F$, 直线$l$交抛物线于不同的$A$、$B$两点.\\\\\n(1) 若直线$l$的方程为$y=x-1$, 求线段$AB$的长;\\\\\n(2) 若直线$l$经过点$P(-1,0)$, 点$A$关于$x$轴的对称点为$A'$, 求证: $A'$、$F$、$B$三点共线;\\\\\n(3) 若直线$l$经过点$M(8,-4)$, 抛物线上是否存在定点$N$, 使得以线段$AB$为直径的圆恒过点$N$? 若存在, 求出点$N$的坐标, 若不存在, 请说明理由.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -187532,7 +188070,8 @@
"content": "求二项式$(2x-\\dfrac 3{2x^2})^7$展开式的第四项的二项式系数和笫四项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187553,7 +188092,8 @@
"content": "求$(1+x)+(1+x)^2+(1+x)^3+\\cdots +(1+x)^{2n}$($n\\in \\mathbf{N}$)的展开式中念$x^n$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187576,7 +188116,8 @@
"content": "在$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^{100}$的展开式中, 有多少项是有理项?",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187597,7 +188138,8 @@
"content": "求$(x^2+\\dfrac 1{x^2}-2)^3$展开式中含$x^2$项的表达式.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187620,7 +188162,8 @@
"content": "求$(1+x+x^2)(1-x)^{10}$展开式中含$x^4$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187643,7 +188186,8 @@
"content": "求$(ax+by+cz)^n$的展开式中含$x^py^qz^r$项的系数, 其中$p+q+r=n$($p,q,r,n\\in \\mathbf{N}$).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187664,7 +188208,8 @@
"content": "求$(x+\\dfrac 1x-1)^5$展开式中的常数项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -187878,7 +188423,8 @@
"content": "在$(a-b)^n$($n\\in \\mathbf{N}$)的展开式中, 笫$r$项的二项式系数为\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^r$}{$\\mathrm{C}_n^{r-1}$}{$(-1)^r\\mathrm{C}_n^r$}{$(-1)^{r-1}\\mathrm{C}_n^{r-1}$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -187899,7 +188445,8 @@
"content": "$(\\sqrt 3i-x)^{10}$展开式的第$8$项是\\bracket{20}.\n\\fourch{$-360\\sqrt 3x^7i$}{$-135x^3$}{$360\\sqrt 3x^7i$}{$3240\\sqrt 3x^3i$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -187920,7 +188467,8 @@
"content": "$(\\dfrac 1{\\sqrt 3}-\\sqrt[3]x)^{20}$的展开式中, 不含$x$的项是\\bracket{20}.\n\\fourch{第$11$项}{第$12$项}{第$13$项}{第$7$项或第$13$项}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -187941,7 +188489,8 @@
"content": "若二项式$(\\sqrt[3]x-\\dfrac 2x)^n$展开式中第$8$项是含$\\sqrt[3]x$的项, 则自然数$n$的值等于\\bracket{20}.\n\\fourch{$27$}{$28$}{$29$}{$30$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -187962,7 +188511,8 @@
"content": "在$(1+x)^n$的二项展开式中, 若第$9$项的系数与第$13$项的系数相等, 则第$20$项的系数等于\\bracket{20}.\n\\fourch{$19$}{$20$}{$21$}{$22$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -187983,7 +188533,8 @@
"content": "若$(1+x)^8$展开式的中间三项依次成等差数列, 则$x$的值等于\\bracket{20}.\n\\fourch{$\\dfrac 12$或$2$}{$\\dfrac 12$或$4$}{$2$或$4$}{$2$或$\\dfrac 14$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188004,7 +188555,8 @@
"content": "在$(x-1)^9$按$x$降幂排列的展开式中, 系数最大的项是\\bracket{20}.\n\\fourch{第$4$项和第$5$项}{第$5$项}{第$5$项和第$6$项}{第$6$项}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188025,7 +188577,8 @@
"content": "在$(x+\\dfrac 2{x^2})^n$的展开式中, 第3项为常数, 则中间项的表达式为\\bracket{20}.\n\\fourch{$60$}{$160x^{-3}$}{$672$}{$960x^{-3}$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188067,7 +188620,8 @@
"content": "在$(x+y)^n$的展开式中, 若第$7$项的系数最大, 则$n$等于\\bracket{20}.\n\\fourch{$11, 12, 13$}{$13, 14$}{$11, 15$}{$12, 13$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188088,7 +188642,8 @@
"content": "在$(x-\\dfrac 1x)^9$的展开式中, $x^3$的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188112,7 +188667,8 @@
"content": "在$(ax+1)^7$的展开式中, 若$x^3$的系数是$x^2$的系数与$x^4$的系数的等差中项, 且$a>1$, 则$a$的值等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188135,7 +188691,8 @@
"content": "在$(x+1+\\mathrm{i})^{10}$的展开式中, $x^6$的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188156,7 +188713,8 @@
"content": "若$a>0$, $n\\in \\mathbf{N}$, 且$(ax+1)^{2n}$和$(x+a)^{2n+1}$展开式的$x^n$的系数相等, 则$a$的収值范围是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188177,7 +188735,8 @@
"content": "$(\\sqrt x+\\sqrt [3]{x^2})^{12}$的展开式的第5项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188198,7 +188757,8 @@
"content": "若二项式$(z-2)^6$展开式中的第$5$项是$-480$, 则复数$z$的值是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188219,7 +188779,8 @@
"content": "若$(x+\\dfrac 1x)^n$展开式中的第$3$项和第$7$项系数相等, 则系数的最大项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188240,7 +188801,8 @@
"content": "在$(\\sqrt[3]a-\\dfrac 1{\\sqrt a})^{15}$的展开式中, 不含$a$的项是第\\blank{50}项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188261,7 +188823,8 @@
"content": "$(\\dfrac{\\sqrt x}3+\\dfrac 3{\\sqrt x})^{12}$展开式的中间一项等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188282,7 +188845,8 @@
"content": "$(2x^2+\\dfrac 1x)^{12}$展开式的常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188303,7 +188867,8 @@
"content": "若$(\\dfrac 1{x\\sqrt[3]x}+x)^n$展开式中第$5, 6, 7$项的系数成等差数列, 则展开式中不含$x$的项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188324,7 +188889,8 @@
"content": "在$(\\sqrt[3]2+\\sqrt 3)^{12}$的展开式中, 有理项是第\\blank{50}项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188345,7 +188911,8 @@
"content": "在$(1-3x)^{12}$的展开式中, 各项的二项式系数之和为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188366,7 +188933,8 @@
"content": "在$(1-x)^9$的展开式中, $x$的奇次项系数之和等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188432,7 +189000,8 @@
"content": "在$(2x-1)^5$的展开式中, 各项系数的绝对值之和等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188457,7 +189026,8 @@
"content": "在$(x+2y)(2x+y)^2(x+y)^3$的展开式中, 各项系数的和是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -188562,7 +189132,8 @@
"content": "若$(2x^2-\\dfrac 1{\\sqrt[3]x})^n$的展开式中含有非零常数项, 则正整数$n$的最小值是\\bracket{20}.\n\\fourch{$8$}{$6$}{$5$}{$4$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188583,7 +189154,8 @@
"content": "在$(\\sqrt[5]3+\\sqrt[7]5)^{24}$的展开式中, 整数项是\\bracket{20}.\n\\fourch{第$12$项}{第$13$项}{第$14$项}{第$15$项}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188604,7 +189176,8 @@
"content": "在$(\\sqrt 3x+\\sqrt[3]2)^{100}$的展开式中, $x$的系数为有理数的项共有\\bracket{20}.\n\\fourch{$15$项}{$16$项}{$17$项}{$18$项}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188625,7 +189198,8 @@
"content": "在$(1-x)^n(1+x)^n$的展开式中, 若含$x^4$项的系数是$10$, 则自然数$n$的值等于\\bracket{20}.\n\\fourch{$3$}{$4$}{$5$}{$6$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188646,7 +189220,8 @@
"content": "在二项式$(1+x)^n$的展开式中, 若相邻两项的系数之比为$8:15$, 则$n$的最小值是\\bracket{20}.\n\\fourch{$21$}{$22$}{$23$}{$24$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188688,7 +189263,8 @@
"content": "在$(2-3x)^n$的展开式中, 各项系数之和是\\bracket{20}.\n\\twoch{$1$}{$n$为偶数时是$2$, $n$为奇数时是$-2$}{$-1$}{$n$为偶数时是$1$, $n$为奇数时是$-1$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188709,7 +189285,8 @@
"content": "在$(1+x)^3+(1+x)^4+\\cdots +(1+x)^{n+2}$的展开式中, 含$x^2$项的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_{n+3}^3$}{$\\mathrm{C}_{n+3}^3-1$}{$\\mathrm{C}_{n+2}^3-1$}{$\\mathrm{C}_{n+2}^3$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188730,7 +189307,8 @@
"content": "$(a+b+c)^{10}$展开式的项数共有\\bracket{20}.\n\\fourch{$11$项}{$66$项}{$121$项}{$132$项}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188772,7 +189350,8 @@
"content": "在$(1+x_1)(1+x_2)^2\\cdots (1+x_{n-1})^{n-1}(1+x_n)^n$展开式中, 各项系数之和是\\bracket{20}.\n\\fourch{$2^{n(n+1)}$}{$2^{\\frac{n(n+1)}2}$}{$2^{n+1}+2$}{$2(2^n-1)$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -188814,7 +189393,8 @@
"content": "求$(x^2+\\dfrac 4{x^2}-4)^5$展开式中含$x^4$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188837,7 +189417,8 @@
"content": "求$(x^2+3x+2)^5$展开式中含$x$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188861,7 +189442,8 @@
"content": "求$(1-x)^5(1+x+x^2)^4$展开式中含$x^7$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188882,7 +189464,8 @@
"content": "求$(x-2)^4(1+x)^5$展开式中含$x^6$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188906,7 +189489,8 @@
"content": "求$(x^2+x-2)^4$展开式中含$x^2$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188930,7 +189514,8 @@
"content": "求$(2\\sqrt x-\\dfrac 1{\\sqrt x})^6$展开式中, $x$的一次幂的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188951,7 +189536,8 @@
"content": "求$(x+y-3z)^8$的展开式中含$x^5yz^2$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188972,7 +189558,8 @@
"content": "求$(x+2y+z)^9$展开式中含$x^2y^3z^4$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -188995,7 +189582,8 @@
"content": "求$(1-2x)^5(2+x)$展开式中含$x^3$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189016,7 +189604,8 @@
"content": "求$(1+x+x^2)(1-x)^{10}$展开式中含$x^4$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189039,7 +189628,8 @@
"content": "求$(1+x)^{2n}+x(1+x)^{2n-1}+x^2(1+x)^{2n-2}+\\cdots +x^n\\cdot (1+x)^n$展开式中含$x^n$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189060,7 +189650,8 @@
"content": "求$(x-1)-(x-1)^2+(x-1)^3-(x-1)^4+(x-1)^5$的展开式中含$x^2$项的系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189081,7 +189672,8 @@
"content": "若$(x+x^{\\lg x})^5$的展开式的第$4$项为$10^6$, 求$x$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189102,7 +189694,8 @@
"content": "若$x(1-x)^4+x^2(1+2x)^k+x^3(1+3x)^{12}$的展开式中$x^4$的系数是$144$, 求$k$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189123,7 +189716,8 @@
"content": "若$(x^{\\lg x}+1)^n$展开式中最后$3$项的二项式系数的和是$22$, 而它的中间项是$20000$, 求$x$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189144,7 +189738,8 @@
"content": "已知$(x\\sin \\alpha +1)^6$的展开式中$x^2$项的系数与$(x-\\dfrac{15}2\\cos \\alpha)^4$的展开式中$x^3$项的系数相等, 求$\\alpha$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189165,7 +189760,8 @@
"content": "已知$(a+b)^n$展开式的末$3$项系数之和为$22$, 又$(x^{\\lg x}-3)^n$展开式的中间项等于$-540000$, 求$x$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189186,7 +189782,8 @@
"content": "求$(|x|+\\dfrac 1{|x|}-2)^3$展开式中的常数项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189209,7 +189806,8 @@
"content": "求$[(1+\\log _3x)(1+\\log _x3)]^n$的展开式中不含$x$的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189230,7 +189828,8 @@
"content": "已知$(\\sqrt x+\\dfrac 2{x^2})^n$展开式中的第$5$项系数与第$3$项系数之比是$56:3$, 求展开式中不含$x$的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189251,7 +189850,8 @@
"content": "已知$(\\sqrt x+\\dfrac 1{2\\cdot \\sqrt[4]x})^n$展开式中前$3$项的系数依次成等差数列, 求展开式中所有的有理项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189272,7 +189872,8 @@
"content": "已知$(x\\cdot \\sqrt x-\\dfrac 1x)^6$展开式的第$5$项等于$\\dfrac{15}2$, 求$\\displaystyle\\lim_{n\\to\\infty} (x^{-1}+x^{-2}+\\cdots +x^{-n})$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189295,7 +189896,8 @@
"content": "已知多项式$f(x)=(1+x)^m+(1+x)^n$($m\\in \\mathbf{N}$, $n\\in \\mathbf{N}$)的展开式中$x$项的系数为$19$.\\\\\n(1) 求$f(x)$中含$x^2$项的系数的最小值;\\\\\n(2) 对于使$f(x)$的$x^2$项的系数取最小值时的$m$, $n$, 求$f(x)$中含$x^7$的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189358,7 +189960,8 @@
"content": "求多项式$(x^2+x-1)^9(2x+1)^4$展开式中$x$的奇次项系数之和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189379,7 +189982,8 @@
"content": "求多项式$(x^2+2x+2)^{1993}+(x^2-3x-3)^{1993}$展开式中$x$的偶次项系数之和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189400,7 +190004,8 @@
"content": "求$(2-5x+2x^2)^5(2-x)^7$展开后各项系数的和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189421,7 +190026,8 @@
"content": "求$(x^3+2x+1)(5x^2+4)$展开后各项系数的和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189442,7 +190048,8 @@
"content": "已知$(1+x)^n$展开式中奇数项之和为$A$, 偶数项之和为$B$, 试证: $A^2-B^2=(1-x^2)^n$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189463,7 +190070,8 @@
"content": "若$(a+b)^n$展开式的所有奇数项的二项式系数之和为$1024$, 则展开式中间项的系数是\\bracket{20}.\n\\fourch{$330$}{$462$}{$682$}{$792$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -189484,7 +190092,8 @@
"content": "在$(x-\\dfrac 1x)^n$的展开式中, 若奇数项的系数之和为32, 则含$x^2$项的系数是\\bracket{20}.\n\\fourch{$-20$}{$-15$}{$15$}{$20$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -189526,7 +190135,8 @@
"content": "记$(1+2x)^n$展开式中各项系数和为$a_n$, 其二项式系数和为$b_n$, 则$\\displaystyle\\lim_{n\\to\\infty}\\dfrac{b_n-a_n}{b_n+a_n}$为\\bracket{20}.\n\\fourch{$1$}{$0$}{$-1$}{不存在}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -189568,7 +190178,8 @@
"content": "在$(x-1)^{11}$的展开式中, $x$的偶次幂项的系数和为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -189675,7 +190286,8 @@
"content": "在$(x^2+x-1)^{100}+(x^2-x-1)^{100}$的展开式中, $x$的偶次项系数之和为\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$8$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -189759,7 +190371,8 @@
"content": "已知$x$为实数, $i$为虚数单位, 则$(1+ix)^{50}$展开式中实系数项的系数和为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -189780,7 +190393,8 @@
"content": "设$a$是$\\sqrt 2$的整数部分, $b$是$\\sqrt 2$的小数部分, 则$(a-\\dfrac 1b)^6$展开式的中间项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -189801,7 +190415,8 @@
"content": "设$(2x+x^{\\lg x})^n$展开式各项的二项式系数之和为$256$, 且二项式系数最大项的值为$1120$, 求$x$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -189822,7 +190437,8 @@
"content": "已知$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^n$展开式系数之和比$(a+b)^{2n}$展开式的系数之和小240, 求$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^n$展开式中系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -190118,7 +190734,8 @@
"content": "求$(1-x)+(1-x)^2+(1-x)^3+\\cdots +(1-x)^n$展开式中所有奇次项系数的和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -190358,7 +190975,8 @@
"content": "已知$\\mathrm{C}_{18}^n=\\mathrm{C}_{18}^{n+2}$, $4\\mathrm{P}_m^2=\\mathrm{P}_{m+1}^4$, 求$(1+\\sqrt m\\mathrm{i})^n$展开式中所有实数项的和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -190757,7 +191375,8 @@
"content": "求$(\\sqrt x+2)^{2n+1}$的展开式中$x$的整数次幂的各项系数之和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -190778,7 +191397,8 @@
"content": "求$(1+i)^{4k-2}$($k\\in \\mathbf{N}$)展开式中奇数项之和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -219628,7 +220248,9 @@
"008900": {
"id": "008900",
"content": "写出分别满足下列条件的双曲线的标准方程.\\\\\n(1) 曲线上的点$P$到点$F_1(4,0)$的距离与它到点$F_2(-4,0)$的距离的差的绝对值等于$6$;\\\\\n(2) 曲线上的点$P$到点$F_1(-10,0)$的距离与它到点$F_2(10,0)$的距离的差等于$16$;\\\\\n(3) 焦点在$x$轴上, 且双曲线经过点$(-\\sqrt 2,-\\sqrt 3)$、$(\\dfrac{\\sqrt {15}}3,\\sqrt 2)$.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219650,7 +220272,9 @@
"008901": {
"id": "008901",
"content": "设方程$\\dfrac{x^2}{m+2}-\\dfrac{y^2}{m+1}=1$表示焦点在$y$轴上的双曲线, 求实数$m$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219676,7 +220300,9 @@
"008902": {
"id": "008902",
"content": "已知双曲线的对称轴为坐标轴, 焦点为$(-6,0)$、$(6,0)$, 且双曲线经过点$(-5,2)$, 求此双曲线的标准方程.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219698,7 +220324,9 @@
"008903": {
"id": "008903",
"content": "过双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$的右焦点$F_2$作$x$轴的垂线, 求此垂线与双曲线的交点$M$到左焦点$F_1$的距离.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219720,7 +220348,9 @@
"008904": {
"id": "008904",
"content": "已知双曲线关于原点对称, 它的焦点在坐标轴上, 焦距为$10$, 且此双曲线经过点$(3,4\\sqrt 2)$, 求它的标准方程.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219742,7 +220372,9 @@
"008905": {
"id": "008905",
"content": "已知双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$的左、右焦点分别为$F_1,F_2$, 直线$l$过点$F_1$, 交双曲线的左支于$A,B$两点, 且$|AB|=m$, 求$\\triangle ABF_2$的周长.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219764,7 +220396,10 @@
"008906": {
"id": "008906",
"content": "如果中心在原点, 对称轴在坐标轴上的等轴双曲线的一个焦点为$F_1(0,-6)$, 那么此双曲线的标准方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716003X",
+ "K0717002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219786,7 +220421,10 @@
"008907": {
"id": "008907",
"content": "双曲线$2x^2-y^2=8$的焦点坐标是\\blank{50}, 两条渐近线的夹角为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219808,7 +220446,9 @@
"008908": {
"id": "008908",
"content": "若双曲线的中心在坐标原点, 它的一个焦点的坐标是$(-5,0)$, 两个顶点间的距离为$6$, 则此双曲线的方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{36}-\\dfrac{y^2}{11}=1$}{$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$}{$\\dfrac{x^2}{11}-\\dfrac{y^2}{36}=1$}",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219830,7 +220470,9 @@
"008909": {
"id": "008909",
"content": "在下列双曲线中, 以$y=\\pm \\dfrac 12x$为渐近线的是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}2-y^2=1$}{$x^2-\\dfrac{y^2}2=1$}",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219854,7 +220496,9 @@
"008910": {
"id": "008910",
"content": "若方程$4x^2+ky^2=4k$表示双曲线, 则此双曲线的虚轴长等于\\bracket{20}.\n\\fourch{$2\\sqrt k$}{$2\\sqrt {-k}$}{$\\sqrt k$}{$\\sqrt {-k}$}",
- "objs": [],
+ "objs": [
+ "K0717002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219897,7 +220541,10 @@
"008912": {
"id": "008912",
"content": "已知双曲线的虚轴的长为$6$, 一条渐近线的方程为$3x-y=0$, 求此双曲线的标准方程.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219919,7 +220566,10 @@
"008913": {
"id": "008913",
"content": "求与双曲线$x^2-\\dfrac{y^2}4=1$有共同渐近线, 且过点$M(2,2)$的双曲线的标准方程.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219941,7 +220591,10 @@
"008914": {
"id": "008914",
"content": "已知双曲线$\\dfrac{x^2}8-\\dfrac{y^2}{b^2}=1$的右焦点为点$F$, 若直线$x-y-3=0$经过点$F$, 求此双曲线渐近线的方程.",
- "objs": [],
+ "objs": [
+ "K0716003X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219963,7 +220616,9 @@
"008915": {
"id": "008915",
"content": "已知双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$的两个焦点分别为$F_1,F_2$, 点$P$为此双曲线上一点, $|PF_1|\\cdot|PF_2|=32$, 求证: $PF_1\\perp PF_2$.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -219989,7 +220644,8 @@
"content": "求以椭圆$\\dfrac{x^2}8+\\dfrac{y^2}5=1$的焦点为顶点, 以椭圆的顶点为焦点的双曲线的方程.",
"objs": [
"K0713002X",
- "K0714003X"
+ "K0714003X",
+ "K0716003X"
],
"tags": [
"第七单元",
@@ -220056,7 +220712,9 @@
"008919": {
"id": "008919",
"content": "已知直线$l$: $y=ax+1$与双曲线$C$: $3x^2-y^2=1$相交于$AB$两点.\\\\\n(1) 求实数$a$的取值范围;\\\\\n(2) 求当实数$a$为何值时, 以线段$AB$为直径的圆经过坐标原点.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -220078,7 +220736,9 @@
"008920": {
"id": "008920",
"content": "写出分别满足下列条件的抛物线的标准方程.\\\\\n(1) 焦点是$F(1,0)$;\\\\\n(2) 准线方程是$x=-2$.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220102,7 +220762,10 @@
"008921": {
"id": "008921",
"content": "在抛物线$y^2=20x$上求一点$P$, 使点$P$与焦点的距离等于$15$.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0719002X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220124,7 +220787,9 @@
"008922": {
"id": "008922",
"content": "求抛物线$y^2=x$的一组斜率为$2$的平行弦的中点的轨迹方程.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220146,7 +220811,9 @@
"008923": {
"id": "008923",
"content": "抛物线$y^2=2x$上的$AB$两点到焦点$F$的距离之和是$5$, 求线段$AB$的中点的横坐标.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220170,7 +220837,9 @@
"008924": {
"id": "008924",
"content": "求抛物线$x=ay^2(a>0)$的焦点坐标与准线方程.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220194,7 +220863,9 @@
"008925": {
"id": "008925",
"content": "过抛物线$y^2=2px(p>0)$的焦点的一条直线与抛物线相交于两个不同的点, 两个交点的纵坐标分别为$y_1,y_2$, 求证: $y_1y_2=-p^2$.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220218,7 +220889,9 @@
"008926": {
"id": "008926",
"content": "抛物线$x^2=-32y$的焦点坐标是\\blank{50}, 准线方程是\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220242,7 +220915,9 @@
"008927": {
"id": "008927",
"content": "已知抛物线的顶点在原点, 对称轴为$x$轴, 且过点$(-2,3)$, 求此抛物线的标准方程.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220264,7 +220939,10 @@
"008928": {
"id": "008928",
"content": "已知抛物线$y^2=8x$的焦点为$F$, $P$在此抛物线上, 且$|PF|=5$, 求点$P$的坐标.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0719002X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220286,7 +220964,9 @@
"008929": {
"id": "008929",
"content": "已知一隧道的顶部是抛物拱形, 拱高是$1$米, 跨度为$2$米, 建立适当的直角坐标系, 求相应坐标系下此拱形的抛物线方程.",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220308,7 +220988,9 @@
"008930": {
"id": "008930",
"content": "已知直线$l$: $y=kx-4$与抛物线$y^2=8x$有且只有一个公共点, 求实数$k$的值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220334,7 +221016,9 @@
"008931": {
"id": "008931",
"content": "已知正三角形$ABC$的顶点$A$位于坐标原点, 顶点$B$与$C$均在抛物线$y^2=2x$上, 求$\\triangle ABC$的边长.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220356,7 +221040,9 @@
"008932": {
"id": "008932",
"content": "已知直线$l$垂直于$x$轴, 且交抛物线$y^2=4x$于点$AB$, 且$|AB|=4\\sqrt 3$, 求直线$AB$的方程.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220378,7 +221064,9 @@
"008933": {
"id": "008933",
"content": "在抛物线$x^2=\\dfrac 14y$上求一点$M$, 使点$M$到直线$y=4x-5$的距离最短.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220400,7 +221088,9 @@
"008934": {
"id": "008934",
"content": "过点$Q(4,1)$作抛物线$y^2=8x$的弦$AB$, $AB$恰好被点$Q$平分, 求$AB$所在直线的方程.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220422,7 +221112,9 @@
"008935": {
"id": "008935",
"content": "已知过抛物线$y^2=4x$的焦点$F$的直线交抛物线于$AB$两点, 过原点$O$作$\\overrightarrow {OM}$, 使$\\overrightarrow {OM}\\perp \\overrightarrow {AB}$, 垂足为$M$, 求点$M$的轨迹方程.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220444,7 +221136,9 @@
"008936": {
"id": "008936",
"content": "抛物线$y^2=8x$的动弦$AB$的长为$16$, 求弦$AB$的中点$M$到$y$轴的最短距离.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220466,7 +221160,9 @@
"008937": {
"id": "008937",
"content": "``太阳火''(sunfire)是一种利用太阳能的装置, 其截面是抛物线形状, 如图所示.``太阳火''依靠抛物线形状的镜面把反射的太阳光聚焦于抛物线的焦点处的锅炉, 加热产生的蒸汽推动汽轮发电机产生电能.根据图中所注尺寸, 求``太阳火''装置中截面抛物线的方程(其中抛物线截面深$10$英尺, 抛物线开口宽$37$英尺).\n\\begin{center}\n \\begin{tikzpicture}[scale = 0.15,>=latex]\n \\draw [domain = -18.5:18.5] plot (\\x,{\\x*\\x/40});\n \\filldraw [fill = gray!50, draw = black] (0,10) ellipse (2 and 1);\n \\draw (-2,10) -- (2,10);\n \\draw [->] (20,7) -- (20,10);\n \\draw [->] (20,3) -- (20,0);\n \\draw (19.5,0) -- (20.5,0) (19.5,10) -- (20.5,10);\n \\draw [dashed] (2,10) -- (19.5,10);\n \\draw (20,5) node {$10$英尺};\n \\draw [->] (-6,-2) -- (-18.5,-2);\n \\draw [->] (6,-2) -- (18.5,-2);\n \\draw (0,-2) node {$37$英尺};\n \\draw (-18.5,-1.5) -- (-18.5,-2.5) (18.5,-1.5) -- (18.5, -2.5);\n \\draw [dashed] (0,0) -- (0,10);\n \\draw (-2,10) node [left] {锅炉};\n \\end{tikzpicture}\n\\end{center}",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220801,7 +221497,10 @@
"008951": {
"id": "008951",
"content": "已知$F_1F_2$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0,b>0)$的两个焦点, 过点$F_2$且垂直于$x$轴的直线交双曲线于点$P$, 且$\\angle F_1PF_2=60^{\\circ }$, 求此双曲线的渐近线的方程.",
- "objs": [],
+ "objs": [
+ "K0716002X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -220823,7 +221522,9 @@
"008952": {
"id": "008952",
"content": "过点$B(1,1)$能否作直线$l$, 使它与双曲线$x^2-\\dfrac{y^2}2=1$交于$Q_1,Q_2$两点, 且点$B$是线段$Q_1Q_2$的中点? 如果存在, 求此直线的方程; 如果不存在, 请说明理由.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -220845,7 +221546,9 @@
"008953": {
"id": "008953",
"content": "已知抛物线的焦点在$y$轴上, 抛物线上一点$M(a,-4)$到焦点$F$的距离为$5$, 求此抛物线的标准方程及实数$a$的值.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220867,7 +221570,9 @@
"008954": {
"id": "008954",
"content": "过抛物线$y^2=4x$的焦点作直线交抛物线于$A(x_1,y_1)$、$B(x_2,y_2)$两点, 且$x_1+x_2=6$, 求$|AB|$的值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220889,7 +221594,9 @@
"008955": {
"id": "008955",
"content": "已知直线$y=x-2$与抛物线$y^2=ax$相交于$AB$两点, 且$OA\\perp OB$, 求实数$a$的值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220911,7 +221618,10 @@
"008956": {
"id": "008956",
"content": "已知抛物线的顶点是双曲线$16x^2-9y^2=144$的中心, 它的焦点是双曲线的左顶点, 求此抛物线的方程.",
- "objs": [],
+ "objs": [
+ "K0717002X",
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"双曲线",
@@ -220934,7 +221644,9 @@
"008957": {
"id": "008957",
"content": "如图, 一位运动员在距篮下$4$米处跳起投篮, 球运行的路线是抛物线, 当球运行的水平距离为$2.5$米时, 达到最大高度为$3.5$米, 然后准确落入篮框, 已知篮框中心到地面的距离为$3.05$米.\\\\\n(1) 建立如图所示的平面直角坐标系, 求抛物线的方程;\\\\\n(2) 该运动员身高为$1.8$米, 在这次跳投中, 球在头顶上方$0.25$米处出手, 问球出手时, 他跳离地面的高度是多少.",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -220981,7 +221693,8 @@
"id": "008959",
"content": "若椭圆$\\dfrac{x^2}4+\\dfrac{y^2}{a^2}=1$与双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}2=1$有相同的焦点, 则实数$a$为\\bracket{20}.\n\\fourch{1}{$-1$}{$\\pm 1$}{不确定}",
"objs": [
- "K0713002X"
+ "K0713002X",
+ "K0716002X"
],
"tags": [
"第七单元",
@@ -221007,7 +221720,9 @@
"008960": {
"id": "008960",
"content": "填若抛物线$y^2=2px(p>0)$上一点$M$到焦点的距离为$a(a>\\dfrac p2)$, 则点$M$到准线的距离为\\blank{50}, 点$M$的横坐标为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -221052,7 +221767,8 @@
"id": "008962",
"content": "命题: 椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$与双曲线$\\dfrac{x^2}{11}-\\dfrac{y^2}5=1$的焦距相等.试将此命题推广到一般情形, 使已知命题成为推广后命题的一个特例:\\blank{50}.",
"objs": [
- "K0713002X"
+ "K0713002X",
+ "K0716002X"
],
"tags": [
"第七单元",
@@ -221099,7 +221815,10 @@
"008964": {
"id": "008964",
"content": "求渐近线方程为$3x\\pm 4y=0$, 焦点为椭圆$\\dfrac{x^2}{10}+\\dfrac{y^2}5=1$的一对顶点的双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -221147,7 +221866,10 @@
"008966": {
"id": "008966",
"content": "已知双曲线的渐近线方程为$y=\\pm x$, 它的两个焦点都在抛物线$x^2=y+2$上, 求此双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线",
@@ -221191,7 +221913,9 @@
"008968": {
"id": "008968",
"content": "(1) 已知直线$l$: $4x-y-1=0$与抛物线$x^2=2y$交于$A(x_A,y_A)$、$B(x_B,y_B)$两点, 直线$l$与$x$轴相交于点$C(x_C,0)$, 求证: $\\dfrac 1{x_A}+\\dfrac 1{x_B}=\\dfrac 1{x_C}$;\\\\\n(2) 试将第(1)题中的命题加以推广, 使得第(1)题中的命题是推广后得到的命题的特例, 并证明推广后得到的命题正确.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -223544,7 +224268,9 @@
"009077": {
"id": "009077",
"content": "若点$A(-2,3)$在抛物线$y^2=2px(p>0)$的准线上, 则实数$p$的值为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -223632,7 +224358,9 @@
"009081": {
"id": "009081",
"content": "经过点$P(4,-2)$的抛物线的标准方程是\\bracket{20}.\n\\fourch{$y^2=x$或$x^2=y$}{$y^2=x$或$x^2=8y$}{$x^2=y$或$y^2=-8x$}{$y^2=x$或$x^2=-8y$}",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -223748,7 +224476,10 @@
"009086": {
"id": "009086",
"content": "已知双曲线的中心是坐标原点, 它的一条渐近线方程为$3x-4y=0$, 且此双曲线经过点$(2,1)$, 求此双曲线的标准方程.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -223794,7 +224525,9 @@
"009088": {
"id": "009088",
"content": "已知双曲线的中心在原点, 且它的一个焦点为$F(\\sqrt 7,0)$, 直线$y=x-1$与其相交于$MN$两点, 线段$MN$中点的横坐标为$-\\dfrac 23$, 求此双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0718001X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -223817,7 +224550,8 @@
"id": "009089",
"content": "直线$x-y+1=0$与椭圆$mx^2+ny^2=1(m,n>0)$相交于$AB$两点, 弦$AB$的中点的横坐标是$-\\dfrac 13$.求双曲线$\\dfrac{y^2}{m^2}-\\dfrac{x^2}{n^2}=1$的两条渐近线所夹锐角的大小.",
"objs": [
- "K0715003X"
+ "K0715003X",
+ "K0717004X"
],
"tags": [
"第七单元",
@@ -223841,7 +224575,9 @@
"009090": {
"id": "009090",
"content": "如图, 双曲线$x^2-\\dfrac{y^2}4=1$的左、右两个焦点为$F_1,F_2$, 第二象限内的一点$P$在双曲线上, 且$\\angle F_1PF_2=\\dfrac{\\pi }3$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex, scale = 0.5]\n \\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n \\draw [->] (0,-4) -- (0,4) node [left] {$y$};\n \\draw (0,0) node [below left] {$O$};\n \\foreach \\i in {-3,-2,-1,1,2,3}\n {\n \\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n \\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};\n };\n \\filldraw ({-sqrt(5)},0) circle (0.05) node [above] {$F_1$} coordinate (F1);\n \\filldraw ({sqrt(5)},0) circle (0.05) node [above] {$F_2$} coordinate (F2);\n \\draw [domain = -3.5:3.5] plot ({sqrt(\\x*\\x/4+1)},\\x);\n \\draw [domain = -3.5:3.5] plot ({-sqrt(\\x*\\x/4+1)},\\x);\n \\draw ({-3/sqrt(5)},{4/sqrt(5)}) node [left] {$P$} coordinate (P);\n \\draw (F1) -- (P) -- (F2);\n \\end{tikzpicture}\n\\end{center}\n(1) 求$|PF_1|\\cdot|PF_2|$;\\\\\n(2) 求点$P$的坐标.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -224000,7 +224736,11 @@
"009097": {
"id": "009097",
"content": "已知双曲线的中心为原点, 两条渐近线方程是$y=\\pm \\dfrac 23x$.若这条双曲线过点$M(\\dfrac 92,-1)$, 则这条双曲线的焦距为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717006X",
+ "K0716003X",
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -224022,7 +224762,9 @@
"009098": {
"id": "009098",
"content": "若抛物线$x^2=y$上的点到直线$y=2x+b$的最短距离为$\\sqrt 5$, 则实数$b=$\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -224044,7 +224786,9 @@
"009099": {
"id": "009099",
"content": "若$\\theta \\in \\mathbf{R}$, 则方程$x^2+y^2\\sin \\theta =1$所表示的曲线一定不是\\bracket{20}.\n\\fourch{直线}{圆}{抛物线}{双曲线}",
- "objs": [],
+ "objs": [
+ "K0719002X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -224108,7 +224852,9 @@
"009102": {
"id": "009102",
"content": "已知双曲线的中心在原点, 且它的一个焦点为$F_1(-\\sqrt 5,0)$.若点$P$位于此双曲线上, 线段$PF_1$的中点坐标为$(0,2)$, 则此双曲线的方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}4-\\dfrac{y^2}1=1$}{$x^2-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}2-\\dfrac{y^2}3=1$}{$\\dfrac{x^2}3-\\dfrac{y^2}2=1$}",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -224176,7 +224922,8 @@
"id": "009105",
"content": "已知抛物线$y^2=4x$与椭圆$\\dfrac{x^2}9+\\dfrac{y^2}k=1$有公共焦点$F_1$, 椭圆的另一焦点为$F_2$, $P$是这两条曲线的一个交点, 求$\\triangle PF_1F_2$的周长.",
"objs": [
- "K0713002X"
+ "K0713002X",
+ "K0719003X"
],
"tags": [
"第七单元",
@@ -224200,7 +224947,9 @@
"009106": {
"id": "009106",
"content": "已知抛物线$y=2x^2$上有$A(x_1,y_2)$、$B(x_2,y_2)$两点, 且$,B$关于直线$y=x+m$对称, $x_1x_2=-\\dfrac 12$, 求实数$m$的值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -224289,7 +225038,9 @@
"009110": {
"id": "009110",
"content": "如图, 直线$y=\\dfrac 12x$与抛物线$y=\\dfrac 18x^2-4$交于$AB$两点, 线段$AB$的垂直平分线与直线$y=-5$交于点$Q$.\n\\begin{center}\n \\begin{tikzpicture}[>=latex,scale = 0.2]\n \\draw [->] (-10,0) -- (10,0) node [below] {$x$};\n \\draw [->] (0,-10) -- (0,10) node [left] {$y$};\n \\draw (0,0) node [above left] {$O$};\n \\draw [name path = line, domain = -8:10] plot (\\x,{\\x/2});\n \\draw [name path = para, domain = -9:9] plot (\\x,{pow(\\x,2)/8-4});\n \\draw (-10,-5) -- (10,-5);\n \\draw (0,-5) node [below left] {$-5$};\n \\draw (-4,-2) node [below] {$A$} coordinate (A);\n \\draw (8,4) node [below right] {$B$} coordinate (B);\n \\filldraw (6,{36/8-4}) circle (0.1) node [right] {$P$};\n \\draw (-2,9) coordinate (T) -- (6,-7) (5,-5) node [below left] {$Q$};\n \\draw (2,1) coordinate (V);\n \\draw (V) pic [draw, scale = 0.3] {right angle = B--V--T};\n \\end{tikzpicture}\n\\end{center}\n(1) 求点$Q$的坐标;\\\\\n(2) 当$P$为抛物线上位于线段$AB$下方(含点$AB$)的动点时, 求$\\triangle OPQ$面积的最大值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -228561,7 +229312,8 @@
"content": "用二项式定理展开下列两式:\\\\\n(1) $(a+2b)^6$;\\\\\n(2) $(1-\\dfrac 1x)^5$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228603,7 +229355,8 @@
"content": "(1) 求$(x-1)^{15}$的二项展开式中的前$4$项;\\\\\n(2) 求$(2a^3-3b^2)^{10}$的二项展开式中的第$8$项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228624,7 +229377,8 @@
"content": "求下列各式的二项展开式中指定的项的系数:\\\\\n(1) $(1-\\dfrac 1{2x})^{10}$二项展开式中含$\\dfrac 1{x^4}$的项;\\\\\n(2) $(3x^3-\\dfrac 1{3x^3})^{10}$的二项展开式中的常数项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228645,7 +229399,8 @@
"content": "在$(3x-2y)^9$的展开式中, 求二项式系数的和以及各项系数的和.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228666,7 +229421,8 @@
"content": "(1) 用二项式定理证明: $(n+1)^n-1$能被$n^2$整除;\\\\\n(2) 用二项式定理证明: $99^{10}-1$能被$1000$整除.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228687,7 +229443,8 @@
"content": "已知$(1+x)^n$的二项展开式中第$4$项与第$8$项的二项系数相等, 求这两项的二项式系数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228771,7 +229528,8 @@
"content": "在$(x^2-\\dfrac 3x)^n$的二项展开式中, 有且只有第五项的二项式系数最大, 求$\\mathrm{C}_n^0-\\dfrac 12\\mathrm{C}_n^1+\\dfrac 14\\mathrm{C}_n^2-\\cdots +(-1)^n\\cdot \\dfrac 12\\mathrm{C}_n^n$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228813,7 +229571,8 @@
"content": "求$(\\dfrac{\\sqrt x}2-\\dfrac 2{\\sqrt x})^{10}$的二项展开式的中间一项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228834,7 +229593,8 @@
"content": "求$(x\\sqrt y-y\\sqrt x)^{11}$的二项展开式的中间两项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -228855,7 +229615,8 @@
"content": "在$(1+3x)^n$的二项展开式中, 末三项的二项式系数之和等于$631$.\\\\\n(1) 求二项展开式中二项式系数最大的项是第几项;\\\\\n(2) 求二项展开式中系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -229025,7 +229786,8 @@
"content": "$(x^2-\\dfrac 1{2\\sqrt x})^3$的二项展开式的第$3$项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -229151,7 +229913,8 @@
"content": "$(x-1)^n$的二项展开式中第$m$项($m\\le n$, $n\\in \\mathbf{N}^*$)的二项式的系数是\\bracket{20}.\n\\fourch{$\\mathrm{C}_n^{m-1}$}{$(-1)^{m-1}\\mathrm{C}_n^m$}{$\\mathrm{C}_n^m$}{$(-1)^m\\mathrm{C}_n^m$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -229214,7 +229977,8 @@
"content": "已知$(x\\sin \\theta +1)^6$的二项展开式$x^2$项的系数与$(x-\\dfrac{15}2\\cos \\theta)^4$的二项展开式中$x^3$项的系数相等, 求$\\cos \\theta$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -229319,7 +230083,8 @@
"content": "若$(1+\\sqrt x)^n$的展开式的系数和大于$8$且小于$32$, 则系数最大的项是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -229424,7 +230189,8 @@
"content": "已知$(2^{\\lg x}-1)^n$的二项展开式中, 最后三项的二项式系数和等于$22$, 中间项为$-1280$, 求$x$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -230807,7 +231573,8 @@
"content": "已知$(x\\sqrt x-\\dfrac 1x)^6$的二项展开式的第$5$项为$\\dfrac{15}2$, 求$\\displaystyle\\lim_{n\\to\\infty}(x^{-1}+x^{-2}+\\cdots +x^{-n}) $的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -231069,7 +231836,8 @@
"content": "利用二项式定理证明: $3^n>2^{n-1}(n+2)$($n\\in \\mathbf{N}^*$, $n\\ge 2$).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -231114,7 +231882,8 @@
"content": "已知$(\\sqrt[3]x-\\dfrac 1{\\sqrt x})^n$的二项展开式中, 第三项与第二项的二项式系数之比为$11:2$ , 求正整数$n$及二项展开式中的所有的有理项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -231135,7 +231904,8 @@
"content": "已知$(x^{\\lg x}+1)^n$的二项展开式中, 求三项的二项式系数的和为22. 二项式系数最大的项为20000, 求实数$x$的值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -240498,7 +241268,9 @@
"009827": {
"id": "009827",
"content": "已知双曲线$\\dfrac{x^2}9-\\dfrac{y^2}m=1$的焦点在$x$轴上, 焦距为$10$. 求实数$m$的值.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240520,7 +241292,9 @@
"009828": {
"id": "009828",
"content": "已知双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$的两个焦点分别为$F_1$、$F_2$, $P$为双曲线上一点, 且$\\angle F_1PF_2=\\dfrac \\pi 2$. 求$\\triangle PF_1F_2$的面积.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240542,7 +241316,10 @@
"009829": {
"id": "009829",
"content": "分别写出下列双曲线的实半轴长、虚半轴长、离心率、焦点坐标、顶点坐标和渐近线方程:\\\\\n(1) $9x^2-16y^2=144$;\\\\\n(2) $\\dfrac{y^2}4-\\dfrac{x^2}3=1$.",
- "objs": [],
+ "objs": [
+ "K0717002X",
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240564,7 +241341,9 @@
"009830": {
"id": "009830",
"content": "在下列双曲线中, 以$y=\\pm \\dfrac 12x$为渐近线的是\n\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}2-y^2=1$}{$x^2-\n\\dfrac{y^2}2=1$}",
- "objs": [],
+ "objs": [
+ "K0717004X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240588,7 +241367,9 @@
"009831": {
"id": "009831",
"content": "判断双曲线$\\dfrac{x^2}4-\\dfrac{y^2}5=1$与双曲线$\\dfrac{y^2}5-\\dfrac{x^2}4=1$的四个焦点是否共圆.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240610,7 +241391,10 @@
"009832": {
"id": "009832",
"content": "求适合下列条件的双曲线的标准方程:\\\\\n(1) 顶点在$x$轴上, 两顶点间的距离是$10$, 且经过点$(10,3)$;\\\\\n(2) 一个焦点的坐标为$(5,0)$, 一条渐近线方程为$3x-4y=0$.",
- "objs": [],
+ "objs": [
+ "K0716003X",
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240632,7 +241416,9 @@
"009833": {
"id": "009833",
"content": "给定一对直线$y=\\pm\\dfrac ba x$($a>0$, $b>0$), 写出所有以这对直线为渐近线的、实轴在$x$轴上的双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240654,7 +241440,9 @@
"009834": {
"id": "009834",
"content": "联系双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的性质, 讨论并叙述双曲线$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1$($a>0$, $b>0$)的性质(不要求推理过程).",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -240697,7 +241485,9 @@
"009836": {
"id": "009836",
"content": "分别写出满足下列条件的抛物线的标准方程:\\\\\n(1) 焦点是$F(-2,0)$;\\\\\n(2) 准线方程是$y=1$.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -240721,7 +241511,9 @@
"009837": {
"id": "009837",
"content": "求抛物线$y^2=4x$上到焦点的距离等于$9$的点的坐标.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -240743,7 +241535,9 @@
"009838": {
"id": "009838",
"content": "过点$P(2,4)$且与抛物线$y2=8x$有且只有一个公共点的直线有\n\\bracket{20}.\n\\fourch{$1$条}{$2$条}{$3$条}{$4$条}",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -240765,7 +241559,9 @@
"009839": {
"id": "009839",
"content": "求抛物线$y^2=4x$上的点到直线$4x+3y+7=0$的最短距离.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -240787,7 +241583,9 @@
"009840": {
"id": "009840",
"content": "由抛物线的标准方程知, 函数$y=\\sqrt x$的图像是某条抛物线的一部分. 求这条抛物线的焦点坐标和准线方程.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -243293,7 +244091,8 @@
"content": "(1) 求$(x-\\sqrt2y)^8$的二项展开式;\\\\\n(2) 求$(x-x^{-\\frac 13})^{12}$的二项展开式中的常数项;\\\\\n(3) 求$(x-\\dfrac 2x)^9$的二项展开式中$x^3$的系数;\\\\\n(4) 在$(1-x^2)^{20}$的二项展开式中, 如果第$4r$项和第$r+2$项的系数的绝对值相等, 求此展开式的第$4r$项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -243314,7 +244113,8 @@
"content": "利用二项式定理证明: $7^{100}-1$是$8$的倍数.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -243356,7 +244156,8 @@
"content": "(1) 求$(1+2x)^7$的二项展开式中系数最大的项;\\\\\n(2) 求$(1-2x)^7$的二项展开式中系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -244135,7 +244936,9 @@
"009985": {
"id": "009985",
"content": "双曲线$\\dfrac{x^2}{9}-y^2=1$的实轴长为\\blank{50}.",
- "objs": [],
+ "objs": [
+ "K0717002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -244254,7 +245057,8 @@
"content": "二项式$(3+x)^n$的展开式中, $x^2$项的系数是常数项的$5$倍, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -259172,7 +259976,7 @@
},
"010630": {
"id": "010630",
- "content": "若直线$x+my+5=0$与直线$x+y+1=0$的夹角为$\\pi 4$, 求实数$m$的值.",
+ "content": "若直线$x+my+5=0$与直线$x+y+1=0$的夹角为$\\dfrac\\pi 4$, 求实数$m$的值.",
"objs": [
"K0707004X"
],
@@ -260250,7 +261054,9 @@
"010674": {
"id": "010674",
"content": "双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$上一点$P$到焦点$F_1$的距离等于$6$, 求点$P$到另一焦点$F_2$的距离.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260272,7 +261078,9 @@
"010675": {
"id": "010675",
"content": "已知双曲线以坐标轴为对称轴, 两个顶点间的距离为$2$, 焦点到渐近线的距离为$\\sqrt 2$. 求该双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260294,7 +261102,9 @@
"010676": {
"id": "010676",
"content": "如果双曲线关于原点对称, 它的焦点在坐标轴上, 实轴的长为$8$, 焦距为$10$. 写出此双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260316,7 +261126,9 @@
"010677": {
"id": "010677",
"content": "如果方程$\\dfrac{x^2}{m+2}-\\dfrac{y^2}{m+1}=1$表示焦点在$y$轴上的双曲线, 求实数$m$的取值范围.",
- "objs": [],
+ "objs": [
+ "K0716002X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260340,7 +261152,9 @@
"010678": {
"id": "010678",
"content": "已知双曲线经过点$(1, 1)$, 其渐近线方程为$y=\\pm\\sqrt 2x$. 求此双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0717006X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260362,7 +261176,9 @@
"010679": {
"id": "010679",
"content": "已知双曲线的中心在原点, 焦点在$y$轴上, 并且双曲线上两点$P_1$、$P_2$的坐标分别为$(3, -4\\sqrt 2)$、$(\\dfrac 94, 5)$. 求该双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260384,7 +261200,10 @@
"010680": {
"id": "010680",
"content": "已知离心率为$\\dfrac 53$的双曲线与椭圆$\\dfrac{x^2}{40}+\\dfrac{y^2}{15}=1$有公共焦点, 求此双曲线的方程.",
- "objs": [],
+ "objs": [
+ "K0717004X",
+ "K0716003X"
+ ],
"tags": [
"第七单元",
"双曲线"
@@ -260427,7 +261246,9 @@
"010682": {
"id": "010682",
"content": "求抛物线$y^2=ax$($a\\ne 0$)的焦点坐标和准线方程.",
- "objs": [],
+ "objs": [
+ "K0719003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260451,7 +261272,9 @@
"010683": {
"id": "010683",
"content": "若抛物线$y^2=2x$上的$A$、$B$两点到焦点$F$的距离之和是$5$, 求线段$AB$的中点的横坐标.",
- "objs": [],
+ "objs": [
+ "K0719006X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260475,7 +261298,9 @@
"010684": {
"id": "010684",
"content": "求以坐标原点为顶点, 以$y$轴为对称轴, 并经过点$P(-6, -3)$的抛物线的标准方程.",
- "objs": [],
+ "objs": [
+ "K0719004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260497,7 +261322,9 @@
"010685": {
"id": "010685",
"content": "已知直线$y=kx-4$与抛物线$y^2=8x$有且只有一个公共点, 求实数$k$的值.",
- "objs": [],
+ "objs": [
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260523,7 +261350,9 @@
"010686": {
"id": "010686",
"content": "已知一隧道的顶部是抛物拱形, 拱高是$5\\text{m}$, 跨度为$10\\text{m}$. 建立适当的平面直角坐标系, 求此拱形所在的抛物线方程.",
- "objs": [],
+ "objs": [
+ "K0720004X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260566,7 +261395,10 @@
"010688": {
"id": "010688",
"content": "过抛物线$y^2=2px$($p>0$)焦点的一条直线与抛物线相交于两个不同的点, 求证: 这两个点的纵坐标$y_1$、$y_2$满足$y_1y_2=-p^2$.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260588,7 +261420,10 @@
"010689": {
"id": "010689",
"content": "过抛物线$y^2=2px$的焦点且倾斜角为$\\alpha$的直线$l$与抛物线交于$A$、$B$两点, 求证:$|AB|=\\dfrac{2p}{\\sin^2\\alpha}$.",
- "objs": [],
+ "objs": [
+ "K0719003X",
+ "K0720003X"
+ ],
"tags": [
"第七单元",
"抛物线"
@@ -260909,7 +261744,10 @@
"id": "010703",
"content": "点$P$到定点$F(2, 0)$的距离与它到直线$x=8$的距离之比为$k$, 请分别给出$k$的某个值, 使得轨迹是椭圆、双曲线和抛物线.",
"objs": [
- "K0713002X"
+ "K0713002X",
+ "K0716001X",
+ "K0719001X",
+ "K0719005X"
],
"tags": [
"第七单元",
@@ -264164,7 +265002,8 @@
"content": "乘积$(a_1+a_2)(b_1+b_2+b_3)(c_1+c_2+c_3+c_4)$的展开式中有多少项?",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -264964,7 +265803,8 @@
"content": "求$(2x^2-\\dfrac 1x)^6$的二项展开式中的中间项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -264985,7 +265825,8 @@
"content": "求$(x+\\dfrac 1x)^{10}$的二项展开式中的常数项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -265006,7 +265847,8 @@
"content": "在$(\\sqrt x+\\dfrac 1{\\sqrt[3]x})^{24}$的二项展开式中, $x$的幂指数是负数的项一共有多少个?",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -265027,7 +265869,8 @@
"content": "求$(x+\\dfrac 12)^8$的二项展开式中系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -265048,7 +265891,8 @@
"content": "已知$x>0$, 且$(x+\\dfrac 1{x^3})^9$的二项展开式中, 第二项不大于第三项. 求实数$x$的取值范围.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -265090,7 +265934,8 @@
"content": "求$(3-2x)^9$的二项展开式中系数最大的项.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -265111,7 +265956,8 @@
"content": "设$f(x)=(1+x)^m+(1+x)^n$($m$、$n$为正整数). 若二项展开式中关于$x$的一次项系数之和为$11$, 则当$m$、$n$为何值时, 含$x^2$项的系数取得最小值?",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -265132,7 +265978,8 @@
"content": "在$(1+x)^n$的二项展开式中, 设奇数项之和为$A$, 偶数项之和为$B$. 求证: $A^2-B^2=(1-x^2)^n$.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "",
@@ -267922,7 +268769,8 @@
"content": "在$(1-2x)^6$的二项展开式中, $x^3$项的系数为\\blank{50}. (用数字作答)",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -268808,7 +269656,8 @@
"content": "已知$(2x^2-\\dfrac 1x)^n$($n\\in \\mathbf{N}^*)$的展开式中各项的二项式系数之和为$128$, 则其展开式中含$\\dfrac 1x$项的系数是\\blank{50}.(结果用数值表示)",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -269328,7 +270177,8 @@
"content": "已知二项式$(2x+\\dfrac 1x)^6$, 则其展开式中的常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -271239,7 +272089,8 @@
"content": "二项式$(x-\\dfrac 1x)^6$的展开式中的常数项为\\blank{50}. (用数字作答)",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -273305,7 +274156,8 @@
"content": "若$(x-\\dfrac ax)^9$的展开式中$x^3$的系数是$-84$, 则$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -274242,7 +275094,8 @@
"content": "若$(ax^2+\\dfrac 1{\\sqrt x})^{5}$的展开式中的常数项为$-\\dfrac 5{2}$, 则实数$a$的值为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -274762,7 +275615,8 @@
"content": "设$n\\in \\mathbf{N}^*$, 若$(2+\\sqrt x)^n$的二项展开式中, 有理项的系数之和为$365$, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -275089,7 +275943,8 @@
"content": "已知二项式$(x^2+\\dfrac ax)^6$的展开式中含$x^3$项的系数是$160$, 则实数$a$的值是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -275789,7 +276644,8 @@
"content": "在$(\\sqrt x+\\dfrac 2{x^2})^n$($n\\in \\mathbf{N}^*$)的展开式中, 设含有$(\\sqrt x)^{n-r}(\\dfrac 2{x^2})^r$的项为第$r+1$($0\\le r\\le n$, $n\\in \\mathbf{N}$)项. 若第$3$项与第$5$项的系数之比为$3:56$, 则展开式中的常数项为\\bracket{20}.\n\\fourch{$180$}{$160$}{$120$}{$100$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -275936,7 +276792,8 @@
"content": "$(x^2+\\dfrac 1x)^8$的展开式中$x^4$项的系数是\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -276406,7 +277263,8 @@
"content": "$(x+\\dfrac 1x)^n$的展开式中的第$3$项为常数项, 则正整数$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -277036,7 +277894,8 @@
"content": "$(1+\\dfrac 1{x^2})(1+x)^6$展开式中$x^2$项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -277455,7 +278314,8 @@
"content": "在$(1-2x)^{6}$的二项展开式中, $x^{3}$项的系数为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -278056,7 +278916,8 @@
"tags": [
"第八单元",
"第四单元",
- "第七单元"
+ "第七单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -278448,7 +279309,8 @@
"content": "在$(x^2+\\dfrac 2x)^6$的二项展开式中, 常数项等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -279309,7 +280171,8 @@
"content": "已知二项式$(2x+\\dfrac 1x)^6$, 则其展开式中的常数项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -279983,7 +280846,8 @@
"content": "在$(\\dfrac 2x-x)^6$的二项展开式中, 常数项等于\\bracket{20}.\n\\fourch{$-160$}{$160$}{$-150$}{$150$}",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "选择题",
"ans": "",
@@ -282236,7 +283100,8 @@
"content": "在$(x-\\dfrac 2x)^6$的二项展开式中, 常数项等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -282755,7 +283620,8 @@
"content": "设常数$a\\in \\mathbf{R}$, 若$(x^2+\\dfrac ax)^5$的二项展开式中$x^7$项的系数为$-10$, 则$a=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -283885,7 +284751,8 @@
"content": "在$(1+x+\\dfrac 1{x^{2015}})^{10}$的展开式中, $x^2$项的系数为\\blank{50}(结果用数值表示).",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -284313,7 +285180,8 @@
"content": "在$(\\sqrt[3]x-\\dfrac 2x)^n$的二项式中, 所有项的二项式系数之和为$256$, 则常数项等于\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "",
@@ -290123,7 +290991,8 @@
"content": "$(1+x)^n$的二项展开式中, 若第9项与第13项系数相等, 则第20项为\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$20x^{19}$",
@@ -290994,7 +291863,8 @@
"content": "已知代数式$(\\dfrac 2m+\\dfrac mx)^n$($m>0$,$x>0$).\\\\\n(1) 当$m=2$, $n=6$时, 求二项展开式中二项式系数最大的项;\\\\\n(2) 若$(\\dfrac 2m+\\dfrac mx)^{10}=a_0+\\dfrac{a_1}x+\\dfrac{a_2}{x^2}+\\cdots +\\dfrac{a_{10}}{x^{10}}$, 且$a_2=180$, 求$a_i$($0\\le i \\le 10$,$i\\in \\mathbf{N}$)的最大值.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "解答题",
"ans": "(1) $\\dfrac{160}{x^3}$; (2) $15360$",
@@ -291147,7 +292017,9 @@
"id": "012033",
"content": "集合$A=\\{x|x\\ge 0\\}$, $B=\\{x|x\\ge a\\}$, 若$A\\subseteq B$, 则实数$a$的取值范围为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291166,7 +292038,9 @@
"id": "012034",
"content": "函数$y=\\lg(2-x)$的定义域为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291185,7 +292059,9 @@
"id": "012035",
"content": "陈述句``$a\\ge 1$且$a\\le 3$''的否定形式为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291204,7 +292080,9 @@
"id": "012036",
"content": "已知$A,B$是独立事件, $P(A)=0.3$, $P(B)=0.5$, 则$P(A\\cap B)=$\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第八单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291223,7 +292101,9 @@
"id": "012037",
"content": "若圆锥的轴截面是边长为$1$的正三角形, 则圆锥的侧面积是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第六单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291242,7 +292122,9 @@
"id": "012038",
"content": "若$z=\\dfrac{1-a\\mathrm{i}}{2+\\mathrm{i}}$($\\mathrm{i}$为虚数单位)为纯虚数, 则实数$a$的值为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第五单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291261,7 +292143,9 @@
"id": "012039",
"content": "已知$\\overrightarrow{a}=(2,1)$, $\\overrightarrow{b}$在$\\overrightarrow{a}$上的投影为$-2\\overrightarrow{a}$, 则$\\overrightarrow{a}\\cdot\\overrightarrow{b}=$\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第五单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291280,7 +292164,9 @@
"id": "012040",
"content": "如果幂函数$y=f(x)$的图像经过点$(2,\\dfrac 12)$, 那么$y=f(x)$的单调减区间是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291299,7 +292185,9 @@
"id": "012041",
"content": "某医院对某学校高三年级的$600$名学生进行身体健康调查, 采用男女分层抽样法抽取一个容量为$50$的样本, 已知女生比男生少抽了$10$人, 则该年级的女生人数是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第九单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291318,7 +292206,9 @@
"id": "012042",
"content": "偶函数$y=f(x)$在区间$[0,+\\infty)$上是严格减函数, 若$f(1)=0$, 则关于$x$的不等式$f(x)-x^2>-1$的解集是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291337,7 +292227,9 @@
"id": "012043",
"content": "已知$a,b\\in \\mathbf{R}$且$a\\ne 0$, 则$|a+b|+|\\dfrac 4a-b|$的最小值是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291356,7 +292248,9 @@
"id": "012044",
"content": "已知函数$y=\\sin x+\\sin 2x$在$(-a,a)$上恰有$5$个零点, 则实数$a$的最大值为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291375,7 +292269,9 @@
"id": "012045",
"content": "设$x\\in \\mathbf{R}$, 则``$x<1$''是``$x^3<1$''的\\bracket{20}.\n\\twoch{充分而不必要条件}{必要而不充分条件}{充要条件}{既不充分也不必要条件}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291394,7 +292290,9 @@
"id": "012046",
"content": "同时掷两枚骰子, 向上的点数之和是$6$的概率是\\bracket{20}.\n\\fourch{$\\dfrac 1{12}$}{$\\dfrac 19$}{$\\dfrac 16$}{$\\dfrac 5{36}$}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第八单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291413,7 +292311,9 @@
"id": "012047",
"content": "已知某射击爱好者打靶成绩(单位:环)的茎叶图如图所示, 其中整数部分为``茎'', 小数部分为``叶'', 则这组数据的标准差为(精确到$0.01$)\\bracket{20}.\n\\begin{center}\n\\begin{tabular}{c|cccc}\n$5$ & $7$ & $9$ \\\\\n$6$ & $1$ & $2$ & $7$ & $7$ \\\\\n$7$ & $2$ & $5$\n\\end{tabular}\n\\end{center}\n\\fourch{$0.35$}{$0.59$}{$0.40$}{$0.63$}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第九单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291432,7 +292332,9 @@
"id": "012048",
"content": "如图所示, 图中多面体是由两个底面相同的正四棱锥所拼接而成, 且这六个顶点在同一个球面上. 若二面角$M-AB-C$的正切值为$1$, 则二面角$N-AB-C$的正切值为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = (-120:0.5)]\n\\draw (-1,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,0,-1) node [left] {$D$} coordinate (D);\n\\draw (0,1,0) node [above] {$M$} coordinate (M);\n\\draw (0,-2,0) node [below] {$N$} coordinate (N);\n\\draw (A) -- (B) -- (C) (A) -- (N) (B) -- (N) (C) -- (N) (M) -- (D) (M) -- (A) (M) -- (B) (M) -- (C) (A) -- (D);\n\\draw [dashed] (D) -- (C) (D) -- (N);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$1$}{$\\sqrt{2}$}{$2$}{$2\\sqrt{2}$}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第六单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291451,7 +292353,9 @@
"id": "012049",
"content": "已知$O$为坐标原点, $\\overrightarrow{OA}=(2,3)$, $\\overrightarrow{OB}=(4,2)$, $\\overrightarrow{OC}=(x,3)$.\\\\\n(1) 若$A,B,C$三点共线, 求$x$的值;\\\\\n(2) 若$\\overrightarrow{AB}$与$\\overrightarrow{OC}$夹角为钝角, 求$x$的取值范围.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第五单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291470,7 +292374,9 @@
"id": "012050",
"content": "已知函数$f(x)=ax^2+x-1$. ($a>0$)\\\\\n(1) 若关于$x$的不等式$f(x)<0$的解集为$(-1,b)$, 求实数$a$和$b$的值;\\\\\n(2) 若函数$y=f(x)$在$[-3,-1]$上的最大值为$2$, 求实数$a$的值.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291489,7 +292395,10 @@
"id": "012051",
"content": "如图, 一辆汽车在水平公路上向正西直线行驶, 到$A$处测得公路北侧远处一山顶$D$($D$在水平面上的射影为点$C$)在西偏北$30^\\circ$的方向上, 仰角为$30^\\circ$, 行驶$1\\text{km}$后到达$B$处, 测得山顶在西偏北$45^\\circ$的方向上.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = (-115:0.5)]\n\\draw (-3,0) -- (3,0);\n\\draw (1,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [below] {$A$} coordinate (A);\n\\draw ({(sqrt(3)-1)/2},0,{1/(1-sqrt(3))}) node [left] {$C$} coordinate (C);\n\\draw [dashed] (A) -- (C) (B) -- (C);\n\\draw (C) ++ (0,{1/sin(15)*sin(45)*sin(30)}) node [above] {$D$} coordinate (D);\n\\draw [dashed] (D) -- (A) (D) -- (B) (D) -- (C);\n\\draw (D) .. controls (0.5,2) .. (-2,1);\n\\draw (D) .. controls (1,2) .. (2,1.3);\n\\end{tikzpicture}\n\\end{center}\n(1) 求此山的高度(单位: $\\text{km}$, 精确到$0.01\\text{km}$);\\\\\n(2) 求汽车行驶过程中仰望山顶$D$的仰角$\\theta$的最大值(精确到$1^\\circ$).",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元",
+ "第六单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291508,7 +292417,9 @@
"id": "012052",
"content": "如图, 三棱柱$ABC-A_1B_1C_1$中, $\\angle CAB=90^\\circ$, $AB=AC=A_1B=A_1C=2\\sqrt{2}$, $AA_1=2$, 点$M,F$分别为$BC,A_1B_1$的中点, 点$E$为$AM$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = (-45:0.5),scale = 1.25]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{2*sqrt(2)}) node [below] {$B$} coordinate (B);\n\\draw ({2*sqrt(2)},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ ({sqrt(2)/2},{sqrt(3)},{sqrt(2)/2}) node [above left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ ({sqrt(2)/2},{sqrt(3)},{sqrt(2)/2}) node [below right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ ({sqrt(2)/2},{sqrt(3)},{sqrt(2)/2}) node [above right] {$C_1$} coordinate (C_1);\n\\draw ($(B)!0.5!(C)$) node [right] {$M$} coordinate (M);\n\\draw ($(A_1)!0.5!(B_1)$) node [above] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(M)$) node [below left] {$E$} coordinate (E);\n\\draw (A) -- (B) -- (C) (A) -- (A_1) (B) -- (B_1) (C) -- (C_1) (A_1) -- (B_1) -- (C_1) (A_1) -- (C_1) (A_1) -- (B);\n\\draw [dashed] (A) -- (M) (E) -- (F) (A_1) -- (M) (A_1) -- (C) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $AA_1\\perp BC$;\\\\\n(2) 证明: $EF\\parallel$平面$BCC_1B_1$;\\\\\n(3) 求直线$EF$与平面$A_1BC$所成角的正弦值.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第六单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291527,7 +292438,10 @@
"id": "012053",
"content": "已知对任意正整数$n$, 都存在$n$次多项式函数$y=f_n(x)$, 使得$\\cos nx=f_n(\\cos x)$对一切$x\\in \\mathbf{R}$恒成立. 例如``$y=f_2(x)=2x^2-1$, $\\cos 2x=2\\cos^2 x-1=f_2(\\cos x)$''.\\\\\n(1) 求$f_n(0)$;\\\\\n(2) 求证: 当$n$为偶数时, 不存在函数$y=g_n(x)$使得$\\sin nx=g_n(\\sin x)$对一切$x\\in \\mathbf{R}$恒成立;\\\\\n(3) 求证: 当$n$为奇数时, 存在多项式函数$y=h_n(x)$使得$\\sin nx=h_n(\\sin x)$对一切$x\\in \\mathbf{R}$恒成立, 并求其最高次项系数.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元",
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291546,7 +292460,9 @@
"id": "012054",
"content": "若集合$A=\\{2,a^2-a+1\\}$, $B=\\{3,a+3\\}$, 且$A\\cap B=\\{3\\}$, 则实数$a=$\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291565,7 +292481,9 @@
"id": "012055",
"content": "若复数$z=(1+m\\mathrm{i})(2-\\mathrm{i})$($\\mathrm{i}$是虚数单位)是纯虚数, 则实数$m$的值为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第五单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291584,7 +292502,9 @@
"id": "012056",
"content": "已知全集$U=\\mathbf{R}$, 集合$A=\\{x|\\dfrac{x+1}{x-2}\\le 0\\}$, 则集合$\\overline{A}=$\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291603,7 +292523,9 @@
"id": "012057",
"content": "已知$\\{a_n\\}$为等差数列, 其前$n$项和为$S_n$, 若$a_1=1$, $a_3=5$, $S_n=64$, 则\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第四单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291622,7 +292544,9 @@
"id": "012058",
"content": "已知复数$z_0=3+\\mathrm{i}$($\\mathrm{i}$为虚数单位), 复数$z$满足$z\\cdot z_0=3z+z_0$, 则$|z|=$\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第五单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291641,7 +292565,9 @@
"id": "012059",
"content": "已知$\\tan \\theta =3$, 则$\\sin 2\\theta -2\\cos ^2\\theta$的值为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291660,7 +292586,9 @@
"id": "012060",
"content": "已知$\\{a_n\\}$是各项均为正数的等比数列,\n且$a_6=2$, 则$\\log_2(a_1\\cdot a_2\\cdot a_3\\cdot \\cdots \\cdot a_{11})=$\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第四单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291679,7 +292607,9 @@
"id": "012061",
"content": "已知函数$f(x)=A\\sin (\\omega x+\\varphi)(A,\\omega ,\\varphi$为常数且$A>0,\\omega >0$)的部分图像如图所示, 则$f(0)$的值是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-pi/6}:{11*pi/12}, samples = 100] plot (\\x,{sqrt(2)*sin(2*\\x/pi*180+60)});\n\\draw [dashed] ({pi/12},0) -- ({pi/12},{sqrt(2)}) -- (0,{sqrt(2)});\n\\draw [dashed] ({7*pi/12},0) -- ({7*pi/12},{-sqrt(2)}) -- (0,{-sqrt(2)});\n\\draw (0,{-sqrt(2)}) node [left] {$-\\sqrt{2}$};\n\\draw ({pi/3},0) node [below] {$\\dfrac{\\pi}{3}$};\n\\draw ({7*pi/12},0) node [above] {$\\dfrac{7\\pi}{12}$};\n\\end{tikzpicture}\n\\end{center}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291698,7 +292628,9 @@
"id": "012062",
"content": "设$f(x)$是$\\mathbf{R}$上的奇函数, $g(x)$是$\\mathbf{R}$上的偶函数, 若函数$f(x)+g(x)$的值域为$[-1,4]$, 则$f(x)-g(x)$的值域为\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291717,7 +292649,9 @@
"id": "012063",
"content": "若函数$y=\\log_a(x^2-ax+1)$有最小值, 则$a$的取值范围是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291736,7 +292670,9 @@
"id": "012064",
"content": "已知关于$x$的方程$|x+a^2|+|x-a^2|=-x^2+2x-1+2a^2$有解, 则实数$a$的取值范围是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291755,7 +292691,9 @@
"id": "012065",
"content": "如果数列$\\{a_n\\}$满足: $a_1=1$,$ a_{2021}=2017$, 且对于任意$n\\in \\mathbf{N}$, $n\\ge 1$, 存在实数$a$使得$a_n$, $a_{n+1}$是方程$x^2-(2a+1)x+a^2+a=0$的两个根, 则$a_{100}$的所有可能值构成的集合是\\blank{50}.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第四单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291774,7 +292712,9 @@
"id": "012066",
"content": "若$\\cos \\theta >0$, 且$\\sin 2\\theta <0$, 则角$\\theta$的终边所在象限是\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291793,7 +292733,9 @@
"id": "012067",
"content": "记$S_n$为数列$\\{a_n\\}$的前$n$项和, ``$\\{a_n\\}$是递增数列''是``$\\{S_n\\}$是递增数列''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分也非必要条件}",
"objs": [],
- "tags": [],
+ "tags": [
+ "第四单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291812,7 +292754,9 @@
"id": "012068",
"content": "有四个命题:\n\\textcircled{1} 若$0>a>b$, 则$\\dfrac 1a<\\dfrac 1b$; \\textcircled{2} 若$ab^2$; \\textcircled{3} 若$\\dfrac 1a>1$, 则$1>a$; \\textcircled{4} 若$10$时, 求解该不等式.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第一单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291869,7 +292817,9 @@
"id": "012071",
"content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对的边长分别为$a$、$b$、$c$,\n且$2\\sqrt 3\\sin B\\cos B-2\\cos ^2B=1$.\\\\\n(1) 求角$B$的大小;\\\\\n(2) 若$b=2$, 求$\\triangle ABC$的面积的最大值.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第三单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291888,7 +292838,9 @@
"id": "012072",
"content": "某地博物馆整体理念是将生态自然与人文历史有机的融合, 与周边环境自然过渡连接. 为了减少能源损耗, 馆顶和外墙需要建造隔热层. 博物馆每年节省的能源费用$h$(单位: 万元)与隔热层厚度$x$(单位: $\\text{cm}$)满足关系: $h(x)=32-\\dfrac{32}{x+k}$($0\\le x\\le 20$). 当不建造隔热层时, 每年节省费用为$0$, 但是隔热层自身需要消耗能源, 每年隔热层自身消耗的能源费用$g$(单位: 万元)与隔热层厚度$x$(单位: $\\text{cm}$)满足关系: $g(x)=2x$.\\\\\n(1) \\textcircled{1} 求$k$的值; \\textcircled{2} 为了使得每年隔热层节省的能源费用不低于隔热层自身消耗的能源费用, 隔热层建造的厚度$x$应该满足什么条件?\\\\\n(2) 在建造厚度为$x$的隔热层后, 每年博物馆真正节省的能源费用$f(x)=h(x)-g(x)$, 求每年博物馆真正节省的能源费用的最大值.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -291907,7 +292859,9 @@
"id": "012073",
"content": "设函数$f(x)=\\dfrac{2x+1}x$($x>0$), 数列$\\{a_n\\}$满足$a_1=1$, $a_n=f(\\dfrac 1{a_{n-1}})$($n\\in \\mathbf{N}$, $n\\ge 2$).\\\\\n(1) 求数列$\\{a_n\\}$的通项公式;\\\\\n(2) 设$T_n=a_1a_2-a_2a_3+a_3a_4-a_4a_5+\\cdots -a_{2n}a_{2n+1}$, 若$T_n\\ge tn^2$对$n\\in \\mathbf{N}^*$恒成立, 求实数$t$的取值范围;\\\\\n(3) 是否存在以$1$为首项, 公比为$q$($0g(x)$恒成立, 则称该函数满足性质$M$.\\\\\n(1) 判断函数$f_1(x)=\\sin x$, $f_2(x)=x^2$是否满足性质$M$(无需说明理由);\\\\\n(2) 若函数$f(x)$满足性质$M$, 求证: $f(x)$不是奇函数;\\\\\n(3) 若函数$f(x)$满足性质$M$, 求证: 当$\\lambda >0$, $x_1\\ne x_2$时, 不等式\n$\\dfrac{f(x_1)+\\lambda f(x_2)}{1+\\lambda }>f(\\dfrac{x_1+\\lambda x_2}{1+\\lambda })$恒成立,\n并求函数$h(x)=f(x)+f(2021-x),x\\in [1,2020]$的最大值.",
"objs": [],
- "tags": [],
+ "tags": [
+ "第二单元"
+ ],
"genre": "",
"ans": "",
"solution": "",
@@ -297795,7 +298751,8 @@
],
"tags": [
"第四单元",
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$\\dfrac 12 (1 - (\\dfrac 13)^{100})$",
@@ -299578,7 +300535,8 @@
"content": "已知$(a+3b)^n$的展开式中, 各项系数的和为$2^{n+6}$, 则$n=$\\blank{50}.",
"objs": [],
"tags": [
- "第八单元"
+ "第八单元",
+ "二项式定理"
],
"genre": "填空题",
"ans": "$6$",