名校
1 . 在几何学常常需要考虑曲线的弯曲程度,为此我们需要刻画曲线的弯曲程度.考察如图所示的光滑曲线C:
上的曲线段
,其弧长为
,当动点从A沿曲线段
运动到B点时,A点的切线
也随着转动到B点的切线
,记这两条切线之间的夹角为
(它等于
的倾斜角与
的倾斜角之差).显然,当弧长固定时,夹角越大,曲线的弯曲程度就越大;当夹角固定时,弧长越小则弯曲程度越大,因此可以定义
为曲线段
的平均曲率;显然当B越接近A,即
越小,K就越能精确刻画曲线C在点A处的弯曲程度,因此定义
(若极限存在)为曲线C在点A处的曲率.(其中y',y''分别表示
在点A处的一阶、二阶导数)
(2)求椭圆
在
处的曲率;
(3)定义
为曲线
的“柯西曲率”.已知在曲线
上存在两点
和
,且P,Q处的“柯西曲率”相同,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/505d83f4d34a8cd385577a6ce93a4b11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aea61ddc41f927684c6dfaacdd7f8e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0636a11a086df66133bd50e43481a546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/427eceadd7bb569ff140ea73d650db1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0636a11a086df66133bd50e43481a546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aea61ddc41f927684c6dfaacdd7f8e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92877ce3543f19dc565dbeff9777ecc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/505d83f4d34a8cd385577a6ce93a4b11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a6ffded1e8b3dd5ef03b57aa2beacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
(2)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82e7d9f4f7ace849e09e9adcb786b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67a345c0ab9bc098efa03e17ea556fcb.png)
(3)定义
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/117c39fe1b37a6862ad0e46282488210.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6004e46d022f4976a52dc949691da232.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75def50794f0b3c42765b1e43334fcd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a0cc87bade827b694da4e6e5c020eea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7add187842d3ee824ed3a501f392735f.png)
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9卷引用:湖北省武汉市武钢三中2024届高三下学期开学考试数学试题
湖北省武汉市武钢三中2024届高三下学期开学考试数学试题浙江省宁波市镇海中学2024届高三上学期期末数学试题(已下线)第四套 九省联考全真模拟(已下线)压轴题函数与导数新定义题(九省联考第19题模式)练(已下线)新题型01 新高考新结构二十一大考点汇总-3(已下线)微考点2-5 新高考新试卷结构19题压轴题新定义导数试题分类汇编湖南省长沙外国语学校2023-2024学年高二下学期3月月考数学试题山东省菏泽市定陶区第一中学2023-2024学年高二下学期第一次月考数学试题(已下线)拔高点突破05 函数与导数背景下的新定义压轴解答题(九大题型)
2 . 棱长为10cm的密闭正四面体容器内装有体积为
的水,翻转容器,使得水面至少与2条棱平行,且水面是三角形,不考虑容器厚度及其它因素影响,则水面面积的最小值为______
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d762aad87f41c486312d8ae0bbe31c97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ce13774b09ff2edddaf21a072cf60a.png)
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4卷引用:辽宁省沈阳市第二中学2024届高三下学期开学考试数学试题
辽宁省沈阳市第二中学2024届高三下学期开学考试数学试题湖北省武汉市武昌区2024届高三上学期期末质量检测数学试题(已下线)第三章 折叠、旋转与展开 专题一 平面图形的翻折、旋转 微点4 翻折、旋转问题中的最值(一)上海市金山中学2023-2024学年高二下学期3月月考数学试卷
名校
解题方法
3 . 已知
为有穷正整数数列,且
,集合
.若存在
,使得
,则称
为
可表数,称集合
为
可表集.
(1)若
,判定31,1024是否为
可表数,并说明理由;
(2)若
,证明:
;
(3)设
,若
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67905ad53186bb2908b603bc14005d10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/702dcfe2523f774f6bc4f075f3d24fe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80566aaf96db9c785cda10dc0935c1c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84076d0854ef7c1a99a937fd50b25843.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c6985405452b5d04bd0d3305544cc2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c93e3391890fc877c761121b68cb927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54119668d2f6cbc9ce0cb92310037713.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c93e3391890fc877c761121b68cb927.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b83efe191fb8adaf89737c03ef34d1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c93e3391890fc877c761121b68cb927.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ebfe653088b1a534d0731947db43d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/562441c2767a65f3671afa93b190126b.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffceb52b543819898a9a6fc96d7337e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7eab142f716f69be57d3f4ca2197894.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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7卷引用:北京市海淀区北京一零一中2023-2024学年高三下学期统考四(开学考)数学试题
23-24高三上·北京西城·期末
名校
解题方法
4 . 给定正整数
,已知项数为
且无重复项的数对序列
:
满足如下三个性质:①
,且
;②
;③
与
不同时在数对序列
中.
(1)当
,
时,写出所有满足
的数对序列
;
(2)当
时,证明:
;
(3)当
为奇数时,记
的最大值为
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad2477167a02872167b2a3760f09d6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dc25d4213ca2eadce49e6d8ba805e15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b730e2023809495f2bd7fbf48f07a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca986e62ec3a6e50e4e2cad639aa9201.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd869b784314b8278f5d144b2d3a9fb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698c4d4e50062b4a7dd70fe1b4ab4fd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/302391681aa37ac20d6f533dbae9e137.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a215612787e43d28bfebc840c3903b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8a3cc8c48bf54ec8252e5dce6867754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60302649eb940748da818199e55da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/241d587c2e6f2f109a4e41b79f1c800f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/926c16dd072c9ff8a560b003cfb47053.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4546b12ff89d1599427da82294afc09b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4546b12ff89d1599427da82294afc09b.png)
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7卷引用:高三数学开学摸底考 (北京专用)
(已下线)高三数学开学摸底考 (北京专用)(已下线)北京市西城区2024届高三上学期期末数学试题北京市西城区2024届高三上学期期末数学试题2024年普通高等学校招生全国统一考试数学冲刺卷二(九省联考题型)江西省南昌市第二中学2024届高三“九省联考”考后适应性测试数学试题(四)江苏省连云港高级中学2023-2024学年高二下学期第一次月考数学试卷广东省深圳市外国语学校2024届高三教学情况测试(一)数学试题
5 . 已知
和数表
,其中
.若数表
满足如下两个性质,则称数表
由
生成.
①任意
中有三个
,一个3;
②存在
,使
中恰有三个数相等.
(1)判断数表
是否由
生成;(结论无需证明)
(2)是否存在数表
由
生成?说明理由;
(3)若存在数表
由
生成,写出
所有可能的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1b76c6898e230717d3daed334b0303.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbda44091b0da7321b26722d6ab78845.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac56300140ed9e27f8dff86ef1eaea0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24c7d6627a568c6eaae35260d53dfb29.png)
①任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f29210b9144737a127a428679c58f406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
②存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dd600b451b2b7f1680cbbcf36a49703.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5137f97e66d136940d82a4027cd4fa2b.png)
(1)判断数表
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a97e4a4a351df2053a3cab244213d41c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e88b1329f12c3b53e86627d04f5e5a3.png)
(2)是否存在数表
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79dc44942df9856c903cd70e4776e86b.png)
(3)若存在数表
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0808a749c7fe9d45bea1edbd3ee96e20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f44f67ab69be2217f7884536cfa53aa.png)
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6卷引用:辽宁省沈阳市第二中学2024届高三下学期开学考试数学试题
辽宁省沈阳市第二中学2024届高三下学期开学考试数学试题北京市第一次普通高中2023-2024学年高二上学期学业水平合格性考试数学试题(已下线)第一套 新高考新结构全真模拟1(艺体生)(模块二)(已下线)微考点8-1 新高考新题型19题新定义题型精选北京市第二中学2023-2024学年高一下学期期中考试数学试题2024届河北省名校联盟高考三模数学试题
名校
6 . 图一所示,在平面直角坐标系中,抛物线
与x轴交于
,
两点,与y轴交于点C.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/22/60680793-2af4-4c55-a4e2-b4f9dc9eba59.png?resizew=120)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/22/f93df309-e8e7-4e20-b34d-80be466866ee.png?resizew=119)
(1)求抛物线的函数表达式及顶点坐标;
(2)点P为第三象限内抛物线上一动点,作直线AC,连接PA,PC,求
面积的最大值及此时点P的坐标;
(3)设直线
:
交抛物线于点M,N,求证:无论k为何值,平行于x轴的直线
:
上总存在一点E,使得
为直角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/818ccf8040f189fff5665ec93892b2ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a525534689bd2701205d4ab17574c39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c850811ba59a05e945a665196539a048.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/22/60680793-2af4-4c55-a4e2-b4f9dc9eba59.png?resizew=120)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/22/f93df309-e8e7-4e20-b34d-80be466866ee.png?resizew=119)
(1)求抛物线的函数表达式及顶点坐标;
(2)点P为第三象限内抛物线上一动点,作直线AC,连接PA,PC,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fbfcae2cecc98e2d6c16dde6d3ec1c1.png)
(3)设直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff7cd2f34acbf4884ac6ea40b5d5be19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/216be086e1952cf1e782a7a3c132496e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/435f30095577dc714634354f5ad27715.png)
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名校
解题方法
7 . 已知定义在
的函数
满足:①对
恒有
;②对任意的正数
,
恒有
.则下列结论中正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b6a8984aa398bf767ccd9a601d77983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52a66d161030ad93973d77b1c53dec8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5386d0aff1b962f99b5565adb9a58cbf.png)
A.![]() |
B.过点![]() ![]() |
C.对![]() ![]() |
D.若![]() ![]() ![]() |
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|
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6卷引用:湖南省长沙市明德中学2023-2024学年高二下学期开学考试数学试卷
湖南省长沙市明德中学2023-2024学年高二下学期开学考试数学试卷湖北省黄冈市部分普通高中2024届高三上学期阶段性教学质量监测数学试题广东省广州市广东实验中学2024届高三上学期大湾区数学冲刺卷(三)江西省抚州市临川第一中学2024届高三“九省联考”考后适应性测试数学试题(一)(已下线)第五章 一元函数的导数及其应用(压轴题专练,精选34题)-2023-2024学年高二数学单元速记·巧练(人教A版2019选择性必修第二册)2024届高三新高考改革数学适应性练习(九省联考题型)
名校
解题方法
8 . 已知实数m,n满足
,且
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d4b55b21e776a254f07f56fab5c958.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8c294fe813bfa5e8ef0c612c7d1d1a9.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2023-11-28更新
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|
5卷引用:广东省佛山市第一中学2024届高三下学期开学预测数学试题(一)
广东省佛山市第一中学2024届高三下学期开学预测数学试题(一)河南省部分名校2023-2024学年高三上学期阶段性测试(三)(11月)数学试题江苏省扬州市扬州中学2024届新高考一卷数学模拟测试一(已下线)模块五 全真模拟篇 基础1 期末终极研习室(2023-2024学年第一学期)高三江西省赣州市南康中学2024届高三上学期新高考“七省联考”考前数学猜题卷(一)
名校
解题方法
9 . 已知函数
给出下列四个结论:
①若
有最小值,则
的取值范围是
;
②当
时,若
无实根,则
的取值范围是
;
③当
时,不等式
的解集为
;
④当
时,若存在
,满足
,则
.
其中,所有正确结论的序号为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff4e0cfa86f6d91ff0b3cdb3251393b.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c6cc71d0c988d725b25c55c2672919c.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/976d18a5396ba232f0aa38d136f1d749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da1606791e5eeefbf298210543e01dd4.png)
③当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e5bcfb3bafe8373dd907e0e55d08f8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168fef6477a494abceae56fb6c2e4c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b61bb7cb94b4d06f0090df1e365667.png)
④当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ede78fd7ac619ea597856254bb5d75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26d8dafc71b106f39f4e15442220897b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/339de01d9636343c484391b421c31301.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7ec808ad60dbf016632ec816eaca1df.png)
其中,所有正确结论的序号为
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2023-11-02更新
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836次组卷
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5卷引用:北京市第二中学2023-2024学年高三下学期开学考试数学试卷
10 . 抛物线
与y轴交于点A,
,顶点为D.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/18/5fd8d323-e143-4e18-9ba1-f23ac31b4cbf.png?resizew=132)
(1)若点A的坐标为(0,-2),求抛物线的顶点D和点P的坐标;
(2)如图,在(1)的条件下,连接AD,PD,在直线AD下方的抛物线上是否存在点N,满足
,若存在,请求出点N的坐标;若不存在,请说明理由;
(3)已知点
,若线段PQ与抛物线恰有1个交点,求m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e7025380bef07df5f62056160e6ea85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cab321e76d7c4c521cdc77d2d27c407.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/18/5fd8d323-e143-4e18-9ba1-f23ac31b4cbf.png?resizew=132)
(1)若点A的坐标为(0,-2),求抛物线的顶点D和点P的坐标;
(2)如图,在(1)的条件下,连接AD,PD,在直线AD下方的抛物线上是否存在点N,满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4742f5710314af955b8dd4e6ddfd7151.png)
(3)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5649ae59934da963ba37a37fdd0977ea.png)
您最近一年使用:0次