来源:
题型:
分类:
解题方法
1 . 在平面直角坐标系
中,点
,点a为动点,以线段
为直径的圆与
轴相切,记a的轨迹为
,直线
交
于另一点b.
(1)求
的方程;
(2)
的外接圆交
于点
(不与o,a,b重合),依次连接o,a,c,b构成凸四边形
,记其面积为
.
(i)证明:
的重心在定直线上;
(ii)求
的取值范围.
![](/uploads/image/squformula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](/uploads/image/squformula/1f449cadb49859b80c31ef1f68bfe81b.png)
![](/uploads/image/squformula/20a541b81584a032f571159ea152c85a.png)
![](/uploads/image/squformula/d053b14c8588eee2acbbe44fc37a6886.png)
![](/uploads/image/squformula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](/uploads/image/squformula/20a541b81584a032f571159ea152c85a.png)
![](/uploads/image/squformula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(1)求
![](/uploads/image/squformula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
(2)
![](/uploads/image/squformula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](/uploads/image/squformula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/3ea16ceca816f7d3d50650af141baf42.png)
![](/uploads/image/squformula/cf231f8f86fb922df4ca0c87f044cec3.png)
(i)证明:
![](/uploads/image/squformula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(ii)求
![](/uploads/image/squformula/cf231f8f86fb922df4ca0c87f044cec3.png)
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解题方法
2 . 已知如图,点
为椭圆
的短轴的两个端点,且
的坐标为,椭圆
的离心率为.
![](/uploads/image/idqe2124/acef4fa0-271a-4bd0-814d-7deb9069cc85.png)
(1)求椭圆
的标准方程;
(2)若直线
不经过椭圆
的中心,且分别交椭圆
与直线
于不同的三点
(点
在线段
上),直线
分别交直线
于点
.求证:四边形
为平行四边形.
![](/uploads/image/squformula/04d468be20b4d43f5de75416de20e8ee.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/43a71fc9c0068109dad1382354570665.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/idqe2124/acef4fa0-271a-4bd0-814d-7deb9069cc85.png)
(1)求椭圆
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](/uploads/image/squformula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/eefa44964db83759aff6fc8dd7ef8f28.png)
![](/uploads/image/squformula/6684304a7537da9517c889c9cbf90a48.png)
![](/uploads/image/squformula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](/uploads/image/squformula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](/uploads/image/squformula/ef49a3ca580a144cc65a609c167facc1.png)
![](/uploads/image/squformula/14d0ff4224f475ab37c6f96d00506f69.png)
![](/uploads/image/squformula/7789a500686c7a73770404ead6af0590.png)
![](/uploads/image/squformula/0d107710e7aff959395ca6f8d23c52c7.png)
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2卷引用:辽宁省沈阳市2023-2024学年高三上学期教学质量监测(一)数学试题
解答题-问答题
|
困难(0.15)
|
名校
解题方法
3 . 已知直线
方程为
,点
,点
到点
的距离与到直线
的距离之比为
,
.
(1)求点
的轨迹
的方程(用
表示);
(2)若斜率为
的动直线
与(1)中轨迹
交于点
,
,其中
,
.点
(
)在轨迹
上,且直线
、
与
轴分别交于
、
两点,若恒有
,求
的值.
![](/uploads/image/squformula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](/uploads/image/squformula/54e756c89b600e37dfb36bb22ef28eb6.png)
![](/uploads/image/squformula/24550b13dbecf7d86c7054250e987274.png)
![](/uploads/image/squformula/ac047e91852b91af639feec23a9598b2.png)
![](/uploads/image/squformula/a0ed1ec316bc54c37c4286c208f55667.png)
![](/uploads/image/squformula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](/uploads/image/squformula/071a7e733d466949ac935b4b8ee8d183.png)
![](/uploads/image/squformula/3e4cc00c283519973f7f8e1274b5c733.png)
(1)求点
![](/uploads/image/squformula/ac047e91852b91af639feec23a9598b2.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/071a7e733d466949ac935b4b8ee8d183.png)
(2)若斜率为
![](/uploads/image/squformula/274a9dc37509f01c2606fb3086a46f4f.png)
![](/uploads/image/squformula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/12a3efb79f35db8448f3391252ab7d4e.png)
![](/uploads/image/squformula/8df332f01628130c084fd46aaca0a4b7.png)
![](/uploads/image/squformula/6270bb08b90f72d5671ab8225f356c43.png)
![](/uploads/image/squformula/c2fe3251e054fe97089806ba7033f802.png)
![](/uploads/image/squformula/891fe97a26ca688e22e5d704432f764b.png)
![](/uploads/image/squformula/0b170470d02c85c1be9a3faff5eca0de.png)
![](/uploads/image/squformula/c5db41a1f31d6baee7c69990811edb9f.png)
![](/uploads/image/squformula/bd33764ff4efddfe11a98a609753715c.png)
![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](/uploads/image/squformula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](/uploads/image/squformula/8455657dde27aabe6adb7b188e031c11.png)
![](/uploads/image/squformula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](/uploads/image/squformula/374e2198b495b86b0f8308d28035a3db.png)
![](/uploads/image/squformula/071a7e733d466949ac935b4b8ee8d183.png)
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解题方法
4 . 给出下列两个定义:
ⅰ.对于函数
,定义域为
,且其在
上是可导的,其导函数定义域也为
,则称该函数是“同定义函数”.
ⅱ.对于一个“同定义函数”
,若有以下性质:
①
;②
,其中,
为两个新的函数,
是
的导函数.
我们将具有其中一个性质的函数
称之为“单向导函数”,将两个性质都具有的函数
称之为“双向导函数”,将称之为“自导函数”.
(1)判断下列两个函数是“单向导函数”,或者“双向导函数”,说明理由.如果具有性质①,则写出其对应的“自导函数”.ⅰ.
;ⅱ.
.
(2)给出两个命题
,
,判断命题
是
的什么条件,证明你的结论.
:
是“双向导函数”且其“自导函数”为常值函数,
:
.
(3)已知函数
.
①若
的“自导函数”是
,试求
的取值范围.
②若
,且定义
,若对任意
,
,不等式
恒成立,求
的取值范围.
ⅰ.对于函数
![](/uploads/image/squformula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](/uploads/image/squformula/8455657dde27aabe6adb7b188e031c11.png)
![](/uploads/image/squformula/8455657dde27aabe6adb7b188e031c11.png)
![](/uploads/image/squformula/8455657dde27aabe6adb7b188e031c11.png)
ⅱ.对于一个“同定义函数”
![](/uploads/image/squformula/51c530f4b7491b95acb8ce3eef9aa09d.png)
①
![](/uploads/image/squformula/88e0de48d522d75246890e4c645ab772.png)
![](/uploads/image/squformula/3cc3b32ebc2f47448b01793b08d362c3.png)
![](/uploads/image/squformula/e915b67f8f747698b8b46d37bc453667.png)
![](/uploads/image/squformula/851c68ef2e0703706f3b528daa902eb8.png)
![](/uploads/image/squformula/51c530f4b7491b95acb8ce3eef9aa09d.png)
我们将具有其中一个性质的函数
![](/uploads/image/squformula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](/uploads/image/squformula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(1)判断下列两个函数是“单向导函数”,或者“双向导函数”,说明理由.如果具有性质①,则写出其对应的“自导函数”.ⅰ.
![](/uploads/image/squformula/457eb5e0000350b102d387a80cf3476b.png)
![](/uploads/image/squformula/64263fe2ca48e694c87496d61e63fb9f.png)
(2)给出两个命题
![](/uploads/image/squformula/b1010846eeec6c9da29640f5aa3f8738.png)
![](/uploads/image/squformula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](/uploads/image/squformula/b1010846eeec6c9da29640f5aa3f8738.png)
![](/uploads/image/squformula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](/uploads/image/squformula/b1010846eeec6c9da29640f5aa3f8738.png)
![](/uploads/image/squformula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](/uploads/image/squformula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](/uploads/image/squformula/b4dde8c4dc004c420d9a8d70e42a4326.png)
(3)已知函数
![](/uploads/image/squformula/5ab4e6f11638cab0d6cfbb3b3fe398bd.png)
①若
![](/uploads/image/squformula/0eb7df298a9364b36e079a61caec815c.png)
![](/uploads/image/squformula/d77f5191798242b7b9b88a75e17e4425.png)
![](/uploads/image/squformula/0a6936d370d6a238a608ca56f87198de.png)
②若
![](/uploads/image/squformula/48adb8a59b5c02fad5eada1b35171cf3.png)
![](/uploads/image/squformula/49ce204f241f7d34d051ede4f489fa62.png)
![](/uploads/image/squformula/3b7319db0b8ed08f2cdf97fa404e1f85.png)
![](/uploads/image/squformula/b7443e9644eb8e20b5b52721f37d63a2.png)
![](/uploads/image/squformula/186b9f08f5b59b7e038eede1e4f90bab.png)
![](/uploads/image/squformula/071a7e733d466949ac935b4b8ee8d183.png)
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2卷引用:上海市普陀区桃浦中学2022-2023学年高二上学期12月月考数学试题
5 . 若有穷数列
满足:
,则称此数列具有性质
.
(1)若数列
具有性质
,求
的值;
(2)设数列a具有性质
,且
为奇数,当
时,存在正整数,使得
,求证:数列a为等差数列;
(3)把具有性质
,且满足
(
为常数)的数列a构成的集合记作
.求出所有的
,使得对任意给定的
,当数列
时,数列a中一定有相同的两项,即存在
.
![](/uploads/image/squformula/ef94592b70bea840c747393959c71b39.png)
![](/uploads/image/squformula/c735a110f4cf68dea9133c78e205b43f.png)
![](/uploads/image/squformula/bf404363fe0f057007b8e8d90a775d90.png)
(1)若数列
![](/uploads/image/squformula/030ef2d631bb39945bb752932146364b.png)
![](/uploads/image/squformula/bf404363fe0f057007b8e8d90a775d90.png)
![](/uploads/image/squformula/cc9927f218d1b9cd9d7a8b979da6c669.png)
(2)设数列a具有性质
![](/uploads/image/squformula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](/uploads/image/squformula/88e6153f9e3bfe84d3a61f388c7fa2b8.png)
![](/uploads/image/squformula/cd00e20f967cb2bdce939165abd38440.png)
![](/uploads/image/squformula/c1c87acdb6ce8286ea7d256b96801507.png)
(3)把具有性质
![](/uploads/image/squformula/bf404363fe0f057007b8e8d90a775d90.png)
![](/uploads/image/squformula/b835321cd8b7cf192f9e0af0d2f1239b.png)
![](/uploads/image/squformula/fe83b9b62b3511e37f9726042964db5e.png)
![](/uploads/image/squformula/e57696e6509aebe3a8444525b702050e.png)
![](/uploads/image/squformula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](/uploads/image/squformula/1ce021a66a6856d5078186cffe13f2c6.png)
![](/uploads/image/squformula/80baa977f2523242a5a3f9a2ac364ad4.png)
![](/uploads/image/squformula/1b38760e49cb2b3b7bf23410fc189e93.png)
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1卷引用:北京市东城区2024届高三上学期期末统一检测数学试题
解题方法
6 . 设函数
,对于下列四个判断:
①函数
的一个周期为
;
②函数
的值域是
;
③函数
的图象上存在点
,使得其到点
的距离为;
④当
时,函数
的图象与直线
有且仅有一个公共点.
正确的判断是( )
![](/uploads/image/squformula/3c2f37b025f77be6c1cb307e3d5b4683.png)
①函数
![](/uploads/image/squformula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](/uploads/image/squformula/70f5389990c3a0c5373f3bd9fb2454c9.png)
②函数
![](/uploads/image/squformula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](/uploads/image/squformula/2e270ab87afa5958e8d226f535be2474.png)
③函数
![](/uploads/image/squformula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](/uploads/image/squformula/aee82283f06cedef32eb15b87964f5d2.png)
![](/uploads/image/squformula/1d7a999c36de5c9a9ce876a4a56fa34c.png)
④当
![](/uploads/image/squformula/31aba8ca22579a6d5eed632aecff4548.png)
![](/uploads/image/squformula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](/uploads/image/squformula/107babba45f110012183dc4dc54490f7.png)
正确的判断是( )
a.① | b.② | c.③ | d.④ |
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1卷引用:北京市东城区2024届高三上学期期末统一检测数学试题
解答题-证明题
|
困难(0.15)
|
解题方法
7 . 离散对数在密码学中有重要的应用.设
是素数,集合
,若
,记
为
除以
的余数,
为
除以
的余数;设
,
两两不同,若
,则称
是以
为底
的离散对数,记为
.
(1)若
,求
;
(2)对
,记
为
除以
的余数(当
能被
整除时,
).证明:
,其中
;
(3)已知
.对
,令
.证明:
.
![](/uploads/image/squformula/b1010846eeec6c9da29640f5aa3f8738.png)
![](/uploads/image/squformula/f05bea470ae14b90937f6f71dc9a6242.png)
![](/uploads/image/squformula/99b2b0dcbc27df9950b26028e46f6c17.png)
![](/uploads/image/squformula/1e5865fd0fb7c35e8a4a1d311163290b.png)
![](/uploads/image/squformula/7cbe6ebc6c1d1a214f5ca478ae666cdb.png)
![](/uploads/image/squformula/b1010846eeec6c9da29640f5aa3f8738.png)
![](/uploads/image/squformula/b67a1f88ae28ecdb67c7f9c4ae61481b.png)
![](/uploads/image/squformula/dae890dd5b6300cf23b4905e86410317.png)
![](/uploads/image/squformula/b1010846eeec6c9da29640f5aa3f8738.png)
![](/uploads/image/squformula/ff99d1615f90ff71b56ca1dfebd626d1.png)
![](/uploads/image/squformula/420a12638f77a27c696f63ff946e8684.png)
![](/uploads/image/squformula/d2b0087ea124b6fd98fbbcb9bc4c2e09.png)
![](/uploads/image/squformula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](/uploads/image/squformula/0a6936d370d6a238a608ca56f87198de.png)
![](/uploads/image/squformula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](/uploads/image/squformula/9ce071bd0d6fa72ff4ba4e72d810d11f.png)
(1)若
![](/uploads/image/squformula/ac54185ed8bb89c774ceb685408156c4.png)
![](/uploads/image/squformula/ca7b54c31c5ab3831f260012758ffa12.png)
(2)对
![](/uploads/image/squformula/099d389a1c0e5877350e62c52c4a724c.png)
![](/uploads/image/squformula/28ab2ad5d8b72e3f26bef4be0697ec70.png)
![](/uploads/image/squformula/3b6b09b60bf1bf8403c49bc17e365cd8.png)
![](/uploads/image/squformula/77ebb2233e8492cf61fe9f9bc68af470.png)
![](/uploads/image/squformula/3b6b09b60bf1bf8403c49bc17e365cd8.png)
![](/uploads/image/squformula/77ebb2233e8492cf61fe9f9bc68af470.png)
![](/uploads/image/squformula/34fc26e532b65641a53eaa7e127aa683.png)
![](/uploads/image/squformula/b4d45dbe0a914249371aed3641515123.png)
![](/uploads/image/squformula/53ace23b21d7b119ad7ac5cf877c19f0.png)
(3)已知
![](/uploads/image/squformula/9ce071bd0d6fa72ff4ba4e72d810d11f.png)
![](/uploads/image/squformula/2793be26b839ae9f8f83cf2b5a597cd3.png)
![](/uploads/image/squformula/c1a6740a4f2378965bc019bc6aacd44a.png)
![](/uploads/image/squformula/f278b4fd6ed264265e3ccfac4ab7ef02.png)
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3卷引用:2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题
(已下线)2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题变式题16-19
2024·全国·模拟预测
名校
解题方法
8 . 如果有且仅有两条不同的直线与函数
的图象均相切,那么称这两个函数
为“
函数组”.
(1)判断函数
与
是否为“
函数组”,其中
为自然对数的底数,并说明理由;
(2)已知函数
与
为“
函数组”,求实数
的取值范围.
![](/uploads/image/squformula/f8fd1e808e015f4cb43d2e3a0529ac6a.png)
![](/uploads/image/squformula/f8fd1e808e015f4cb43d2e3a0529ac6a.png)
![](/uploads/image/squformula/0c88d9142df6ba8e43c1a93bd04a1362.png)
(1)判断函数
![](/uploads/image/squformula/4bea75e6fa8f587c7afff0ffb563b921.png)
![](/uploads/image/squformula/5db192285632d1991b4ee7a003a52205.png)
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(2)已知函数
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9 . 学习几何体结构素描是学习素描的重要一步.如图所示,这是一个用来练习几何体结构素描的石膏几何体,它是由一个圆柱
和一个正三棱锥
穿插而成的对称组合体.棱
和面
与圆柱侧而相切,点
是棱
与圆柱侧而的切点.直线
分别与面
,面交于点
,圆柱
在面
,面上分别截得椭圆
.在平面
和平面中,椭圆
上分别有两组不重合的两点
和
(图中未画出).且满足关系
.已知三棱锥
的外接球表面积为
,圆柱的底面直径为
,请问平面
,平面上是否分别存在点
,使得对于满足
的直线
分别恒过定点
.若存在,试求
和
夹角的余弦值:若不存在,请说明理由.
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10 . 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.
![](/uploads/image/squformula/ac56300140ed9e27f8dff86ef1eaea0c.png)
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①任意
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②存在
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(1)判断数表
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(2)是否存在数表
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(3)若存在数表
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1卷引用: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
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