1 . 已知椭圆
短轴长为2,椭圆
上一点
到
距离的最大值为3.
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
的取值范围;
(2)当椭圆
的离心率达到最大时,过原点
斜率为
的直线
与
交于
两点,
分别与椭圆
的另一个交点为
.
①是否存在实数
,使得
的斜率
等于
?若存在,求出
的值;若不存在,说明理由;
②记
与
交于点
,求线段
长度的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e5578ca83f5bd5c285994061b9c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9059420851d219dba2189975f8eb5178.png)
![](https://img.xkw.com/dksih/QBM/2024/5/26/3504894981390336/3520242638839808/STEM/32c129353263403684caca44141869ef.png?resizew=151)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7708c156249bb475564027f73313fa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2ec2e454afe8452a6f1714a986aa38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235557c7e459c7100787e436826c198e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f963b16dcebd808b41842febc329a6.png)
①是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7e08d5c04f0431fb57b33a01717b599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4b2e1a496f0df7918efed76a51f34c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
②记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
您最近一年使用:0次
昨日更新
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2卷引用:江苏省扬州市2024届高三下学期高考考前调研测试数学试题
2 . 已知双曲线
的左、右焦点分别为
,点
在双曲线
上,且直线
的倾斜角是直线
的倾斜角的2倍.
(1)求双曲线
的标准方程;
(2)若
,
是双曲线
上的两个动点,且恒有
,是否存在定圆与直线
相切?若存在,求出定圆的方程,若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40f98254a6193566587a70c7d95fdabf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/786ab40492d54c5a17fd385df50f0de4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/183b6a0cef4256c9696a5bca31053da5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b67528f875a6d4bac8bbf784f7b66a0.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f788fb0059b7356dc6c7811f46057e66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
您最近一年使用:0次
解题方法
3 . 已知函数
.
(1)若曲线
在
处的切线的方程为
,求实数
的值;
(2)若函数
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987ac6a091c08c22d0c1988afb516ff4.png)
(1)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/707ea658f3a9359f5740d5aab48f7948.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec25d99e167935341292c69465a4cc5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb3229b5416d45ec449f2bd1983d29d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
4 . 在数列
中,
.
(1)求数列
的通项公式;
(2)已知数列
满足
;
①求证:数列
是等差数列;
②若
,设数列
的前n项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76e09df3126ec131431e8dda971dccf9.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d245c2ee1b810663233fd252f33b61d2.png)
①求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627e48c5ab76f5d1874c57a40d32d89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ad7169f1ea7b8297330acae740ec3ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db770bbfced53ba0173863c0d0849bbc.png)
您最近一年使用:0次
解题方法
5 . 已知函数
.
(1)求函数
的极值;
(2)函数
.
①讨论函数
的单调性;
②函数
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68117fb4d78db5e23066bc0ddb54f51d.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f93343f8df0cb98770feade165543cf.png)
(2)函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54dfe135d195f2aee1cc3335117c23d8.png)
①讨论函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
②函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c6f2c7d17856f344d8214f7a5043b18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
解题方法
6 . 已知函数
.
(1)
时,求
的零点个数;
(2)若
恒成立,求实数
的最大值;
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cf8a50553b1b9040eb5fd8149602f0b.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f3c2be7482719651bcf491949681e05.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43920f5171ed31db2520ef00e4c5fc24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e7c3a84d5c3ab338a6b95d1e4a7ce8a.png)
您最近一年使用:0次
解题方法
7 . 在数列
的第
项与第
项之间插入
个1,称为变换
.数列
通过变换
所得数列记为
,数列
通过变换
所得数列记为
,以此类推,数列
通过变换
所得数列记为
(其中
).
(1)已知等比数列
的首项为1,项数为
,其前
项和为
,若
,求数列
的项数;
(2)若数列
的项数为3,
的项数记为
.
①当
时,试用
表示
;
②求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd6f81b87ebc6a949329520a6e42ed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd6f81b87ebc6a949329520a6e42ed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b536225194ade4db8e99df253041e034.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/819f6ef3827f73ee7dea7883be3bdfdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3788785078f4cbc29eb4b3c202d1587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(1)已知等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1752474698cd5466dd180df0a00ba9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e5577ef4b9b5764af5e21eef60ebca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fd6f81b87ebc6a949329520a6e42ed8.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3788785078f4cbc29eb4b3c202d1587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e9d41db805f5235ec4b69e436690222.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/147a33b715da5466c2d4f9b18cb925d7.png)
您最近一年使用:0次
解题方法
8 . 已知抛物线
,动直线
与抛物线
交于
,
两点,分别过点
、点
作抛物线
的切线
和
,直线
与
轴交于点
,直线
与
轴交于点
,
和
相交于点
.当点
为
时,
的外接圆的面积是
.
(1)求抛物线
的方程;
(2)若直线
的方程是
,点
是抛物线
上在
,
两点之间的动点(异于点
,
),求
的取值范围;
(3)设
为抛物线
的焦点,证明:若
恒成立,则直线
过定点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf92a1ba410263d4f68b7e0432b19aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf5f0df2579e567038baa206cb0b016f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed283a253b61df01f2a1cdc0cd8003f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b3a91ccf6028608cd03df7072f6536.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/912bae66a2b7cc1caa4cd0333796ec41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59880e470359d8e9faf6ae5ce155cf2a.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32a07da1731feb22da90850241b3580f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
您最近一年使用:0次
解题方法
9 . 已知圆
,直线
,直线
和圆交于A,B两点,过A,B分别做直线
的垂线,垂足为C,D.
(1)求实数b的取值范围;
(2)若
,求四边形ABDC的面积取最大值时,对应实数
的值;
(3)若直线AD和直线BC交于点
,问是否存在实数
,使得点
在一条平行于
轴的直线上?若存在,求出实数
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79382ba44ba669b5d43fdd5427adf16c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13b673b7961af5c4b51e67d6ffe00912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa941aa5d428f5d78ec3f281464f826.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
(1)求实数b的取值范围;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdbf6506e9de40da0f3c51b81b35a901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(3)若直线AD和直线BC交于点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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10 . 设
,函数
.
(1)当
时,求过点
且与曲线
相切的直线方程:
(2)
是函数
的两个极值点,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41cbf8bd2bb0cbf80c2c2aa450045fa1.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953bfeb398bab2b2ba61b3e6bf0a22e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb7050bb45d4e623dcbdf0be7a00e09f.png)
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