名校
1 . 已知函数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5edbbc3f9dcc564f13325250a361235.png)
(1)求曲线
在点
处的切线方程;
(2)求证:函数
的图象位于直线
的下方;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5edbbc3f9dcc564f13325250a361235.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
您最近一年使用:0次
2 . 已知椭圆
:
(
)的半长轴的长度与焦距相等,且过焦点且与
轴垂直的直线被椭圆截得的弦长为3.
(1)求椭圆
的方程;
(2)已知直线
:
与椭圆
交于
,
两点,过点
的直线交椭圆
于
,
两点(
在靠近
的一侧)
(ⅰ)求
的取值范围;
(ⅱ)在直线
上是否存在一定点
,使
恒成立?若存在,求出
点坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7aea48c44781a844b5c19191f70f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0c4c098615c6bc7e6dcf72e5b5201a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)求椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c34517f479fb08f6096d2fb0362f3ad0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0accaaa66de4dfd9ed8257fa942c2cff.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/9c7f6a03826a5ad351c1f7ca553a6945.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(ⅰ)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591ccce336af06e51e7f4887c287b80f.png)
(ⅱ)在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c34517f479fb08f6096d2fb0362f3ad0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3d14da8892d33751940b69a790bc5c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
您最近一年使用:0次
名校
解题方法
3 . 已知函数
,
.(注:
是自然对数的底数)
(1)若
无极值点,求实数
的取值范围;
(2)当
时,
恒成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94c33ef652f63f1b43282aedbf71311e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5f7f23e7f20dd8bc65a4967cd306782.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797bbd18359c9a29842b39109b3a0aac.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a01be1031ca8543348a8dddc833db2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
名校
4 . 泊松分布是一种重要的离散型分布,用于描述稀有事件的发生情况.如果随机变量
的所有可能取值为0,1,2…,且
,
其中
,则称
服从泊松分布,记作
.
(1)设
,且
,求
;
(2)已知当
,
时,可以用泊松分布
近似二项分布
,即对于
,
,当
不太大时,有
.
(ⅰ)已知甲地区共有100000户居民,每户居民每天有0.00010的概率需要一名水电工.试估计某天需要至少2名水电工的概率;
(ⅱ)在(ⅰ)的基础上,已知乙地区共有200000户居民,每户居民每天有0.00004的概率需要一名水电工.试估计某天两个地区一起至少需要3名水电工的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d9506436f1f7db2b7c20a84f9a5f34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4988011c1e60a6256c25c3bdff4bd352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be362dec96173f246ff747264007817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce3a0f8d6451376d85c0f432c74faf33.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce3a0f8d6451376d85c0f432c74faf33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a80cdc7e5d4067d00dff0a0b347b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e39326381d0fbd83c8156c3b33e74eb.png)
(2)已知当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c4fe5a95acf4db3241c6cba652e1589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9529cc68a2f219ea5e6f467af4b6e8d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9edd73921106b5c092f6b685ada1f5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3821cada1948964a9741005833f52d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/989602dd198d2cb52fc1875921d56ad8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/761ddcffd715d75da2c739fe67fa3a96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e6c010e17368e545d63217810ffc8fb.png)
(ⅰ)已知甲地区共有100000户居民,每户居民每天有0.00010的概率需要一名水电工.试估计某天需要至少2名水电工的概率;
(ⅱ)在(ⅰ)的基础上,已知乙地区共有200000户居民,每户居民每天有0.00004的概率需要一名水电工.试估计某天两个地区一起至少需要3名水电工的概率.
您最近一年使用:0次
名校
解题方法
5 . 在等差数列
(
)中,
,
.
(1)求
的通项公式;
(2)若
,数列的
前
项和为
,证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2771c5f04582c545e0f9afc8a2cb9597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cff0abcfc9ed6c1afdd6d4ca61a19897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a38c5d56a437f36bde9ce723233e0063.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856314ac1667503a2dc071f5a6018424.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47bcba5b14edadd8fb6a18bf6efdb54c.png)
您最近一年使用:0次
名校
解题方法
6 . 如图1,在矩形
中,
,
,将
沿矩形的对角线
进行翻折,得到如图2所示的三棱锥
,且
.
的长;
(2)点
满足
,求
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ffc2817fa590affb5a760a25dc65308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a77e3c1c236141d6118429fade0a9b9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db8679b329b17bc65022bd1a1418632b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
您最近一年使用:0次
名校
7 . 已知函数
.
(1)当
时,求
在
处的切线方程;
(2)若函数
在
上单调递增,求实数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662704fdd021f1cc3c239cb0362b4017.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5828873f8369183faf71181cda5b61d2.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
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解题方法
8 . 已知抛物线
的焦点为
,
为
上任意一点,且
的最小值为1.
(1)求抛物线
的方程;
(2)已知
为平面上一动点,且过
能向
作两条切线,切点为
,记直线
的斜率分别为
,且满足
.
①求点
的轨迹方程;
②试探究:是否存在一个圆心为
,半径为1的圆,使得过
可以作圆
的两条切线
,切线
分别交抛物线
于不同的两点
和点
,且
为定值?若存在,求圆
的方程,不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9158f21b372fd0390fab040ad65c586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/493875c56db5cf73d6417923c6cecb9d.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4099122dfb15704bb265197e1bc74e49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dca1726d463bd741c904abd9b6589056.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c30af83767217e6497d2ef64b9793b6.png)
①求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
②试探究:是否存在一个圆心为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2186fca32ea8c51488e6f181fd23366e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0abef37676985bd2340ccfaae2c70192.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e47d15440b1d85569fe5f2298a3536ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a594cd950e1ae2eb2c19eccf309f71d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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9 . 从抛物线
上各点向
轴作垂线段,垂线段中点的轨迹为
.
(1)求
的轨迹方程;
(2)
是
上的三点,过三点的三条切线分别两两交于点
,
①若
,求
的值;
②证明:三角形
与三角形
的面积之比为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ac62b1ade07205ae2693ec1ab135def.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/927456b0989846a2f1573844bbaa2105.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b05cc4297b34393d18222e7299e8f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb4638124355b7be66231a604f667f0c.png)
②证明:三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134ef0b1a2669a09f05bd4dc2496f706.png)
您最近一年使用:0次
10 . 第二十五届中国国际高新技术成果交易会(简称“高交会”)在深圳闭幕.会展展出了国产全球首架电动垂直起降载人飞碟.观察它的外观造型,我们会被其优美的曲线折服.现代产品外观特别讲究线条感,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
上的曲线段
,其弧长为
,当动点从
沿曲线段
运动到
点时,
点的切线
也随着转动到
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
越接近
,即
越小,
就越能精确刻画曲线
在点
处的弯曲程度,因此定义
(若极限存在)为曲线
在点
处的曲率.(其中
,
分别表示
在点
处的一阶、二阶导数)
的焦点到准线的距离为3,则在该抛物线上点
处的曲率是多少?
(2)若函数
,不等式
对于
恒成立,求
的取值范围;
(3)若动点
的切线沿曲线
运动至点
处的切线,点
的切线与
轴的交点为
.若
,
,
是数列
的前
项和,证明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d0dd437ad522066cc7d4b7d208e2866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/505d83f4d34a8cd385577a6ce93a4b11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.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/aea61ddc41f927684c6dfaacdd7f8e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0636a11a086df66133bd50e43481a546.png)
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(2)若函数
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(3)若动点
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