名校
解题方法
1 . 已知抛物线
的准线
与圆
相切.
(1)求
的方程;
(2)点
是
上的动点,且
,过点
作圆
的两条切线分别与
交于
两点,求
面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ab7408ffcefcb8e5e1ad4a9c58f1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/560adea7b0d4fbe4131fc41f3fcbd871.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7775aa57ca0e62216f3039ed88dceed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75f702bfe1a376398286f1dc3daf8c67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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2 . 已知函数
,其中
为自然对数的底数.
(1)讨论
的单调性;
(2)若方程
有两个不同的根
.
(i)求
的取值范围;
(ii)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ca6a47d8e2b8ff86f955f401ed47acc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28638f8c054a7bb4d9b46fde330bc76f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(ii)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04133adb6f1562d859510c9771b2e545.png)
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3 . 某大型商场的所有饮料自动售卖机在一天中某种饮料的销售量
(单位:瓶)与天气温度
(单位:
)有很强的相关关系,为能及时给饮料自动售卖机添加该种饮料,该商场对天气温度
和饮料的销售量
进行了数据收集,得到下面的表格:
经分析,可以用
作为
关于
的经验回归方程.
(1)根据表中数据,求
关于
的经验回归方程(结果保留两位小数);
(2)若饮料自动售卖机在一天中不需添加饮料的记1分,需添加饮料的记2分,每台饮料自动售卖机在一天中需添加饮料的概率均为
,在商场的所有饮料自动售卖机中随机抽取3台,记总得分为随机变量
,求
的分布列与数学期望.
参考公式及数据:对于一组数据
,经验回归方程
的斜率和截距的最小二乘估计公式分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de96374212b9a5df820d78d10e7d1291.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![]() | 10 | 15 | 20 | 25 | 30 | 35 | 40 |
![]() | 4 | 16 | 64 | 256 | 2048 | 4096 | 8192 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8029f44c190cd8ced93ed27ba0c18b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)根据表中数据,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)若饮料自动售卖机在一天中不需添加饮料的记1分,需添加饮料的记2分,每台饮料自动售卖机在一天中需添加饮料的概率均为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
参考公式及数据:对于一组数据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eafbe64ec9acf78f3624abbd06d516e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1db6103cb0f1d2bd6b19235d53ee7e98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/666e172502a39172a8379934128101f8.png)
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4 . 已知有穷正项数列
,若将每个项依次围成一圈,满足每一项的平方等于相邻两项平方的乘积,则称该数列可围成一个“HL-Circle”.例如:数列
都可围成“HL-Circle”.
(1)设
,当
时,是否存在
使该数列可围成“HL-Circle”,并说明理由:
(2)若
的各项不全相等,且可围成“HL-Circle”.
(i)求
的取值集合;
(ii)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a2723483435ce9c39acae4032d12e9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ec293ea5d248edf3b66a54f81b5e5f0.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7a1d739890a8951586e23b78b035bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(ii)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9989990eebe2cc2bdddafd5e1d6c0a85.png)
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解题方法
5 . 某社团对男女学生是否喜欢书法进行了一个随机调查,调查的数据如下表所示.
根据调查数据回答:
(1)有
的把握认为性别与是否喜欢书法有关吗?
(2)若该社团某小组有男生4人,女生2人,现从中随机选取2人作为志愿者参加活动,记参加活动的女生人数为
,求
的分布列及期望
.
参考公式:
,其中
.
参考数据:
喜欢书法 | 不喜欢书法 | |
男学生 | 24 | 32 |
女学生 | 16 | 24 |
(1)有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90198de4171921876c6a76f880377f46.png)
(2)若该社团某小组有男生4人,女生2人,现从中随机选取2人作为志愿者参加活动,记参加活动的女生人数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
参考公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2187714e660234f0b72f2b47d3ea685a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356b05e46b10ee51c3e43546d73ec96c.png)
参考数据:
![]() | 0.10 | 0.05 | 0.010 | 0.005 |
![]() | 2.706 | 3.841 | 6.635 | 7.879 |
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解题方法
6 . 在四棱锥
中,
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4cd8ba7eb52e38857830162e770f534.png)
(2)当点
到平面
的距离为
时,求直线
与平面
所成的角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a11deacc70c6f6708ff40204f2eb106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4cd8ba7eb52e38857830162e770f534.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d560a1ffda9ab80a28cd31d29e0637.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
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解题方法
7 . 已知椭圆
的左右焦点分别是
,双曲线
的顶点恰好是
、
,且一条渐近线是
.
(1)求
的方程:
(2)若
上任意一点
(异于顶点),作直线
交
于
,作直线
交
于
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57cae0e98fe3950ee530d84f53f20f7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d2a97987f71835f519b462f5b8f5957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a7f312e75377fc50f516f5e74a2505.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a72fc93dac9c5fab4e24f460d58d09f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85235808a0c9eb3683cbd50eeeaa2daa.png)
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解题方法
8 . 已知函数
.
(1)当
时,求
的极值;
(2)若
恒成立,求实数
的取值范围;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8d0dffe17c575a4feb86c28d97182a7.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b108ab31cc093f03cf48ad65429889e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9c599e8d420006448905acec2b8234.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b245ba1efcc5776b03e8669fa30da8aa.png)
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9 . 已知双曲线
的实轴长为2,离心率为2,右焦点为
,
为
上的一个动点,
(1)若点
在双曲线
右支上,在
轴的负半轴上是否存在定点
.使得
?若存在,求出点
的坐标;若不存在,请说明理由.
(2)过
作圆
的两条切线
,若切线
分别与
相交于另外的两点
、
,证明:
三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bf4fd84818abac17a9d21237ac5ce5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(1)若点
![](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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a6b915acceebea47b3fc9e902755557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdcbaa80a48affc347be5a1bb5079a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50297ad9f7256b4d2efc3462289f18b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50297ad9f7256b4d2efc3462289f18b7.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/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aaf867a11ac4014b51f564478f01f4b.png)
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10 . 已知函数
.
(1)若
是函数
的极值点,求
的值,并求其单调区间;
(2)若函数
在
上仅有2个零点,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e955fbccbc9dafb4b3fd3f293c2c664c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c3d2ff5ee566fcc1b69c65712a661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d698d47dc6421a75df1e698b3f0b4f2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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