名校
解题方法
1 . 元荡湖位于长三角一体化示范区内,2018年青浦㨦手吴江启动实施了元荡生态岸线整治,2023年8月实现元荡青浦段岸线全线贯通.如图,为拓展旅游业务,现准备在元荡湖边建造一个观景台
,已知射线
,
为元荡湖两边夹角为
的公路(长度均超过2千米),在两条公路
,
上分别设立游客接送点
,
,从观景台
到
,
建造两条观光线路
,
,测得
千米,
千米.
的长度;
(2)若
,求两条观光线路
与
之和的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/810fee760e06028fb7d37e65db79bfa7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28c65312db84bbe37e0e1b400bf78c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/add60595ecea5d00ff1fbb2a5409be73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
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2 . 2位教师和3名学生站成一排,要求2位教师不相邻,则不同排法的种数为______ .
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解题方法
3 . 2024年4月30日17时46分,神舟十七号返回舱成功着陆,返回舱是宇航员返回地球的座舱.返回舱的轴截面可近似看作是由半个椭圆 和一段圆弧 组成的“果圆”.如图,在平面直角坐标系中,某“果圆”中圆弧经过椭圆的一个焦点
和短轴的两个顶点
与
.
(2)直线
交该“果圆”于A、B两点,求弦AB的长度(精确到0.01).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad7345600383230ab4325bd6310813f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c0573e2af8a0dc8c6a1c0af067a324f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ee8c50793afd59e6ab4a2be5a877759.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
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解题方法
4 . 已知双曲线
的图像经过点
,点
分别是双曲线
的左顶点和右焦点.设过
的直线
交
的右支于
两点,其中点
在第一象限.
的标准方程;
(2)若直线
分别交直线
于
两点,证明:
为定值;
(3)是否存在常数
,使得
恒成立?若存在,求出
的值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32bd59dc2cf63dbd2be894e6131604da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271057b8074eaee352dad52d3a12543c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7d5a99dfaec60d88ad496c1fd184fcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.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/a5b1b15a4605fce993cb13aefbf40360.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53d223a09481973825c5e87ce2f5c03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650820d7bed48ed67a2869ad8c65ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6670479a0083dd2dfd5ad55b47b1ab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f99cc82a7536f92bff713a918415e9.png)
(3)是否存在常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bfe8d34e91e1e98891527ad92a33887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
5 . 在棱长为1的正方体
中,若点P是棱上一点,则满足
的点P的个数为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce6eda9f298400a2f3beb01e935238ea.png)
您最近一年使用:0次
6 . 已知
,函数
,其中
.
(1)若
,
,写出函数
图像的一条水平切线的方程;
(2)若
,
,且满足
,证明:
;
(3)若存在
,使得函数
有唯一零点,求实数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7569cd7e9b31ad838230133b9bc8314.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34e050fbca6ba0d5390263131690a41.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aed39f5aca78934fb383402433fe549.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b140e221ddf537b8964fff8557cca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7562f4853fe96f2054d3f03e4eb207b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73feb6b2d30ca5becd5711582d6ba008.png)
(3)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fcb5d3be7955ac79a959abad606a47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
您最近一年使用:0次
解题方法
7 . 若函数
在区间
上是严格单调函数,则实数
的取值范围为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aed2e979fbe5ccb29a91f649ee0fe12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63f7640aeb7d6ce68904979bd2ce2887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
8 . 如果从甲口袋中摸出一个红球的概率是
,从乙口袋中摸出一个红球的概率是
,现分别从甲乙口袋中各摸出1个球,则2个球都是红球的概率是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
您最近一年使用:0次
9 . 在
的展开式中,含
项的系数为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/005552a7c874c14dd3c1abf924b223a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6a94336d7eb3eb790a3f3521c46401.png)
您最近一年使用:0次
10 . 如左下图1,
是水平放置的矩形,
,将矩形
沿对角线
折起,使得平面
平面
,如右下图2.设O是
的中点,D是
的中点.
与平面
所成角的大小;
(2)连接
,设平面
与平面
的交线为直线l,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c412044acac1a6da3581ed02c979369f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8520a21b909d04f763d0f61dd74bc158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c412044acac1a6da3581ed02c979369f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/566866c39352ba27f4179ac1f3a20c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/787ac5e13622afab5e9f8603afe42356.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80672dda9430cb42b3136bcb1b67bbad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1a2e4e3414aad1eba737fc135d91687.png)
(2)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30513ea48bc1ef3ae78adac83d894f14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/853669d8b1f1fe103cbb35cdb94b758a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/840798a31aba0783f96584e0ad7c0d2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8bdf1dfc5951115bc0cf7fc34add48a.png)
您最近一年使用:0次