1 . 已知
为坐标原点,
,
.
(1)判断
的形状,并给予证明;
(2)若
,求证:
、
、
三点共线;
(3)若
是线段
上靠近点
的四等分点,求
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51116e96f4c35d90677e91e0aa914111.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9612a17c77d5d6ded6123e12f9c8914.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4bdb30cad5418d2b634e697d2d8e46e.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)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
您最近一年使用:0次
名校
2 . ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b57ed792fb63c756aa4372e501f73cf.png)
(1)证明:
存在唯一的零点
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ce4af9a3c5f7987ddef4988ae0a57.png)
(2)若
的零点记为
,设
,求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b57ed792fb63c756aa4372e501f73cf.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ce4af9a3c5f7987ddef4988ae0a57.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e5ad7a134838f6ee246e606a625f7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eb3c14b2ab08a915682646f3377b7b4.png)
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2023-10-01更新
|
159次组卷
|
3卷引用:专题04 指数函数与对数函数2-2024年高一数学寒假作业单元合订本
(已下线)专题04 指数函数与对数函数2-2024年高一数学寒假作业单元合订本福建省漳州实验高级中学2022-2023学年高一创新班上学期期中考试数学试题福建省厦门市厦门二中2023-2024学年高一上学期12月月考数学试题
3 . 证明:
(1)
.
(2)已知
,
,求证:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1fc2d308d990e5771657c9f56a0936b.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fbdc8633a22f3b9fb3a789d3818657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45a173784888adf2946382fa093ba53a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/129f958b08c51df454111d41c6db204f.png)
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名校
4 . 我们知道,“有了运算,向量的力量无限”.实际上,通过向量运算证明某些几何图形的性质比平面几何的“从图形的已知性质推出待证的性质”简便多了.下面请用向量的方法证明“三角形的三条高交于一点”.已知
,
,
是
的三条高,求证:
,
,
相交于一点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
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2021-06-24更新
|
260次组卷
|
5卷引用:专题26 平面向量应用
(已下线)专题26 平面向量应用(已下线)6.4.1 平面几何中的向量方法——课后作业(提升版)江苏省苏州实验中学、木渎中学、太仓中学2020-2021学年高一下学期5月联考数学试题江苏省苏州实验中学2020-2021学年高一下学期5月学情调研数学试题(已下线)专题6.3 平面向量的应用(练)- 2022年高考数学一轮复习讲练测(新教材新高考)
名校
解题方法
5 . 在直三棱柱
中,
,侧棱长为3,侧面积为
.
的体积;
(2)若点D、E分别在三棱柱的棱
上,且
,线段
的延长线与平面
交于
三点,证明:
共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b42eeaa3e80a1e0f298a175bcc0e45e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afaf8822d03a1bc2fa3d8700082e3511.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b083008d31d3f029aa40dbf2a6a1d3.png)
(2)若点D、E分别在三棱柱的棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac3d169c28e3a2cdb9abf322244609d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1699518dd0e565c44cfe7c6318aff824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9e58ac8c84d836aa006a70b20773d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7a8f3a13cb258c61e2a221c2bf09979.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7a8f3a13cb258c61e2a221c2bf09979.png)
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解题方法
6 . 为研究中国工业机器人产量和销量的变化规律,收集得到了
年工业机器人的产量和销量数据,如下表所示.
记
年工业机器人产量的中位数为
,销量的中位数为
.定义产销率为“
”.
(1)从
年中随机取
年,求工业机器人的产销率大于
的概率;
(2)从
年这
年中随机取
年,这
年中有
年工业机器人的产量不小于
,有
年工业机器人的销量不小于
.记
,求
的分布列和数学期望
;
(3)从哪年开始的连续
年中随机取
年,工业机器人的产销率超过
的概率最小.结论不要求证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8c9ab8d7876fbe1160ed976495d7dce.png)
年份 | |||||||||
产量万台 | |||||||||
销量万台 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8c9ab8d7876fbe1160ed976495d7dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88541c92761f06f87a4774bcfe2ff0df.png)
(1)从
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8c9ab8d7876fbe1160ed976495d7dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef9c407a9e79f3612690b9cff43a08e0.png)
(2)从
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4a24d6356957767542cb75b94f3ef8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/678d2b93237d071c6c13e6055fb68497.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60e1ba1988005e5fbf117f35762ff53.png)
(3)从哪年开始的连续
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65d9d0d66a7f8fc34082cf8c45f64839.png)
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2024·全国·模拟预测
名校
7 . 美国数学史家、穆伦堡学院名誉数学教授威廉・邓纳姆在1994年出版的The Mathematical Universe一书中写道:“相比之下,数学家达到的终极优雅是所谓的‘无言的证明’,在这样的证明中一个极好的令人信服的图示就传达了证明,甚至不需要任何解释.很难比它更优雅了.”如图所示正是数学家所达到的“终极优雅”,该图(
为矩形)完美地展示并证明了正弦和余弦的二倍角公式,则可推导出的正确选项为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2024-04-28更新
|
256次组卷
|
3卷引用:2024届新高考数学原创卷3
8 . 进位制是人们为了计数和计算方便而约定的记数方式,通常“满二进一,就是二进制;满八进一,就是八进制;满十进一,就是十进制……;满几进一,就是几进制”.
我们研究的正整数通常是十进制的数,因此,将正整数
的各位上的数字分别记为
,则
表示为关于10的
次多项式,即
,其中
,
,记为
,简记为
.
随着计算机的蓬勃发展,表示整数除了运用十进制外,还常常运用二进制、八进制等等.更一般地,我们可类似给出
进制数定义.
进制数的定义:给出一个正整数
,可将任意一个正整数
,其各位上的数字分别记为
,则
唯一表示为下列形式:
,其中
,
,并简记为
.
进而,给出一个正整数
,可将小数
表示为下列形式:
,其中
,
,并简记为
.
(1)设
在三进制数下可以表示为
,
在十进制数下可以表示为
,试分别将
转化成十进制数,
转化成二进制数;
(2)已知数列
的前
项和为
,且满足
,
,数列
满足,当
时,
;
①当
时,求数列
的通项公式;
②证明:当
时,
.
我们研究的正整数通常是十进制的数,因此,将正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23dd0a10a73674c8a47420d7e91da0cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432851e0d0b7a2924da29b9cc5ca1706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05c95c671f517c02f1deccd2174f7465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f856d17cdd7499eeea956845fe51ad2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ceec6993cc9e39a5d5e6097d6447c6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d5a7b02c9d7bd9e2910849ab591fc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856f717993779ce676195ed3c8074e15.png)
随着计算机的蓬勃发展,表示整数除了运用十进制外,还常常运用二进制、八进制等等.更一般地,我们可类似给出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50e3273deabbf8c7d700d340b71ca2eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c89fe2197a7044536c5c3b52789636.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3f285903ef04eb7a1d2071cb7197521.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c22a19eeff39dff098ea766cf74d3c69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53d2aae1ce527125960613e78e1bb501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2a4ce3547aff6aebdd145d11c566fe.png)
进而,给出一个正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50e3273deabbf8c7d700d340b71ca2eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20277d6743703a3710d6ceb8dc869cef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31ef1766237596595c550af4600c370e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8e6d4140fcd9583fe03f91701ae083f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74304131e01d0e9e68bd8a566b30d529.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99dc317d03cfed7687fad0448c298f9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df1d9b712b639c8b6809c9f3ae03706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19d9abc918557b3bd56cb30dfd20b91f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df1d9b712b639c8b6809c9f3ae03706.png)
(2)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb451756eab7b736e0c576e03f6243b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/931b93e6ee1145e185621ae08dbf170f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26663935b7cb86ca81ad2aa73d49e26a.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/931b93e6ee1145e185621ae08dbf170f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
②证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/931b93e6ee1145e185621ae08dbf170f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac363332016ae0fa59a5e4a1d41e2a0b.png)
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23-24高一下·全国·课前预习
9 . 基本事实4
文字语言 | 平行于同一条直线的两条直线 |
图形语言 | |
符号语言 | 直线a,b,c,a![]() ![]() |
作用 | 证明两条直线平行 |
您最近一年使用:0次
解题方法
10 . 我们知道,利用导数证明基本不等式:
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff9adc4f9272724f0b088f3bf1340639.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53eeb9d5984eb105da2dd992cd694a51.png)
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