名校
解题方法
1 . 设实系数一元二次方程
①,有两根
,
则方程可变形为
,展开得
②,
比较①②可以得到![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71e360b7ba27dfc3e5d401027d5bd8a5.png)
这表明,任何一个一元二次方程的根与系数的关系为:两个根的和等于一次项系数与二次项系数的比的相反数,两个根的积等于常数项与二次项系数的比.这就是我们熟知的一元二次方程的韦达定理.
事实上,与二次方程类似,一元三次方程也有韦达定理.
设方程
有三个根
,则有
③
(1)证明公式③,即一元三次方程的韦达定理;
(2)已知函数
恰有两个零点.
(i)求证:
的其中一个零点大于0,另一个零点大于
且小于0;
(ii)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c3f3db6b7c682450309a6ccba5ac5a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
则方程可变形为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455d33fcfd9a59d6b374e9d25888cd2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e42b42152492cbdfec62c7a02be4055.png)
比较①②可以得到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71e360b7ba27dfc3e5d401027d5bd8a5.png)
这表明,任何一个一元二次方程的根与系数的关系为:两个根的和等于一次项系数与二次项系数的比的相反数,两个根的积等于常数项与二次项系数的比.这就是我们熟知的一元二次方程的韦达定理.
事实上,与二次方程类似,一元三次方程也有韦达定理.
设方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddc5e3c2c7c6f4d2d0ab396b65679a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb8d75a2417827b2c5b09ba9385fe252.png)
(1)证明公式③,即一元三次方程的韦达定理;
(2)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b5e4746c2bd0afb279630698afd3a0.png)
(i)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d6fc9b90f370fbb27552876b650f8f.png)
您最近一年使用:0次
名校
解题方法
2 . (1)
,
,其中x,y均为正实数,比较a,b的大小;
(2)证明:已知
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1af972332e5a9149d66e5a2577e71d42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed2b8f0b94dcf960ca3659536d077d62.png)
(2)证明:已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e56f4504e0f80fd031c8b5f41832e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a0aa068c979c53361d049ce49987a8.png)
您最近一年使用:0次
2022-05-05更新
|
1061次组卷
|
8卷引用:贵州省兴义市顶效开发区顶兴学校2022-2023学年高一上学期期中考试数学试题
贵州省兴义市顶效开发区顶兴学校2022-2023学年高一上学期期中考试数学试题广东省广州市铁一三校2022-2023学年高一上学期期中数学试题内蒙古包头钢铁公司第四中学2022-2023学年高一上学期期中考试数学试题河南省濮阳市油田第二高级中学2021-2022学年高二上学期9月考试文科数学试题(已下线)3.1 不等式的基本性质 (1)(已下线)专题2.2 等式性质与不等式性质-重难点题型检测-2022-2023学年高一数学举一反三系列(人教A版2019必修第一册)2.1 等式性质与不等式性质练习河南省周口市鹿邑县第二高级中学校2023-2024学年高一上学期第一次月考数学试题
3 . 在数列
中,
,
(
).
(1)求
,
,
;
(2)猜想
;并加以证明;
(3)若数列
,设数列
的前
项和
.求证
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e76a35065ee95d9a308d2b439fc57f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
(2)猜想
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/980d51ba3340a31964fbec9e6f243ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.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/dbbb29ff6c20c4d5fd2cb319eb191d77.png)
您最近一年使用:0次
名校
解题方法
4 . 如图,在四棱锥
中,底面
为梯形,
,平面
平面
是
的中点.
![](https://img.xkw.com/dksih/QBM/2020/3/2/2410901481349120/2420895027200000/STEM/9fbf3910e02b4e778bd7a5818ffa9eb8.png?resizew=259)
(1)求证:
平面
;
(2)若
,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/719e5223dc710613a6c223d4145075f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfaefb10f82b89802bb420b3c41de1bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/2020/3/2/2410901481349120/2420895027200000/STEM/9fbf3910e02b4e778bd7a5818ffa9eb8.png?resizew=259)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7200b49682ba4334947b846775b1b841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/822ba132ca9dd0d4a050659aef3c9b26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31b8945c316011287f846c9341d159bd.png)
您最近一年使用:0次
2020-03-16更新
|
688次组卷
|
2卷引用:贵州省思南中学2019-2020学年高一下学期期中考试数学试题
名校
5 . 如图,在边长为
的正方形
中,点
是
的中点,点
是
的中点,点
是
上的点,且
.将△AED,△DCF分别沿
,
折起,使
,
两点重合于
,连接
,
.
![](https://img.xkw.com/dksih/QBM/2018/7/16/1989574276603904/1989768106516480/STEM/88df25c0416e4c0fbd0694a6e48537a8.png?resizew=318)
(Ⅰ)求证:
;
(Ⅱ)试判断
与平面
的位置关系,并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1df842ffd4121b27c70a8d0b1f78b173.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/2018/7/16/1989574276603904/1989768106516480/STEM/88df25c0416e4c0fbd0694a6e48537a8.png?resizew=318)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6969b9971ceae406072933356189a897.png)
(Ⅱ)试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc704b98f4ed2c7359a7a5b6498b5290.png)
您最近一年使用:0次
2018-07-16更新
|
688次组卷
|
3卷引用:贵州省思南中学2019-2020学年高一下学期期中考试数学试题
6 . 如图,四棱锥
的底面是正方形,侧棱
⊥底面
是
的中点.
(Ⅰ)求证:
∥
;
(Ⅱ)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8f449e8cd3075c1de5cae3a57293f38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f6d94889ef44776a1a60586922ee891.png)
(Ⅱ)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/509d8dd6031dc0ef92075877e53fe201.png)
![](https://img.xkw.com/dksih/QBM/2017/11/15/1817625011290112/1819374978760704/STEM/dc9e51a78fac47e59bc20c1aae79dcbe.png?resizew=166)
您最近一年使用:0次
2017-11-17更新
|
936次组卷
|
5卷引用:北京师范大学遵义附属学校2020-2021学年高二第一学期期中考试数学试卷
7 . 已知
为数列
的前
项和,
,且
.
(1)证明数列
是等差数列,并求其前
项和
;
(2)设数列
满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/759caa9a3df7c8381f6600ed3143afdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69c7266d90661cf4467f13c6f5eb670c.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5a28cd035abe6bbf35f7d2b50eb917b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/803608cc91af54afd734afadfd894245.png)
您最近一年使用:0次
解题方法
8 . 如图,在四棱锥
中,
是正方形,
平面
,
,
分别是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/20/329a7886-1b8c-4868-80a5-a789935fbd61.png?resizew=176)
(1)求证:平面
平面
;
(2)在线段
上确定一点
,使
平面
,并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99926bf272cd757f0985c69b390ebcce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b199a99e53d67ff4abf233930961a29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e3a25299c121dbb883fd3c7918d566d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/20/329a7886-1b8c-4868-80a5-a789935fbd61.png?resizew=176)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a281c31b6e501123442d141860908a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe8a84ca3a13f82aff1a022edc66065.png)
(2)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f1897a7e856b42f8cee0f286ad913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5d8073385db872410ca88187bbb0d34.png)
您最近一年使用:0次
9 . 若数列
和
的项数均为
,则将数列
和
的距离定义为
.
(1)求数列1,3,5,6和数列2,3,10,7的距离;
(2)记A为满足递推关系
的所有数列
的集合,数列
和
为A中的两个元素,且项数均为
.若
,
,数列
和
的距离
,求m的最大值;
(3)记S是所有7项数列
(其中
,
或1)的集合,
,且T中的任何两个元素的距离大于或等于3.求证:T中的元素个数小于或等于16.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58eb2bc16ca7ba6db8792eec6e2b48c0.png)
(1)求数列1,3,5,6和数列2,3,10,7的距离;
(2)记A为满足递推关系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23093a3f4c23494a943e3957596fee92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385275d29d8c8a7841eaeaa3dfab2cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2865594c03cd3cfcbf3216cdbf08fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77cb4aa359781e637bd2232813fa8a24.png)
(3)记S是所有7项数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9d7457bc36b80660dc03b668674f065.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31bd42f8e3f220a7b1c6f6945e73bc10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1c458592ba2d5ddd559b8720438a8fe.png)
您最近一年使用:0次
名校
解题方法
10 . 如图,P为圆锥的顶点,
为圆锥底面的直径,
为等边三角形,O是圆锥底面的圆心.
为底面圆O的内接正三角形,且边长为
,点E为线段
中点.
平面
;
(2)M为底面圆O的劣弧
上一点,且
.求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fbfcae2cecc98e2d6c16dde6d3ec1c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
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(2)M为底面圆O的劣弧
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a07657d5498fc951edb413f514eac44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d4a244bd6c29b79ddbf0bbdaf6cd14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
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2024-03-08更新
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4卷引用:贵州省贵阳市第一中学2023-2024学年高二下学期教学质量监测卷(三)数学试题