名校
解题方法
1 . 如图1,等腰
满足
,
,
于
.如图2,将
绕着直线SA旋转时,在BA旋转而成的平面
内总有点
满足
,
,(点
,点
分别在直线BD两侧).
长;
(2)求证:
平面
;
(3)记三棱锥
的体积为
,三棱锥
的体积为
,当四棱锥
的体积最大时,求
值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79482c6de6bbd05affc78f9c625e52f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2affc8264f2e40743bb12dd7ea57177b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fe85217cea14241255ec21200b25b16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fe0bb7d51e559e73aa16a954fe7fa33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79482c6de6bbd05affc78f9c625e52f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c279c8033acb94c3f91be2e05b0a6bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/833cfda415649b832cc136caed392753.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/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10c83f8945042b9c8fb2fbdac9308d62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(3)记三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d535bfd65eb04a29d64425d54b2acf86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69dd9f16a5c7a66e62e52fd66f4449ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c773ac7c3a6575fb7d432c93fe5f2a32.png)
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解题方法
2 . 平面向量是数学中一个非常重要的概念,它具有广泛的工具性,平面向量的引入与运用,大大拓展了数学分析和几何学的领域,使得许多问题的求解和理解更加简单和直观,在实际应用中,平面向量在工程、物理学、计算机图形等各个领域都有广泛的应用,平面向量可以方便地描述几何问题,进行代数运算,描述几何变换,表述物体的运动和速度等,因此熟练掌握平面向量的性质与运用,对于提高数学和物理学的理解和能力,具有非常重要的意义,平面向量
的大小可以由模来刻画,其方向可以由以
轴的非负半轴为始边,
所在射线为终边的角
来刻画.设
,则
.另外,将向量
绕点
按逆时针方向旋转
角后得到向量
.如果将
的坐标写成
(其中
,那么
.根据以上材料,回答下面问题:
,求向量
的坐标;
(2)用向量法证明余弦定理;
(3)如图,点
和
分别为等腰直角
和等腰直角
的直角顶点,连接DE,求DE的中点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91174b2336306191ba275a87864172b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91174b2336306191ba275a87864172b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4293abac93e7633dc4c0fef321347e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8a3b1b11c77ceb7ece55f76d2cd4618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/873c064546108a5bce78bb71bc1e4a1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea99a712a0891faf366d4fec4dde5869.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941b0d76d7b3108df49af338c989dc4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e32257bac4199820ccae5e7bd8377cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0849dbfc3775627925de0fe2e89c1692.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb50427d2e8a7c605bbd18ea8e0c3b79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af5f1b06a56fc382feed28e01f1ad102.png)
(2)用向量法证明余弦定理;
(3)如图,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd99c5000629d7f49499d666e68f40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852b303689c31189cd47bb4a3220f9fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbfa1a2af7e38d33634c462300df381f.png)
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3 . 对于定义在
上的函数
,如果存在一组常数
,
,…,
(
为正整数,且
),使得
,
,则称函数
为“
阶零和函数”.
(1)若函数
,
,请直接写出
,
是否为“2阶零和函数”;
(2)判断“
为2阶零和函数”是“
为周期函数”的什么条件(用“充分不必要条件”“必要不充分条件”“充要条件”或“既不充分也不必要”回答),并证明你的结论;
(3)判断下列函数是否为“3阶零和函数”,并说明理由.
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c7eb49a823f757461cd5260757b088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd84a8f95166367063218ee03ffd5a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f7f4cc0837a4e6dcd0072887e4e2704.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efe6d9f54a34762aadfdf8e2bac977cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96b743603ab1c10330622f16db78dbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892519541cfba6f2763cd29159bf1b02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/329fb959f16f82835aa68fca9d3f08f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcda6a21da79726f8fb3ba6235b9010f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebef85c05f6d84ceb67d92abf77ba2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6ace630100e64ed290d82936ad249c8.png)
(2)判断“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(3)判断下列函数是否为“3阶零和函数”,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab7da79b2400cf8125ef040cd056b76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/321b15db96dc89f136a7421e09fc9814.png)
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解题方法
4 . 设
次多项式
,若其满足
,则称这些多项式
为切比雪夫多项式.例如:由
可得切比雪夫多项式
.
(1)求切比雪夫多项式
;
(2)求
的值;
(3)已知方程
在
上有三个不同的根,记为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bcabb8534436af78551405453864df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/076517385a1ca0aa2d8f7035158f353a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17bd2bc42d15891e0739e1ff3c0993d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b327b904e4d65a88b5adaf4de91694fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/703a4194b9d5650df287fa822cf039cf.png)
(1)求切比雪夫多项式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10608b54173b1b7b559c579f4dc69ae2.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff1e86c5abdaa1ca8599ffa5e933e046.png)
(3)已知方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf211eb82ea0c803eeff551d5819643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455ba3d3e46977fcbe5b71f8bb9df4be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f2c5f7b63a7dd6d0155f9d38158fcf1.png)
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5 . 已知集合
,称
为
的第
个分量.对于
的元素
,定义
与
的两种乘法分别为:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2aa5853722e3158b0f77917726dbc6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e510332893c067eeef1fe76cefa1173.png)
给定函数
,定义
上的一种变换
.
(1)设
,求
和
;
(2)设
,对于
,设
,
对任意
且
,定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c34b66a9efd802f743dcc652aa8820.png)
①当
时,求证:
中为0的分量个数不可能是2个;
②若
的任一分量都只能取
或
,设
的第1个分量为
,求
的最小正周期的最小值,并求出此时所有的
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16531ec81209cad92180eba890c9b137.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70a5369bb892f707c3f0a2ac2fa18f2b.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/a2aa5853722e3158b0f77917726dbc6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e510332893c067eeef1fe76cefa1173.png)
给定函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4c01c6322201e64d7b9442f99728aa7.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/056e669d7afaf70e555f1c4fb9192ca4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12359a62d8ca4edfcecca9909cccfc33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4d3de45518fa839fbf8e2426fa8d1f8.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bc498431c1a7c9e48c3858faf7bfd38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1348da718d01114e3db4355c08531c7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/550a28e39bec7b9d5f8e722f00b44e5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ec3194cabfe4b369b8ff464bda89964.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ff2aa68223dfc02f39d7d10fa005387.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/918893290e48bba154bd5a14a805f10f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c34b66a9efd802f743dcc652aa8820.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1348da718d01114e3db4355c08531c7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7be6f009bfb61b11e4f87edb4132de3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94ea50db79b18d8700cfa2559ff5e2d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/544f91d4fb22c571db9f8481b72a0419.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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6 . 若定义域为
的函数
满足:对于任意
,都有
,则称函数
具有性质
.
(1)设函数
,
的表达式分别为
,
,判断函数
与
是否具有性质
,说明理由;
(2)设函数
的表达式为
,是否存在
以及
,使得函数
具有性质
?若存在,求出
,
的值;若不存在,说明理由;
(3)设函数
具有性质
,且在
上的值域恰为
;以
为周期的函数
的表达式为
,且在开区间
上有且仅有一个零点,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3d641761af730cc20b05a79fad66f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a39800f3595a04a3c9730c531049b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3d641761af730cc20b05a79fad66f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b29a7faa14a6e09d0db2d04f4ced03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc6e69ad1a27916fb5c3d5901ded134.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04afd6b14d712929799c7d092872c354.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342b4871cd7d7766c9054a1dc0b477a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b29a7faa14a6e09d0db2d04f4ced03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc6e69ad1a27916fb5c3d5901ded134.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b29a7faa14a6e09d0db2d04f4ced03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a55838863eacaec3c4f56df61169488d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb8b79682b1872ca13d4d119adc01613.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced695934528674095a9fcf3db511ebe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a555759e23d21c30f1ed29e7d2453fea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/074c228ffc7b1e306f8410afe7bc4b5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
(3)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b29a7faa14a6e09d0db2d04f4ced03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/931db1234c7327aa072f8e96360c96e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6fab7a2597e4d169c942d5c65c98b00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e9fdc1f8ed0ae44b54a9a2a3aca2db4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc6e69ad1a27916fb5c3d5901ded134.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d396d5349f4b2b9b74f01347c242250.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166009a848eadfd8ac7cc83933aa219b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fddb0be24dcd1323c63b8680f5071cdb.png)
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2021-07-12更新
|
1762次组卷
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11卷引用:上海交通大学附属中学2020-2021学年高一下学期期末数学试题
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