1 . 如图所示,在平面直角坐标系
中,点
绕坐标原点
逆时针旋转角
至点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/0f3cb248-ea1c-49b1-864d-55830d1210d1.png?resizew=212)
(1)试证明点的旋转坐标公式:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b519e490e2a73e8b6455f2c6647a713.png)
(2)设
,点
绕坐标原点
逆时针旋转角
至点
,点
再绕坐标原点
旋转角
至点
,且直线
的斜率
,求角
的值;
(3)试证明方程
的曲线
是双曲线,并求其焦点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8701e0cce437edc830438b4fe6277d89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039b02af8364c4714df617a9278eb0fe.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/0f3cb248-ea1c-49b1-864d-55830d1210d1.png?resizew=212)
(1)试证明点的旋转坐标公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b519e490e2a73e8b6455f2c6647a713.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8dd8b1de493e4027839ccdeeac69e14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfac6a734705249042747a8367c5b94c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a949c00526fddf435423272cf10f25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b02f266bd253738e315e84231235f0d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(3)试证明方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3329a078a2704772b46cf74278b7397.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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解题方法
2 . 已知向量
,
是坐标原点,若
,且
方向是沿
的方向绕着
点按逆时针方向旋转
角得到的,则称
经过一次
变换得到
,现有向量
经过一次
变换后得到
,
经过一次
变换后得到
,…,如此下去,
经过一次
变换后得到
,设
,
,
,则
等于( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac66b00e6cb8d30f293f3baeb501333e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb9307fa2c34a86800d60a194747a788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9795e7f5cb9b366776c41d8f3f43942.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d60dcb171bb7fd972aab8294d63acdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d60dcb171bb7fd972aab8294d63acdb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52831e39d2f23629c6699103ff5d1d67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9795e7f5cb9b366776c41d8f3f43942.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f66c2415abe65e3fab5574c08d8d1e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bc22dd6172821ef009d031d466bdd3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7559d65befe0b85c8929f57c9436cd26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7559d65befe0b85c8929f57c9436cd26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1cd05a7d73db8c7a63ba8e3854ef259.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c81c88f84b0533f32d4008a5088b21cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79d7c41caf47f014d2158abe25a876ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acb8f1484640eef1f6439be92f1a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07d8b2f1f9ead1e148c04fb2538421e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a4c7fc2ab3dc022658351fd10a5e07b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4799f1244959c9118346ced85adc4197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/449259bdbfccf519730ad5a0522d80f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3677f2a2a1b4aef3a370a03db832366.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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3 . 已知数列
满足
,则使
成立的正整数
的最小值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d85ed7f421e6fd9460b5ad786a577ca2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/333f8be44bcb49485763f6d71f2fc54a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
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4 . 利用行列式解关于
的二元一次方程组
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6215b48ddcae362f60d5d2623a05f46b.png)
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2020-03-04更新
|
152次组卷
|
3卷引用:上海市曹杨二中2017-2018学年高二上学期期中数学试题
5 . 矩阵乘法运算
的几何意义为平面上的点
在矩阵
的作用下变换成点
,记
,且
.
(1)若平面上的点
在矩阵
的作用下变换成点
,求点
的坐标;
(2)若平面上相异的两点
、
在矩阵
的作用下,分别变换为点
、
,求证:若点
为线段
上的点,则点
在
的作用下的点
在线段
上;
(3)已知△
的顶点坐标为
、
、
,且△
在矩阵
作用下变换成△
,记△
与△
的面积分别为
与
,求
的值,并写出一般情况(三角形形状一般化且变换矩阵一般化)下
与
的关系(不要求证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c158a0dafc1ed72d4cd92ce8be537d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8701e0cce437edc830438b4fe6277d89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10a63c1c60b695b37bec52bfb8bae214.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cba166b3713688f7e6ef0fbfa2890881.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7b45d23c105698406d1299177026285.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367091d7500780500b0eeb961ed19bda.png)
(1)若平面上的点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01fab4a5ee2cd8546c9276d09fcfe346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3346d5fe92ff182b98e2a27b64af0cbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若平面上相异的两点
![](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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee6765a83140d745a6de4c85d9b6b50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb0628cecbfc98d390e5447d52414e8.png)
(3)已知△
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a26e951d6d2369e8da79a793a93a66a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe030690868617d57c0bc82b98ad662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b5956df0cf0f537fe24570a46c7d379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b86d62e7455f30b7aad0d137c6b1403d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a1264a2e3609e1c274acb89b5ea5019.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a1264a2e3609e1c274acb89b5ea5019.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
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6 . 关于
的方程组
的系数矩阵记为
,且该方程组存在非零解,若存在三阶矩阵
,使得
,(0表示零矩阵,即所有元素均为0的矩阵;矩阵
对应的行列式为
),则
(1)
一定为1;
(2)
一定为0;
(3)该方程组一定有无穷多解.
其中正确说法的个数是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/726b31b9e4f085fb2754723f21405d86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d52399e72456de84f0a42dd69da06fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93499d2c5181db3ef5db16b64c2a4911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8f054b30b493c4ce1deecff3468c08.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8f054b30b493c4ce1deecff3468c08.png)
(3)该方程组一定有无穷多解.
其中正确说法的个数是( )
A.0 | B.1 | C.2 | D.3 |
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7 . 已知
阶方阵
中的各元素均为正数,其中每行成等差数列,每列都是公比为2的等比数列,已知
.
(1)求
和
的值;
(2)计算行列式
和
;
(3)设
,证明:当
是3的倍数时,
能被21整除.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/944e7a418f86eef567cf20c196db07dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/425dd6d107574d3aab56558cc08d2648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe3785a7ea5220d7e990f08045f25222.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e84c30444f13d37ada78285dc4f83b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3bb6d9c3e9e00ccda6c5ccafefd630.png)
(2)计算行列式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e027dff5c04c837040f5c46422ad24e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/358b8c349b8fe30ec98d66febfc76db8.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e82ff47d67bfb5ff6310538189ca49a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/963391e1a37b12d40d1bac0b1c9ac4ac.png)
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8 . 在中学阶段,对许多特定的集合(如实数集,平面向量集等)的学习常常是以定义运算(如四则运算)和研究运算律为主要内容,现设集合
由全体二元有序实数对组成,在
上定义一个运算,记为
,对于
中的任意两个元素
,
,规定:
.
(1)计算:
;
(2)请用数学符号语言表述运算
满足交换律和结合律,并证明交换律;
(3)
中是否存在唯一确定的元素
满足:对于任意
,都有
成立,若存在,请求出元素
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75bcf6339e9c978b1109a43e75dbc920.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66e5a3f00c71a8388bddf93eaa74b00a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dfef683de05e5cc1a90635f4a6edd50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85e5848b1798d99663bdeac5c9815666.png)
(1)计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfb93c4b8f07fc687b3119d75fe3c5b1.png)
(2)请用数学符号语言表述运算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75bcf6339e9c978b1109a43e75dbc920.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/561e9a85cd4fc7ba728a6669ce017b82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fff78e2b9da422dff9d4900ddd5d257e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e957a4a634811b1ba8104c21326c7647.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/561e9a85cd4fc7ba728a6669ce017b82.png)
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