名校
解题方法
1 . 已知一个平面内的三个向量
,
,
,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9feedc68e0c2d1b0de51d28e61496e59.png)
(1)若向量
为单位向量,且与
共线,求向量
的坐标;
(2)若
,且
与
垂直,求向量
与
的夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23bd18136e4e71e1e267bc9f8634b932.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec82bd4e0fb7887e5999b8220f60b073.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ecf646bc2e4cdd46fc5cc40d741dab5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9feedc68e0c2d1b0de51d28e61496e59.png)
(1)若向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ecf646bc2e4cdd46fc5cc40d741dab5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23bd18136e4e71e1e267bc9f8634b932.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ecf646bc2e4cdd46fc5cc40d741dab5.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5404940e86fc3db18adcb5be78e2e276.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59feec0a253476e1baa7ac2d196da24c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abb1d6b61205267d523c5cab8ac68cf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23bd18136e4e71e1e267bc9f8634b932.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec82bd4e0fb7887e5999b8220f60b073.png)
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2 . 已知向量
,
,
.
(1)若
,求
的值;
(2)若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fda3f2bbb62d0e0abc140c174944aa89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d88a1983547e8f6d13e5f7f1cb521dec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41c79ec1cb1bcbcc86b2077957ce6c55.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a36228810d0c2f6c6e53584c1ac176b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c66dc9c4258a666e90278e33e084fdba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83634e314df24df73ae37d25a44d20e3.png)
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3 . 已知函数
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
(1)如果点
是角
终边上一点,求
的值;
(2)设
,求
的单调递增区间.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9688a68bd2d1b2c8dcd6568e1a5ac3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
(1)如果点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e7c6b1e8d06e439c1ede19f99286c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/794f2c6bd63355105d179d11306a9cae.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72f3b2cc4fc44eef6a51e1a4ae89ecb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
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解题方法
4 . 已知曲线
.
(1)求与直线
平行,且与曲线
相切的直线方程;
(2)设曲线
上任意一点处切线的倾斜角为
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45a6340e9036762ebbbf27513832c57a.png)
(1)求与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b4b0eba587d0af5c665a8f909df5104.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
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解题方法
5 . 记△ABC的内角A,B,C的对边分别为a,b,c,已知
.
(1)求
:
(2)若
,
.求△ABC的面积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2558ad6191a401842103caa23a5615a.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1311f32edf13f8caee5edb03f24a7ba.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4517e436af5f908b9d322ea6aa7935f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22414bee2bfd17031a9edf61219b6c07.png)
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6 . 已知数列
是公差不为0的等差数列,
是
和
的等比中项.
(1)求数列
的通项公式;
(2)设数列
满足
,求数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1300e77f7c5741b9b5f7af0bdb9ae76d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a26d8ee8bf91cfda41b5b94e6a8415b.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)
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名校
解题方法
7 . 已知复数
.
(1)若复数
为纯虚数,求
;
(2)若复数
在复平面内对应的点在第四象限,求实数m的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4366fa9c602566f93056c558e06e7bcb.png)
(1)若复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41042207515dd2e8349c805e6aee400.png)
(2)若复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
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名校
解题方法
8 . 若
是定义在
上的增函数,其中
,存在函数
,
,且函数
图像上存在两点
,
图像上存在两点
,其中
两点横坐标相等,
两点横坐标相等,且
,则称
在
上可以对
进行“
型平行追逐”,即
是
在
上的“
型平行追逐函数”. 已知
是定义在
上的奇函数,
是定义在
上的偶函数.
(1)求满足
的
的值;
(2)设函数
,若不等式
对任意的
恒成立,求实数
的取值范围;
(3)若函数
是
在
上的“
型平行追逐函数”,求正数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8fd1e808e015f4cb43d2e3a0529ac6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3df718662452f53034b6f702e46dcdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/958a7330503a10fecf9b6b6f4a30ea8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc5ffb9d0b5dc940f53d7370419a08db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3859cd3de4d6d485df050b0f5321d6c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c33c52de7a2c9d148e44283eec3dbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a3e7d1f1890863b7483a98b618119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdb21011ea821b91d539cb763aac649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdeb8cfadb41d94aaa1ba534aa040dcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e257d683ceaca6c6c3ee5ada0e447b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d908675a3ce0661cf6b3d7823143d4b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
(1)求满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfb874a24bc2fa3d232070f16712aa05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)设函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86df4f1b31c51da3551a76606d553f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0bb6dbc79a74521af6338b0140b713b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b8c164755dc2d7cff80fb4c9cffc9be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a86644d5b1157b35cf7b825f108d4c39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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9 . 陀螺是中国民间最早的娱乐工具之一.图1是一种木陀螺,可近似地看作是一个圆锥和一个圆柱的组合体,其直观图如图2所示,其中
分别是上、下底面圆的圆心,且
,现有一箱这种的陀螺共重
(不包含箱子的质量),陀螺的密度为
(
取3)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/4/bd29717f-0d9e-4b4b-acb4-18885d91ec12.png?resizew=238)
(1)试问该箱中有多少个这样的陀螺?
(2)如果要给这箱陀螺的每个表面涂上一种特殊的颜料,试问共需涂多少
的颜料?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d6b1a601cfab0d96a659149a4c3fc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43d00798286f8e07b736da3cea3b471f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/009fb05cda538a62cf0b5b9398600a58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70f5389990c3a0c5373f3bd9fb2454c9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/4/bd29717f-0d9e-4b4b-acb4-18885d91ec12.png?resizew=238)
(1)试问该箱中有多少个这样的陀螺?
(2)如果要给这箱陀螺的每个表面涂上一种特殊的颜料,试问共需涂多少
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88ce13774b09ff2edddaf21a072cf60a.png)
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10 . 已知
,
,在复平面内,复数
,
,
对应的点分别为
,
,
.
(1)求
;
(2)若
,求实数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28e8b708288bd3a38942055519a57708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/845dbecf4d789a2d611a3c39d3fe5d9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34b88f343ca5a4c29057465541b9cf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e4e2b866b0043a32fc78326553841d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe615164ed2995bdeea0f5b0ba94231.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)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c231f5c0eddd418c1e36688321e39ab7.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbc0a1884c6669cec22b714a57d5b2b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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