2023·全国·模拟预测
解题方法
1 . 在锐角
中,内角A,B,C所对的边分别为a,b,c,且
.
(1)证明:
;
(2)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d06493a3dd7808c2747b470b82f242b3.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a2264c134952d41fb9bcb90e6c72c83.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b246aa3b56becc905d3fb64c6d5ec4a.png)
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2 . 已知
,
.
(1)若
,求
的取值范围;
(2)若函数
恰有两个零点,求实数a的取值范围;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b39c5d66018f0736a0457961c91e1c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17a1141aa62d95fc3b75e3d6833aaaf0.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98a3f237b03aaa5f0fb96e572706349c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/732a3c49d8680218bdcc2f39f2b4f601.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08180ea9d4238fd449255a0b47f3bb2f.png)
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3 . 已知有半径为1,圆心角为a(其中a为给定的锐角)的扇形铁皮OMN,现利用这块铁皮并根据下列方案之一,裁剪出一个矩形.
方案1:如图1,裁剪出的矩形ABCD的顶点A,B在线段ON上,点C在弧MN上,点D在线段OM上;
方案2:如图2,裁剪出的矩形PQRS的顶点P,S分别在线段OM,ON上,顶点Q,R在弧MN上,并且满足PQ∥RS∥OE,其中点E为弧MN的中点.
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899185563287552/2921367596277760/STEM/7d7a8b44-6296-4eae-9a0e-4d8607fd42d8.png?resizew=302)
(1)按照方案1裁剪,设∠NOC =
,用
表示矩形ABCD的面积S1,并证明S1的最大值为
;
(2)按照方案2裁剪,求矩形PQRS的面积S2的最大值,并与(1)中的结果比较后指出按哪种方案可以裁剪出面积最大的矩形.
方案1:如图1,裁剪出的矩形ABCD的顶点A,B在线段ON上,点C在弧MN上,点D在线段OM上;
方案2:如图2,裁剪出的矩形PQRS的顶点P,S分别在线段OM,ON上,顶点Q,R在弧MN上,并且满足PQ∥RS∥OE,其中点E为弧MN的中点.
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899185563287552/2921367596277760/STEM/7d7a8b44-6296-4eae-9a0e-4d8607fd42d8.png?resizew=302)
(1)按照方案1裁剪,设∠NOC =
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65b99d8577e4f825adcb79569bab5017.png)
(2)按照方案2裁剪,求矩形PQRS的面积S2的最大值,并与(1)中的结果比较后指出按哪种方案可以裁剪出面积最大的矩形.
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4 . 对于函数
,若存在正常数
,使得对任意的
,都有
成立,我们称函数
为“
同比不减函数”.
(1)求证:对任意正常数
,
都不是“
同比不减函数”;
(2)若函数
是“
同比不减函数”,求
的取值范围;
(3)已知函数
是定义在R上的奇函数,当
时,
.是否存在正常数
,使得对于任意的
,函数
都为“
同比不减函数”,若存在,求
的取值范围;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9298ea50c497b0ad0905c08d72565892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e02cab1add26335b3cb43d5b54c7c853.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c64c9f7e6d921f2f134b832dc87e5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(1)求证:对任意正常数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b1c079afd1b058adc67a50f48f3d466.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3e9c31b39b443a4ac19740ba7dece6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ad72d7565699d1ebb741eb0ce12bac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(3)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad5e5727d78c170a69fade80116d645f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a111628c5ea54305dba24105b84900.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
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解题方法
5 . (1)已知实数
,若函数
满足
,问:这样的函数
是否存在? 若存在,写出一个;若不存在,说明理由;
(2)写出三次函数
,使得
,对一切实数
成立,求
时,
的最大值和取最大值时
的值;
(3)设
,函数
,记M为
在区间[t,t+2]上的最大值,当
变化时,记m(t)为M的最小值.
①证明:m(t)的值是与t无关的常数(记为m)
②求m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b390d4f89c595551244f615b6856bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d889bdd690f84f91abd2c63dcc05139.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e810d7540bf757d1bcdd62bea0f0fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
(2)写出三次函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/932a4f4875c0d88716e36ac7f2eb3288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04ed2e7ae36ecef5de68d8afd668d520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1591d4244dcf5539a4ae98f554e91e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f28badcf9e6e095a9474b5d9fdad58b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9d51db103a0934d764e7f9da43fe6eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae06c488100e31570805778b1d322e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d09a2b7c019dae83e027830b82b3ee8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
①证明:m(t)的值是与t无关的常数(记为m)
②求m的值.
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解题方法
6 . 阅读下面材料:根据两角和与差的正弦公式,有
……………①,
……………②,
由①
②得
…………③,
令
,
,有
,
,
代入③得:
.
(1)利用上述结论,试求
的值.
(2)类比上述推证方法,根据两角和与差的余弦公式,证明:
.
(3)求函数
,
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8b8ee28cf91c5976d074d233c941f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3adf14c530b8090dd2935ff469f829.png)
由①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d4cd9a7068de096606d1ab991f5e6da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4be00cd134162e401f3f62be643be4f9.png)
令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad8df7860f5cc206dd1bae718c5d2b8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/675a8c31fb714cbf331ec2c6b16b65ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c690402ee2cf3fa13a6ae2a8c479a464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b74b931d3078362fed72dd4fcc372cb.png)
代入③得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4839b970e00091890ccd2bcad2f9879b.png)
(1)利用上述结论,试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d108620293aab395f654d8c3d7ab4467.png)
(2)类比上述推证方法,根据两角和与差的余弦公式,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e65b54c997778fe0207955247d6581b.png)
(3)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df4e95925b84b724aab36f1d63ee08cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b63a87545fb8a1913ad16e376d42bd2.png)
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2017-07-23更新
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76次组卷
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2卷引用:安徽省黄山市屯溪第一中学2016-2017学年高二下学期期中考试数学(理)试题1