1 . 称
是
的一个向往集合,当且仅当其满足如下两条性质:(1)任意
,
;(2)任意
和
,有
.任取
,称包含
的最小向往集合称为
的生成向往集合,记为
.
(1)求满足
的正整数
的值;
(2)对两个向往集合
,定义集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbe16b433635b8bc25f303863807b70.png)
(i)证明:
仍然是向往集合,并求正整数
,满足
;
(ii)证明:如果
,则
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/160af7e0b1d01eec9b33474b4d067a76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2077e5032491293f8181c4fc3bcf360a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3ad11a8563df9a39fbe386f746f755c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8104c761c3fac71e51c9a17a154829ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f27e8b43153beb780aa92d61df4b0da4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60cfb0de87efce8d98d89106fd36f61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8060d3a485605dd9fedb3c5ae089c24e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c8f38fd2a2457ab28745c41c0f6b0aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
(1)求满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c248f486fa233098501ba2a64422118.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)对两个向往集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0248166f5a50eb4fe7f8a02a2d8e397e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbe16b433635b8bc25f303863807b70.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a13c9838a7aa389c93dcbaf5ad0449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deb92321829e1fa81061502157411cec.png)
(ii)证明:如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/528af17b6a22c9c808c4231ef395a0c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0161489025ecbc391b1c9affce57b930.png)
您最近一年使用:0次
名校
解题方法
2 . 在数学中,双曲函数是与三角函数类似的函数,最基本的双曲函数是双曲正弦函数与双曲余弦函数,其中双曲正弦函数:
,双曲余弦函数:
.(e是自然对数的底数,
).
(1)计算
的值;
(2)类比两角和的余弦公式,写出两角和的双曲余弦公式:
______,并加以证明;
(3)若对任意
,关于
的方程
有解,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3321510a9eb73909a36c084a8630e89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0099b9b80ed478824fa95677ebe9d5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11204e2fb6e560bf7a4ca26eaebfc526.png)
(1)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e694af0c9f990ecb8b54b1c08bcc578e.png)
(2)类比两角和的余弦公式,写出两角和的双曲余弦公式:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d92c32edc0e000405b7a6b9c48549959.png)
(3)若对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f78f05631a2ecb8bc3d379ca6c81f93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed807cc52eca7b462a3850b5e5e02b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2023-06-21更新
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991次组卷
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7卷引用:山东省济南市山东师大附中2022-2023学年高一下学期数学竞赛选拔(初赛)试题
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