1 . 若关于x的不等式
恒成立,则实数a的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cc1d431cbbb931fabf803a49148f73d.png)
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2 . 函数
是我们最熟悉的函数之一,它是奇函数,且y轴和直线
是它的渐近线,在第一象限和第三象限存在图象,其图象实质是圆锥曲线中的双曲线.
的图象不仅是中心对称图形,而且还是轴对称图形,求其对称轴l的方程;
(2)若保持原点不动,长度单位不变,只改变坐标轴的方向的坐标系的变换,叫坐标系的旋转,简称转轴.
(i)请采用适当的变换方法,求函数
变换后所对应的双曲线标准方程;
(ii)已知函数
图象上任一点到平面内定点
的距离差的绝对值为定值,以线段
为直径的圆与
的图象一个交点为
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
(2)若保持原点不动,长度单位不变,只改变坐标轴的方向的坐标系的变换,叫坐标系的旋转,简称转轴.
(i)请采用适当的变换方法,求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
(ii)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52586ca2a3b783bc8092415e2d4bf6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1305b9abebd7bef3171486df157286b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
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名校
解题方法
3 . 函数
有三个不同极值点
,且
.则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddf42654d900f4ca45ba473d9ba363b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b8ec9d4206ea66a02de5c4a1e1e911.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f7af029ff25c826608dc72348532407.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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|
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2卷引用:山西省晋城市第一中学校2023-2024学年高二下学期第四次调研考试(5月)数学试题
名校
解题方法
4 . 已知偶函数
与其导函数
定义域均为
,
为奇函数,若2是
的极值点,则
在区间
内解的个数最少有( )个.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4a364dbaef735e2530b11d95b0e9c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f168b6811f1da5f09db1d9984ad8664f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1daa67de3b8971d54ced0cac0cd11f2e.png)
A.7 | B.8 | C.9 | D.11 |
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解题方法
5 . 已知椭圆
的焦点为
,点
在
上,且
轴,
.
(1)求
的方程;
(2)求与
有公共焦点的双曲线的方程,使得以它们的交点为顶点的四边形面积最大;
(3)过点
作斜率之积为
的两直线
,若
交
于
,
两点,
交
于
,
两点,
,
分别为
,
的中点,求
面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df7e26bd228679011198cf33e7b8bcb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0803835d6f594a60bd16c823e3ad2cd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19d5292ed97f06ad4423c7214f5642fc.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)求与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(3)过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa7d5fb23330a14a50f6d6a098d086c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44434b647ec546fe787e2164e0be6cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40304b883f3d23bbf066bc0af3c09862.png)
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名校
6 . 某工厂对一条生产线上的产品A和B进行抽检.已知每轮抽到A产品的概率为
,每轮抽检中抽到B产品即停止.设进行足够多轮抽检后抽到A产品的件数与B产品的件数的比例为k,单轮抽检中抽检的次数为x,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95240a8668b864ca18257ef2a8e80932.png)
A.若![]() ![]() |
B.当![]() ![]() |
C.若一轮抽检中x的很大取值为M,![]() |
D.![]() |
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名校
7 . 对于求解方程
的正整数解
(
,
,
)的问题,循环构造是一种常用且有效地构造方法.例如已知
是方程
的一组正整数解,则
,将
代入等式右边,得
,变形得:
,于是构造出方程
的另一组解
,重复上述过程,可以得到其他正整数解.进一步地,若取初始解时满足
最小,则依次重复上述过程 可以得到方程
的所有正整数解 .已知双曲线
(
,
)的离心率为
,实轴长为2.
(1)求双曲线
的标准方程;
(2)方程
的所有正整数解为
,且数列
单调递增.
①求证:
始终是4的整数倍;
②将
看作点,试问
的面积是否为定值?若是,请求出该定值;若不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09e7065aa112872161285c5f3bfc022.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3460cd2f27a53941986606734a9b479a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a79b9eaa5e7ab7a1e5c512b571914dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38ca4e60aab76ce3be3b5ffb9137f163.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ef282595213e6ac1c04b09c8703e176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e62b8fca47d65828c45fc8e38fe8beb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99c804f8356fa6120aa13b2d11bfea10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73f33dbb2fc073ae2d62891732e52dc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e31df193bb9a9b93b02f2daa2fb747c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e1e81a0995ee5492c4281539c65bf00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18483c9c195ecd922772527fa85c0fcb.png)
(1)求双曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb4c637bd4364a8d3b8d13889befd4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba78250f4e67347c7e80c543078d02e6.png)
②将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb4c637bd4364a8d3b8d13889befd4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52b66d595bfea3722fbc56e2fdccd548.png)
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解题方法
8 . 微分中值定理是微积分学中的重要定理,它是研究区间上函数值变化规律的有效工具,其中拉格朗日中值定理是核心,它的内容如下:
如果函数
在闭区间
上连续,在开区间
可导,导数为
,那么在开区间
内至少存在一点
,使得
,其中
叫做
在
上的“拉格朗日中值点”.已知函数
.
(1)若
,求函数
在
上的“拉格朗日中值点”
;
(2)若
,求证:函数
在区间
图象上任意两点
,
连线的斜率不大于
;
(3)若
,且
,求证:
.
如果函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4b04824a308519a61318a82aa97a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f3fcc5073759c73c7a63c8818eca5c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d11582cafaca7560189cf57e70f6a46d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58423a31ef72d8d161b775090c9ed2c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bcc88c47bbbace2c56adced4f781b6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d0d09c61ed9d7289d948d2ab559657f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.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/25a9a2cda21bc7d84330702878e61c63.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1dffe15ea2b4735ad2a274144301328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098345d96f648ba45ad923a79e815496.png)
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解题方法
9 . 已知函数
.
(1)求
在
处的切线方程;
(2)若曲线
与直线
有且仅有一个交点,求
的取值范围;
(3)若曲线
在
处的切线与曲线
交于另外一点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c2e33e637f3ce3f7163031be3bedfd0.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
(2)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1447dbe580ac5c825776995118e75acf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e209a60f1031bc4978c0cfce9bc1358.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2c9a4bbe272cdfd820fbbcfeceea4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc1ae6f6b30f3f422b6d4eb22c2d216.png)
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10 . 在计算机科学中,
维数组
是一种基础而重要的数据结构,它在各种编程语言中被广泛使用.对于
维数组
,
,定义
与
的差为
与
之间的距离为
.
(1)若
维数组
,证明:
;
(2)证明:对任意的数组A,B,C,有
;
(3)设集合
中有
个
维数组,记
中所有两元素间的距离的平均值为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6b93450a25edf994283872816f0a175.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d3f2ae3aa8322ec0e2c36471bfc7f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb963b1d6c794e8d3331214473f44d61.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/90154ab818dc6fd319683db8b642c3f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447cf234057b2b23e80cc0ec6770de2c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/288e149a1d5633a31b2fb684564e1547.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebde3a4de6911a9cf5dcc9641c76f5dd.png)
(2)证明:对任意的数组A,B,C,有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8513f18376e4e456b939d0f1cdb6e602.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/176ff134abb231aa2b562ca6fe240a44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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