名校
解题方法
1 . 在平面直角坐标系
中,设二次函数
的图像与两坐标轴有三个交点,经过这三个交点的圆记为C.
(1)求实数b的取值范围;
(2)请问圆C是否经过某定点(其坐标与b无关)?请证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa8acaf5a9115e6adca8cab1e9faa6e3.png)
(1)求实数b的取值范围;
(2)请问圆C是否经过某定点(其坐标与b无关)?请证明你的结论.
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名校
解题方法
2 . (1)设
均为实数,且
,求证:
.
(2)已知实数
满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83d19a3c7b83608ccc3b47c7d15f4596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/729d91bd444b64e05a046836a7392aab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d60376d38e1eb7bf102743e95bcbb6d7.png)
(2)已知实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/597b657580f9e7669eeb848adba0f4e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e8601794ea87f18b840e3abf951d838.png)
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3 . 设k是正整数,集合A至少有两个元素,且
.如果对于A中的任意两个不同的元素x,y,都有
,则称A具有性质
.
(1)试判断集合
和
是否具有性质
?并说明理由;
(2)若集合
,求证:A不可能具有性质
;
(3)若集合
,且同时具有性质
和
,求集合A中元素个数的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196ee858814f200b4414975cc154f333.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5a33197acadf8491f921a9330f94260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daac43c7675fa411b35028e09b0bad90.png)
(1)试判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb1aca774c04bd0599d77a1c0c9b43d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3aabb4483ddc6e67b4fedb0e09518f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a5d8bc28ee110a9540f383828b7d245.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad04af133fc5d09790cddd1c2c199892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41b85f6960a2a4f988e5ddd9ef48d0e.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d096a06a215ba878c07d26f801498aaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0127f7421ce1839e335f091d730736af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677dcea09d90af109df9cc72cff15cf0.png)
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2023-05-10更新
|
838次组卷
|
3卷引用:北京市清华大学附属中学2022-2023学年高一下学期期中数学试题
北京市清华大学附属中学2022-2023学年高一下学期期中数学试题(已下线)单元高难问题01集合中的新定义问题-【倍速学习法】(人教A版2019必修第一册)北京市第三十五中学2023-2024学年高一上学期期中测试数学试题
4 . 已知数列
满足
,
,
.
(1)猜想数列
的单调性,并证明你的结论;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/195fc747e2fc50cb6df2c844d51e4d80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d8c7493005191506bf32b9e39a5a4c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deda945164283569437cda6976fe35ea.png)
(1)猜想数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d497e59ca415b9973ae8b07cd28e472.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5825807d5e23919ae8dcd6751db947d9.png)
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2023高三·全国·专题练习
名校
5 . 已知由实数组成的数组
满足下面两个条件:
①
;②
.
(1)当
时,求
,
的值;
(2)当
时,求证
;
(3)设
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0766a9dd0058b52abd9ad17ddb04fc2c.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8126d64e2becb117d8d42af22a5919b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c12927cb41f08b3fd18f338623db8d8d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24819c61a0a42291903e3c2f5e1c6e41.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf7310dc844ccb33e0ff0b62aeb47b8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/818dfdff56f53b3ecfb1096a692e914d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/beb74763554fd459d736ed9c7b387e01.png)
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名校
6 . 已知
,
,
,且
.
(1)求证:
;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b74110bc818c2f5a53d63451c5251eb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/436a2732e9c9d5ce401c448cd9de80e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8b804503a0e9910e2ac815a63b020b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df9542750f9551020d70bca880495b4b.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46c83d0a4987bb91d01e192f5c0f82b2.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04574501bde04e90e72ee9f734b83d61.png)
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名校
解题方法
7 . 已知在
中,
.证明:
(1)
;
(2)
在
上恒成立;
(3)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e92a98e220a9a1f2a1caa37e4cf4e213.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba5e4691210486a560c59df09937d9f8.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/991a6e773c41687e5b13d36da7612e01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d01dc2d99655cf7598837cb0886166ed.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb222ce13688da6fc57089ebf5812b0e.png)
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8 . 如图所示,在等腰梯形
中,
,对角线
与
交于
,点
分别是
的中点.求证:
是等边三角形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2deeeceafa44ff745fd28b1fbe918fd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c5970b2d21e88c5ed4fb53584e62f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ab3671a59305d409003138f4d7b8983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f248ba5ffa1df121dc0b037f51e3a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c7d7169ecf9b68fbe9244ceba6724e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/d223e30b-5637-48e6-b039-270105f3bbf8.png?resizew=127)
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名校
9 . 如图,已知
是
的直径,弦
与直径
相交于点
.点
在
外,作直线
,且
.
![](https://img.xkw.com/dksih/PilotRun/2019/10/9/2308152996413440/2314881509220352/EXPLANATION/fb21d114d2184553925dd4cfc07b8e7a.png?resizew=143)
(1)求证:直线
是
的切线.
(2)若
,
,
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0513a30a20818b08dc6ea26e2c86492.png)
![](https://img.xkw.com/dksih/PilotRun/2019/10/9/2308152996413440/2314881509220352/EXPLANATION/fb21d114d2184553925dd4cfc07b8e7a.png?resizew=143)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/154c7f681d6ed9ba848f2dd3ac04a5b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91dad3d14c9ce550215929f248facc73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
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10 . 已知两个均含有
项的有限数列
,
,其中对于
,
且
.定义数列
与
之间的距离:
.定义数列
的“和序列”
,其满足对于
,
,数列
的
项和记为:
;定义数列
的“和序列”
,其满足对于
,
,数列
的
项和记为:
.
(1)已知数列
,
,求
和
;
(2)当
且
时,求
的所有可能取值;
(3)当
且
时,求
的最大值和最小值,并分别列举一对数列
,
,使
取到最大值和最小值;
(4)求证:对于
,当
且
是4的倍数时,
的最小值为0;
(5)当
,
时,直接写出一对数列
,
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5d8aa940a0e54ac8979395fc6dff741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad97e1eb89a88cb2294a7c6715a45b3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff8c536a4f234f76110738d8f0fe343e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a406d53fd6ffd9ee6cd914f5e2b0a9bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/572366af4b70473246d02890f08e5bf4.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/4fdf61eeaedb55fceca383cced5a2a6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b93f46e2f1e4c9b7c1e6b65cb4b71a8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff8c536a4f234f76110738d8f0fe343e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68c347988fa8b9a610e98d8457a7c3c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43d70d5962157e2116e7f1a267153967.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4a3ecd7015e91acdaf5dab17cd4db0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a35dc828d8d4cd583ec2f59fc51cc13c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff8c536a4f234f76110738d8f0fe343e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b6bfb29dcb99958d2f7044f95228a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b3961f5681dca061c2d5eccdfa1d6b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32eda5781c401532afa23e7f7fbbeb38.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7e9f8c9d7cd0c8dcb5b6a29326f3000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4593b4328615e3d4cb4bb409cec24fa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3bcb4828b16c8e845492f1a53ddd9a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a5f4d7aad74922a9af3f9d0c04e4bf.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e79b6edfd5f5bb3866599fcb2aa6f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ce4e30be12ce8f1d3fe5299ba630fb.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe08722cf9300fe188dbbb71989c06c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1ffbdfab9dff3ff41ea474f06375032.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ce4e30be12ce8f1d3fe5299ba630fb.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/93ce4e30be12ce8f1d3fe5299ba630fb.png)
(4)求证:对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86bc44f533c05d764a6f576ca47a4ba5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc930ac7a42a0ec0ce9c65c9e7c24182.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3bcb4828b16c8e845492f1a53ddd9a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93ce4e30be12ce8f1d3fe5299ba630fb.png)
(5)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe08722cf9300fe188dbbb71989c06c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b8758b01cebe286da8592711f614892.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/c7ae7de525dccd6844f5477dac832182.png)
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