1 . 已知数列
为递增等差数列,数列
为等比数列,且
,
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292b6a224fc7539b594c8b73d66f8848.png)
(1)求数列
与
的通项公式:
(2)对任意的正整数
,设
,求数列
的前
项和;
(3)求证
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7c1da88d583a3ae4818b9ccc3b1242c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c25a09929ddd94b69868d3048a79a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292b6a224fc7539b594c8b73d66f8848.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)对任意的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/230238beeb5b5f30228c16247c63ce24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
(3)求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2e9dab25cd410e93bf7de3abb619b11.png)
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2 . 在如图所示的几何体中,
平面
是
的中点,
,
.
(1)求证:
平面
;
(2)求平面
与平面
所成夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f4c3f9dd5d0343597a7f58a1989b537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3deeff8906bc596d92a7f177e854dd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7ad3a578f403b9e6b97fa2dc955fc11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5338f819dae23e41eb8d05cd1c227f45.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/2/a5d9631b-49bf-480a-9899-174a984b0601.png?resizew=125)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1e0bd4b30dc777ac9da80f6baa3eb31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b9d54cbbf601f4583659771eb534997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65277734669566578cbb7d690bb200fb.png)
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3 . 如图,在四棱线中,底面
为矩形,
平面
,点
是棱
的中点.
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30067b7b236d17af8a462f96a58d11bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68a7bf0da4f7c6f739d2e2461ad9b7.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfaa712e64750e3e2843bae68ebad6d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3d9fc62e8d6c61f0b1338a939b2f109.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68a7bf0da4f7c6f739d2e2461ad9b7.png)
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名校
解题方法
4 . 已知数列
满足
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94159bc22104744fe8c8fa782abd0288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3714ee515c531f2c55710673d3c804b4.png)
A.2023 | B.2024 | C.2027 | D.4046 |
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2023-10-26更新
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6卷引用:天津市河东区2023-2024学年高二上学期期末质量检测数学试题
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5 . 已知数列
,其前
项的和为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/614306cc3f34bdee4d5d885b79667645.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb7a45d0c43da7444c03b1566340cc1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/614306cc3f34bdee4d5d885b79667645.png)
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6 . 已知等比数列
的公比
,前
项和为
.证明
,
,
成等比数列,并求这个数列的公比.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354efd9145bb3bd889cf0eaadbcc55f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5188d6760683a860adab0cda195cdf80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd6fe11dfa538e67bfd63478fc428a1.png)
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7 . 过点
作圆
的切线
则切线长为______ ,过切点A,B的直线方程为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/704bdc451ab8f0ffa406993c59ab9913.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8b7615accfdce025efc2d02012c8501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a323c48a9562431cd9a44c1a21cc9e73.png)
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解题方法
8 . 若函数
是定义在
上的增函数,且对一切
,
,满足
,则不等式
的解为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/061813f1ec633c5c4c393c4de7938322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d3e1e248aef013c04f927d442f80997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9963700636f13d11030d7952f4fc145.png)
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9 . 已知
是定义在
上的奇函数,当
时,
,则不等式
的解集为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5749bb82edfb623c63ae4ec6b4d43da8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f72d20f1bf6d42731872b4554cf81a03.png)
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10 . 如图,在长方体
中,
,
.
,M,N分别是棱
,
,
的中点.若点P是平面
内的动点,且满足
平面
,则线段
长度的最小值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6155f82a6f64b20085976cea9b64193.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66fb71b75b63594ebeeeebd1963eed5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0efa6463e3070ffd0db08132ee42087e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28f79db7c270b6ff9fb0a538ee201cfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e06947327f4c41340b8713e8a6b4c87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/12/f569ab9f-ef89-4e37-924c-0dcbb55ab563.png?resizew=176)
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