真题
名校
1 . 请先阅读:
在等式
(
)的两边求导,得:
,由求导法则,得
,化简得等式:
.
(1)利用上题的想法(或其他方法),结合等式
(
,正整数
),证明:
.
(2)对于正整数
,求证:
(i)
; (ii)
; (iii)
.
在等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32eac4b7f177c041219fab18de973c5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dc1e9d6c038e98eb3ced183bb6dcc53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d0035911136a83c7915137c3438e055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92ba7e0c985c673fbb513b4a97d93746.png)
(1)利用上题的想法(或其他方法),结合等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9641914b1dcb9c0097550aebead97810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910adb8a80fceb7949c3526087947220.png)
(2)对于正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba5c659f6e87ab7327ef8c3b3368ab23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbe3f70202a3b38d077fe431a6e63099.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62a002cedddac1e750b5e3f204974078.png)
您最近一年使用:0次
2016-11-30更新
|
2395次组卷
|
4卷引用:2008年普通高等学校招生全国统一考试数学试题(江苏卷)
真题
2 . 从A,B,C,D四个中选做2个,每题10分,共20分
设a,b,c为正实数,求证:
.
A.选修4—1 几何证明选讲 如图,设△ABC的外接圆的切线AE与BC的延长线交于点E,∠BAC的平分线与BC交于点D.求证: ![]() ![]() |
B.选修4—2 矩阵与变换 在平面直角坐标系 ![]() ![]() |
C.选修4—4 参数方程与极坐标 在平面直角坐标系 ![]() ![]() ![]() ![]() |
D.选修4—5 不等式证明选讲 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eba94e4ac26a70c1aefd4743757583b.png)
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真题
3 . 已知
是等差数列,
是公比为q的等比数列,
,
,记
为数列
的前n项和.
(1)若
(m,k是大于2正整数),求证:
;
(2)若
(i是某一正整数),求证:q是整数,且数列
中每一项都是数列
中的项;
(3)是否存在这样的正数q,使等比数列
中有三项成等差数列?若存在,写出一个q的值,并加以说明;若不存在,请说明理由.
![](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/89fe0f4e8a80a2840c0f6929a8a6351b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf92b5d061e45e1c720cdf93409ae75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4fa1d84e33943f4947d4dec19f80f6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fce13be3cf67126a906396ba8ca32721.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac52d20d7bb3a6631f5035ef18b64c19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)是否存在这样的正数q,使等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
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真题
解题方法
4 . 在棱长为4的正方体
中,O是正方形
的中心,点P在棱
上,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/8c2ce9ba-8710-478f-ae32-09829d4b209d.png?resizew=151)
(1)求直线AP与平面
所成的角的大小(结果用反三角函数值表示);
(2)设O点在平面
上的射影是H,求证:
;
(3)求点P到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632f2bf1cd0435041fa04b01901d1c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3082c96cb263ae888242114111baea5c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/8c2ce9ba-8710-478f-ae32-09829d4b209d.png?resizew=151)
(1)求直线AP与平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
(2)设O点在平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714ae984b13488d536f583f610e59945.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/317f727d4d61935571b511f3e3aa6f2b.png)
(3)求点P到平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028856d5101687dd8eaf130846489cfd.png)
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真题
5 . 如图,设
的外接圆的切线
与
的延长线交于点E,
的平分线与
交于点D.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e64237dbda7ac340829f35bc79d86970.png)
![](https://img.xkw.com/dksih/QBM/2022/11/9/3106155121123328/3108296005550080/STEM/ceba556b7fb54b2b8cd1c05d9fadc81c.png?resizew=237)
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真题
6 . 在直角三角形
中,
分别为斜边
上的高和中线,且
与
之比为3∶1,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6806886659f7188ca7791fc1958064ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/727ad3e630a224303d6d3b8ad5c114ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdb8eca20ce2c918ea4034ea15210c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df34369163e511f14028168cb0b21186.png)
![](https://img.xkw.com/dksih/QBM/2022/11/7/3104399899189248/3104530293555200/STEM/196deaba3a8a40cfba7c6cb7862ed35c.png?resizew=220)
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7 . (1)若三角形三内角成等差数列,求证:必有一内角为
.
(2)若三角形三内角成等差数列,而且三边又成等比数列,求证:三角形三内角都是
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
(2)若三角形三内角成等差数列,而且三边又成等比数列,求证:三角形三内角都是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
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真题
8 . 如图,在五棱锥
中,
底面
,
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/dfed0b05-2ba1-4d10-bfd2-7041decb772b.png?resizew=163)
(1)求异面直线
与
所成的角;(用反三角函数值表示)
(2)证明:
平面
;
(3)用反三角函数值表示二面角
的大小.(本小问不必写出解答过程)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/989babd4c8db2422e5d239e03dae94b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10c83f8945042b9c8fb2fbdac9308d62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96c6a57517fb8cdbe016cdee4aa64756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7fc9b1a395ad2e2fb2b207560754dc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73320c736844aaf8e0cc98542a227513.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/16/dfed0b05-2ba1-4d10-bfd2-7041decb772b.png?resizew=163)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d4db9b82b67efe45a02fca32bfcf5dc.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ffc6952e988d04f22f0fb2f7f0ab7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9c9cfa597b444b5c9dbae7a825a695.png)
(3)用反三角函数值表示二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c53f1e79257ff52a0408fdc482488d0.png)
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真题
解题方法
9 . 设数列
满足:
,
,证明:
为等差数列的充分必要条件是
为等差数列且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2249690697d2901e3baf1ff602c366cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dc4124f53a0ebdcc328e5ca6404b6ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ee1767ef20c21af6baaab122f4c834e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64975df9b6b1c5f116edb3d009cca72f.png)
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真题
10 . 已知函数
满足下列条件:对任意的实数
都有
和
,其中
是大于0的常数.设实数
,a,b满足
和
.
(1)证明:
,并且不存在
,使得
;
(2)证明:
;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9dbb70efb53fdd394d7eb8f7720629c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ce7ae90d808f05e86ea063238e4b2f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffb13c8f221c87d9e6eae949405d835d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c2f1ca03ade14de6711c85de8fc5df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f35f7dcce39f3d4dc6b7faf84dc1d0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a17884a2d114eee89f3def58398d2e48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90e8c2c10db5cb8dd9db7a63ef34e655.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c6d2d0d52b0ff7e63d3cfe089786e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a99016f45be584eb484c21efb2a26c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c66ab02a710aff40efd8b09ed714e69f.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039b2d56c9cf6aa38f0c89a932525618.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df6f6b21469f43a953730dee557d8df4.png)
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