解题方法
1 . 已知函数
在R上连续,且存在导函数
,对任意实数x,满足
,当
时,
.若
,则x的取值范围是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4b04824a308519a61318a82aa97a05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c9333496d9d516dfff9a8f79b0b25d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e82c4003d20b36777f7aea584e3dd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07370085c170b6e5cec19da627a3d58c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/377fc3bdedde5c9283e6e0d1dc205d41.png)
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2 . 在一个有穷数列的每相邻两项之间插入这两项的和,形成新的数列,我们把这样的操作称为该数列的一次“和扩充”.如数列1,2第1次“和扩充”后得到数列1,3,2,第2次“和扩充”后得到数列1,4,3,5,2.设数列a,b,c经过第n次“和扩充”后所得数列的项数记为
,所有项的和记为
.
(1)求
;
(2)若
,求n的最小值;
(3)是否存在实数
,使得数列
为等比数列?若存在,求出
满足的条件;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5425108c557f0f21474c045334f97d9c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dcd08431daac10be93e6fafbc5d4a90.png)
(3)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce224c28ca451c4f105dc3b077736cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
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3 . 已知函数
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93208bc770714ae8311ab2ba6274ea8d.png)
A.存在![]() ![]() ![]() |
B.对任意![]() ![]() ![]() |
C.对任意![]() ![]() ![]() |
D.存在![]() ![]() ![]() |
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解题方法
4 . 设直线系
(其中0,m,n均为参数,
,
),则下列命题中是真命题的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b73cf02fb24824f6343afb9d255dd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8453a001673a2850320eebc033c3618d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a0a309ce715cd1a9d503c1a201c71ae.png)
A.当![]() ![]() |
B.存在m,n,使直线系M中所有直线恒过定点,且不过第三象限 |
C.当![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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5 . 已知
,函数
,下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e6404cd8d3bb839cccf6e4176c8fb9a.png)
A.![]() |
B.若![]() ![]() ![]() ![]() |
C.若函数![]() ![]() ![]() |
D.若![]() ![]() ![]() |
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6 . 已知半径为
球与棱长为1的正四面体的三个侧面同时相切,切点在三个侧面三角形的内部(包括边界),记球心到正四面体的四个顶点的距离之和为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
A.![]() | B.![]() |
C.![]() ![]() | D.![]() |
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7 . 函数
满足:当
时,
,
是奇函数.记关于
的方程
的根为
,若
,则
的值可以为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1092d77e2570be5584ebc0cdcdca2ea3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b08d939cee48042d0a565a53dbeadf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57d500a74b865ade2b576720c04becd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ed0b01191624fcc0469ecad0287a0b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fdb4c1ad5fce7cf952767c03b8eb6ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
A.![]() | B.![]() | C.![]() | D.1 |
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8 . 若
表示自然数
的最大奇因数,例如
,
,
,记
(
为自然数),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
______ .,
的通项公式为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91eb1a74ed4eb789a5cf6bf0d08900a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ac85888a9b802aea6eb688edac8f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18443df3018a6806c7ded2bf11ed70a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9bf2837ae4419c83bb69ad10ab7ef14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/640faa4d58105334dafd1cf218f30ae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
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9 . 莫比乌斯函数在数论中有着广泛的应用.所有大于1的正整数
都可以被唯一表示为有限个质数的乘积形式:
(
为
的质因数个数,
为质数,
),例如:
,对应
.现对任意
,定义莫比乌斯函数
(1)求
;
(2)若正整数
互质,证明:
;
(3)若
且
,记
的所有真因数(除了1和
以外的因数)依次为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e046acc0e785892df1ef03a440b0fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5c607987b73502db63f77c9799f4bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08fe943e1acfb453f41bee79119cce60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38261aad19184a74c797b6b88ffd344d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86cb09df4dbbe40a2b7ed54da17346dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09881de0dc186bbcd1e60eb00159ee97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b5872b44498c348c023828ed66e86d1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9b2c4263428e2ee419589171f27e23f.png)
(2)若正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/201e0fbcfb6833c4b1917cfed3096b6f.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a4a887eaea7f0aac8505ed3b3c0c678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4811e3603e8790c25aaf91c41d7c7f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/202a57af91d5be04e95fcbdb8f2b788f.png)
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名校
解题方法
10 . 若数列
满足
,其中
,则称数列
为M数列.
(1)已知数列
为M数列,当
时.
(ⅰ)求证:数列
是等差数列,并写出数列
的通项公式;
(ⅱ)
,求
.
(2)若
是M数列
,且
,证明:存在正整数n.使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a07614926587f57bc5f341c4f97f4d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec574b71bbd7671223f8c833c8c8b61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ec1a744042c32d0a851f98fafaa81f3.png)
(ⅰ)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/362832fa3d3c13c1eafd565349d66dce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115da54f93de5e89d1e7f443fccb61f8.png)
(ⅱ)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0992722f5002aeafa39d25c6b5f4644b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21085fbd6c4b34588f17fc466c845ffe.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a789a9be1723bfbd38ae538a9f39dc1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ce64685821c3e55c07f151996ca8c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/446e8a7985d4d3dd95c70dc4aad67861.png)
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