1 . 已知直线与抛物线交于两点.
(1)求证:若直线
过抛物线的焦点,则
;
(2)写出(1)的逆命题,判断真假,并证明你的判断.
(1)求证:若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/749b17e02ac5325dcfcac745a51b5170.png)
(2)写出(1)的逆命题,判断真假,并证明你的判断.
您最近一年使用:0次
2 . 请仔细阅读以下材料:
已知
是定义在
上的单调递增函数.
求证:命题“设
,若
,则
”是真命题.
证明:因为
,由
得
.
又因为
是定义在
上的单调递增函数,
于是有
. ①
同理有
. ②
由①+ ②得
.
故,命题“设
,若
,则
”是真命题.
请针对以上阅读材料中的
,解答以下问题:
(1)试用命题的等价性证明:“设
,若
,则:
”是真命题;
(2)解关于
的不等式
(其中
).
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
求证:命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/473303324fc54d9fbef44f60c383cdd4.png)
证明:因为
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/8e1dba6af48b4f02a02353cfceac54bc.png)
又因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
于是有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/e138ae2d9d174247aa79ca4be523361f.png)
同理有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/6e01071f3f38469e8e15c3d76700b775.png)
由①+ ②得
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/473303324fc54d9fbef44f60c383cdd4.png)
故,命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/473303324fc54d9fbef44f60c383cdd4.png)
请针对以上阅读材料中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)试用命题的等价性证明:“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/b2a8db4930d64746bee7acb58118f1be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36ce14a18f423fcff11def7512150e.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973809381376/1571973814910976/STEM/add4ea370e3946109d077624424d5f04.png)
(2)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d4c9d254df7fc5169fe8e745a3b74b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c0c6e3ada0970f9a1fefd7200ff677.png)
您最近一年使用:0次
3 . 请仔细阅读以下材料:
已知
是定义在
上的单调递增函数.
求证:命题“设
,若
,则
”是真命题.
证明 :因为
,由
得
.
又因为
是定义在
上的单调递增函数,
于是有
. ①
同理有
. ②
由①+ ②得
.
故,命题“设
,若
,则
”是真命题.
请针对以上阅读材料中的
,解答以下问题:
(1)试用命题的等价性证明:“设
,若
,则:
”是真命题;
(2)解关于
的不等式
(其中
).
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
求证:命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/25072c5bb7274310b540c233b24508ed.png)
证明 :因为
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/63e0cee9c0994ebe9e8c9162a5fd4c58.png)
又因为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a141e697b1a31a9a4e759984e899a5.png)
于是有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/816123fe83654315b9ad464cbbd7d4fd.png)
同理有
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/4cb2879dd5224072a6253ea98f00a84c.png)
由①+ ②得
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/25072c5bb7274310b540c233b24508ed.png)
故,命题“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/25072c5bb7274310b540c233b24508ed.png)
请针对以上阅读材料中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)试用命题的等价性证明:“设
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/24e4835206fe4a69b03e5c5562294155.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36ce14a18f423fcff11def7512150e.png)
![](https://img.xkw.com/dksih/QBM/2015/1/28/1571973654781952/1571973660549120/STEM/8209882c950f4c02a7aa91c6ad4584ae.png)
(2)解关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d4c9d254df7fc5169fe8e745a3b74b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eda48853e8bdb7e266370b4e0d5a258.png)
您最近一年使用:0次
4 . “角股猜想”是“四大数论世界难题”之一,至今无人给出严谨证明.“角股运算”指的是任取一个自然数,如果它是偶数,我们就把它除以2,如果它是奇数,我们就把它乘3再加上1.在这样一个变换下,我们就得到了一个新的自然数.如果反复使用这个变换,我们就会得到一串自然数,该猜想就是:反复进行角股运算后,最后结果为1.我们记一个正整数
经过
次角股运算后首次得到1(若
经过有限次角股运算均无法得到1,则记
),以下说法有误的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99cebc0b8a5e503e1e24cb57dbbde5b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0a1ae2246dcd710cf913417406c2efa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeadb619367f955549a75a4eeb931011.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8a66dd357f643ef976d14e097446fcf.png)
A.![]() ![]() |
B.![]() |
C.对任意正整数![]() ![]() |
D.![]() ![]() |
您最近一年使用:0次
5 . 设
为实数,定义
生成数列
和其特征数列
如下:
(i)
;
(ii)
,其中
.
(1)直接写出
生成数列的前4项;
(2)判断以下三个命题的真假并说明理由;
①对任意实数
,都有
;
②对任意实数
,都有
;
③存在自然数
和正整数
,对任意自然数
,有
,其中
为常数.
(3)从一个无穷数列中抽出无穷多项,依原来的顺序组成一个新的无穷数列,若新数列是递增数列,则称之为原数列的一个无穷递增子列.求证:对任意正实数
生成数列
存在无穷递增子列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/796bb39a2ab23cfdb6e463ab30a7af2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4758b555ca9b157cc074f1e4a092e34a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f61c0bb2370087736c8e00e108b48c8.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c051dc675bcca6a8f70a3dbe922354.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3121951a9b059eef49b4a346d3aa2b94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c400b893304c51631873ded41027cf48.png)
(1)直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4e0c84de10f0f2186313169c3dc997b.png)
(2)判断以下三个命题的真假并说明理由;
①对任意实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81fbdf49cd00af1ff87259836ddd9f6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/508cd31480a898a71472e2d5d22377c7.png)
②对任意实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81fbdf49cd00af1ff87259836ddd9f6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c99515d9952f2f7739fd750a31128f.png)
③存在自然数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a178f2c27906fc74afee1b7d7d52746.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1563da7b0f046a469476668a3686e8f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f59a60eb4d63ebc879ae5c26413bcdcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(3)从一个无穷数列中抽出无穷多项,依原来的顺序组成一个新的无穷数列,若新数列是递增数列,则称之为原数列的一个无穷递增子列.求证:对任意正实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da069077c220af26b9e77b02baeee4a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4758b555ca9b157cc074f1e4a092e34a.png)
您最近一年使用:0次
2023·全国·模拟预测
解题方法
6 . 设点
在椭圆
内,直线
.
(1)求
与
的交点个数;
(2)设
为
上的动点,直线
与
相交于
两点.给出下列命题:
①存在点
,使得
成等差数列;
②存在点
,使得
成等差数列;
③存在点
,使得
成等比数列;
请从以上三个命题中选择一个,证明该命题为假命题.
注:若选择多个命题分别作答,则按所做的第一个计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c993e34db40190e64654a10b0c13c672.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad523e69a1bf925e73a22900b9855df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d6678a1a5cc14704ecf06a7648ff543.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d4aca03910382accfe738520daf689c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
①存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f23bfdeeaa1efc12f64328e962d395b.png)
②存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8d3db975e7888ac13b4448b874b972d.png)
③存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8d3db975e7888ac13b4448b874b972d.png)
请从以上三个命题中选择一个,证明该命题为假命题.
注:若选择多个命题分别作答,则按所做的第一个计分.
您最近一年使用:0次
解题方法
7 . 设
,过
斜率为
的直线与曲线
交于
,
两点(
在第一象限,
在第四象限).
(1)若
为
中点,证明:
;
(2)设点
,若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e34f42b3be15518c29e3689c9fe6d6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62769b7177ef4bc952dc1dd51d6b510.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5db192285632d1991b4ee7a003a52205.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ffbb4e6b92463a41bd9460dac6b1ca7.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c664dcdcf88a834707b415061bed5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d5592a72ca90eeb5a9267340c61c673.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc329b32ecf0f0532d09a8a21343e8cb.png)
您最近一年使用:0次
真题
8 . (1)若四边形
的对角线
将四边形分成面积相等的两个三角形,证明:直线
必平分对角线
;
(2)写出(1)的逆命题,这个逆命题是否正确?为什么?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(2)写出(1)的逆命题,这个逆命题是否正确?为什么?
您最近一年使用:0次
9 . 数列
对任意
,且
,均存在正整数
,满足
.
(1)求
可能值;
(2)命题p:若
成等差数列,则
,证明p为真,同时写出p逆命题q,并判断命题q是真是假,说明理由:
(3)若
成立,求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c703ace0d2c22dd947a19d8afc74eac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d9f79b02c30f810f7d9c661fa7e44c7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
(2)命题p:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26b39cb7d4efd2dd15a1f39ac6ef72c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b30bfc8674948c31b09f824402ebada.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3557b9d9ef8529d963d2cd5962add5e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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解题方法
10 . 若两个函数
和
对任意
,都有
,则称函数
和
在
上是疏远的.
(1)已知命题“函数
和
在
上是疏远的”,试判断该命题的真假.若该命题为真命题,请予以证明;若为假命题,请举反例;
(2)若函数
和
在
上是疏远的,求整数a的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70128385b9ab66ac44614af35a0dcdce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d1226912a2b9d5c7027854fcd762cff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4776c85b79df196f606d3ebf3697fbc3.png)
(1)已知命题“函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/894853b480d4eb048607e45222f9f754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd5f1897f20ab79ba69ec855ba97be08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/894853b480d4eb048607e45222f9f754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9869bc4fdbe558198772d2c70b4466.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0fb0d24064b04be7bb11ae0e5e590de.png)
您最近一年使用:0次
2022-02-22更新
|
211次组卷
|
2卷引用:甘肃省平凉市静宁县两校2022-2023学年高三上学期第一次质检考试数学(理科)试题