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
13-14高二下·福建三明·期中
4 . 已知函数
是
上的增函数.
(1)若
,且
,求证
;
(2)判断(1)中命题的逆命题是否成立,并证明你的结论.
![](https://img.xkw.com/dksih/QBM/2016/11/29/1573186126700544/1573186133049344/STEM/b53443865f9b40ebbec63919508c6e49.png)
![](https://img.xkw.com/dksih/QBM/2016/11/29/1573186126700544/1573186133049344/STEM/a4bd4f4388c24bebb08908c9ae452547.png)
(1)若
![](https://img.xkw.com/dksih/QBM/2016/11/29/1573186126700544/1573186133049344/STEM/73c4b5be20ae45869835a8219f58f908.png)
![](https://img.xkw.com/dksih/QBM/2016/11/29/1573186126700544/1573186133049344/STEM/bcc104d6af0b4990a16b4ed625ba0494.png)
![](https://img.xkw.com/dksih/QBM/2016/11/29/1573186126700544/1573186133049344/STEM/6b931602181d430398881761b853fd51.png)
(2)判断(1)中命题的逆命题是否成立,并证明你的结论.
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2016-12-03更新
|
2604次组卷
|
3卷引用:2013-2014学年福建省三明一中高二下学期期中考试文科数学试卷
名校
5 . 若集合A具有①
,
,②若
,则
,且
时,
这两条性质,则称集合A是“好集”.
(1)分别判断集合
,有理数集Q是否是“好集”,并说明理由.
(2)设集合A是“好集”,求证:若
,则
.
(3)对任意的一个“好集”A,判断命题“若
,
,则
”的真假,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2faf3937abcb6a59071c17bc6bb10f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35a2410ce34b36954ed4923e600d42f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e13a814f8e081078dcf3788177affcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46de01c5104b9112a688df37eadb000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38f0e9c04402a0ffdaa25c3e3c82c7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd77104cc745d1e0e262122da34482d.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9720fd3e90e0f5dedc985310efea84e4.png)
(2)设集合A是“好集”,求证:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e13a814f8e081078dcf3788177affcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
(3)对任意的一个“好集”A,判断命题“若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae4f0ccdfc1206d809e581449d0452e.png)
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6 . 已知函数
.
(1)证明:函数
有且只有两个不同的零点;
(2)已知
,设函数
的两个零点为
,试判断下列四个命题的真假,并说明理由:
①
;②
;③
;④
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edca4db207f4b253d6e9c780e557642f.png)
(1)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bf910f82c3094b267a3d481d23d829f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca579894dad67bc82cb715fd48e0d70.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d3b114eb69ad77a0495468af7bb41b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/885c20eafab97db145af40138279adbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e2095119185f0410bb10cae34f14243.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e4a11440f9199546f719432280176f2.png)
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名校
解题方法
7 . 已知b克糖水中含有a克糖,再添加m克糖也全部溶解了,此时糖水变甜。请将这一事实表示为一个关于不等式的命题,并证明之.
您最近一年使用:0次
8 . 若集合
具有以下性质:①
,
;②若
,
,则
,且
时,
.则称集合A是“好集”.
(1)分别判断集合
,有理数集
是不是“好集”,并说明理由;
(2)设集合
是“好集”,求证:若
,
,则
;
(3)对任意的一个“好集”
,分别判断下面命题的真假,并说明理由.
命题
:若
,
,则必有
;
命题
:若
,
,且
,则必有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2faf3937abcb6a59071c17bc6bb10f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35a2410ce34b36954ed4923e600d42f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46de01c5104b9112a688df37eadb000.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38f0e9c04402a0ffdaa25c3e3c82c7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd77104cc745d1e0e262122da34482d.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a05551b1d4b65f27a932c33ddb1cb6ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
(2)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
(3)对任意的一个“好集”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
命题
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae4f0ccdfc1206d809e581449d0452e.png)
命题
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38f0e9c04402a0ffdaa25c3e3c82c7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa7d8e85b211a6d2aefa223c05c064ca.png)
您最近一年使用:0次
名校
9 . 课上我们学习了“
”符号和数学上陈述句
一些常用的否定形式
,实际上“若
,则
”为假命题可以表述为“至少存在特例
满足性质
,使
”,即我们常说的举反例.
(1)请利用上述逻辑语言说明以下两个命题为假:
①任何集合都不是空集的子集;②若
,则
;
(2)其他教材中有这样一种新命题的表述: 如果把命题“若
,则
”称为原命题,那么将其结论的否定作为条件,将其条件的否定作为结论,可以得到一个新命题“若
,则
”,我们称新命题为原命题的逆否命题.并且有一个非常强有力的结论:原命题与它的逆否命题是同真或同假的.请综合利用上述知识证明:对于正实数
,若
,则
;
(3)证明:原命题“若
,则
”与它的逆否命题“若
,则
”同为真命题或同为假命题.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ef73aff3fe470e367f4af24fdfff3df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d1c79d9d4f43ffb42f22c287058b5f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2303430b989c36a0c5380d64b3182690.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f2c566d4285f887b69c855f31849542.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e31113e042661f75628af5e3b2dc56f1.png)
(1)请利用上述逻辑语言说明以下两个命题为假:
①任何集合都不是空集的子集;②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52ddfcb6c5c9f8b50444386d7221154c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9138d5904f6ff2a48f29e820ce54e0e0.png)
(2)其他教材中有这样一种新命题的表述: 如果把命题“若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/130adfc0b77a1bb4046c19fc52d5fe78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d277dac920ea0456d486ea528332f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/127a0d8c1c7d15ed40ec4b8bca0ebdf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/485a2d99320384a0857b00ce9ab9e990.png)
(3)证明:原命题“若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d1c79d9d4f43ffb42f22c287058b5f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d277dac920ea0456d486ea528332f0.png)
您最近一年使用:0次
2023·全国·模拟预测
解题方法
10 . 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