名校
1 . 观察数列:①
;②正整数依次被4除所得余数构成的数列
;③
.
(1)对以上这些数列所共有的周期特征,请你类比周期函数的定义,为这类数列下一个周期数列的定义:对于数列
,如果________________,对于一切正整数
都满足___________________成立,则称数列
是以
为周期的周期数列;
(2)若数列
满足
,
为
的前
项和,且
,求数列
的周期,并求
;
(3)若数列
的首项,
,且
,判断数列
是否为周期数列,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8d9f608508a65794125b39e67b98eb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb6d15b3f5b6f23a9cb341ff3e43f215.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/078205bbd0d854b6aaf5aa6e0a772723.png)
(1)对以上这些数列所共有的周期特征,请你类比周期函数的定义,为这类数列下一个周期数列的定义:对于数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff13e48f70a467d750be8179c63f534.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6492a4d97fd8f988963cf177ec14fcb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbf62141da783d700923fa2d17b9ae0.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5f61c2e3ee306d0c805f54f83761f85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa9cef966e838bf77be9b00d410741c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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解题方法
2 . 甲、乙、丙三个盒中分别装有大小、形状相同的卡片若干,甲盒中装有2张卡片,分别写有字母A和B;乙盒中装有3张卡片,分别写有字母C,D和E;丙盒中装有2张卡片,分别写有字母H和I.现要从3个盒中各随机取出1张卡片;求:
(1)取出的3张卡片中恰好有1张,2张,3张写有元音字母的概率分别是多少?
(2)取出的3张卡片上全是辅音字母的概率是多少?
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/3/81aec317-9f42-4d3c-9202-2fb4d65ffe61.png?resizew=343)
(1)取出的3张卡片中恰好有1张,2张,3张写有元音字母的概率分别是多少?
(2)取出的3张卡片上全是辅音字母的概率是多少?
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名校
3 . 尝试使用概率的“可加性”解决下面的问题:
(1)设
是同一样本空间中的两个事件,探索
,
,
,
之间的等量关系,并说明理由.
(2)甲、乙各抛郑
枚硬币,证明:“甲得到的正面数比乙得到的正面数少”这一事件的概率小于
.
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88beddb7f2f069cb99a669e12d9ce617.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b6f8cb2faaad82b53b2a66ee817a37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/108ab49f370919e730e3567070deee65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a094432f03455ff0ef89356999f6b5a.png)
(2)甲、乙各抛郑
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
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4 . 定义
,其中
为奇素数.
(1)给出同余方程
的满足
的一组解;
(2)(代数基本定理)设
,且
,求证
在
内至多有
个解;
(3)(
小定理)求证:
;
(4)(原根存在定理)若正整数
满足:
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80754c6842db876978ab0c306640186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411837b4b3078d05b43cb0439259a362.png)
,则记
,则称
为
在
意义下的阶,求证:必定存在
,有
;
(5)求证,存在
,都存在
中必有一者成立;
(6)说明当
时,
必有一组非零解
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa82bc7122ca89b418d33b694350f987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(1)给出同余方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b2afe64159eac83151200719a5c815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a964cf4f45603e28dec030a286786750.png)
(2)(代数基本定理)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc2b731f8e2eaa9ac5e3eff49f586820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a59917a9390d454668b58a0c22c08b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae6227f2f164910966f194fb857d5e6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b75c20efa27f38e93e3ce9e00005ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(3)(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c9362bb4f6acf4bf074d0b7bc7b7aba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34e11a5e27430447b806863c5fdc76e.png)
(4)(原根存在定理)若正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e026ac41c5502a4743b336845bae2d95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1e84f1d28522a63e61283132d1af48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80754c6842db876978ab0c306640186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411837b4b3078d05b43cb0439259a362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/097f1e1905c0034991b00c360c5d28f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58bb4eb36d6c696377fe2e14bbb66858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa75d323fa66f6320034f2c7f45a6c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b657e7e787a785063afc56cb983ca56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e0f45eacebab3cc301c18df8fa0cd1.png)
(5)求证,存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66543500cdc7031573611000b1b5f85c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/107ba654d76c8b754beb5d173c06c6a7.png)
(6)说明当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13b47dded788f1274064925d5f38384e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95730071c97ce5e4d33780626f52e23c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2bad85768515c66beee8beeb0556520.png)
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5 . 已知函数
,其中
,证明:存在
,且
.
的根的实部全部大于0.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70b748237d3ad510f872cb96097e0e32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82f016c950ab496393ba95c709cbc6e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af0e0c37d5595299c1886e6d812a946e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df3ac8c6973c1a97dc47d4fb6737c709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7165f4d49fc207e7cdc35fccdb18e532.png)
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6 . 求符合条件的序列
的个数,满足如下条件:
(1)
;
(2)
,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ed4bbcd82234606bc1b7a4e45f074f.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91c69891459249e3cd9130a4e58aebcd.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b91d5fd3ecbc0873ff2e9c467ed26c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f5f83f22403ce79ca07af950eb399c3.png)
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解题方法
7 . 已知方程
所表示的曲线为E,点
,直线
.
(1)当直线
与曲线
只有一个公共点时,求
的值;
(2)若
,求曲线
上的动点
到点
的距离的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cfda0696f276fff4177533f8c63e349.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78e0b4cce429003557b051ea0fa2f7de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8183da2a1ac0abfb01bd044c0afe198.png)
(1)当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc895959e9bc92294dc9dd2263dbf0c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
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解题方法
8 . 已知函数
的定义域为
,值域为
.若
,则称
为“
型函数”;若
,则称
为“
型函数”.
(1)设
,
,试判断
是“
型函数”还是“
型函数”;
(2)设
,
,若
既是“
型函数”又是“
型函数”,求实数
的值;
(3)设
,
,若
为“
型函数”,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb185cecf0a22b322a82545bdfcb321c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11e685edd2226794e07c27f60acec2c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80177489d687f891f9f3fc5f8860001d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/616f54a98965940ed871b64e55331a53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af07d272da54872ad5a5d2c8181760af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82c69b465c78a4f15e22ec963e8d6655.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be1ce3f01e2b6364f9a9fdaf197d5e29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b84f0063e7e2829fd6169b06ce05f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d141acf95bdc962ad042e694243670c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e1a1611f320c0f358df77aaae3f942.png)
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解题方法
9 . 有一正方形景区
,
所在直线是一条公路,该景区的垃圾可送到位于
点的垃圾回收站或公路
上的流动垃圾回收车,于是,景区分为两个区域
和
,其中
中的垃圾送到流动垃圾回收车较近,
中的垃圾送到垃圾回收站较近,景区内
和
的分界线为曲线
,现如图所示建立平面直角坐标系,其中原点
为
的中点,点
的坐标为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/18/f2081f60-dda2-4d2b-bbf3-8173e9952f7c.png?resizew=150)
(1)求景区内的分界线
的方程;
(2)为了证明
与
的面积之差大于1,两位同学分别给出了如下思路,思路①:求分界线
在点
处的切线方程,借助于切线与坐标轴及景区边界所围成的封闭图形面积来证明;思路②:设直线
:
,分界线
恒在直线
的下方(可以接触),求
的最小值,借助于直线
与坐标轴及景区边界所围成的封闭图形面积来证明.请选择一个思路,证明上述结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589786dd7c3a2679c3230b671cd232d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589786dd7c3a2679c3230b671cd232d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a948d2f7732d7f03e986c63712089b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/18/f2081f60-dda2-4d2b-bbf3-8173e9952f7c.png?resizew=150)
(1)求景区内的分界线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)为了证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07565f10847840e0fb07b05218ad17fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
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解题方法
10 . 设复平面中向量
对应的复数为
,给定某个非零实数
,称向量
为
的
向量.
(1)已知
,求
;
(2)设
的
向量分别为
,已知
,求
的坐标(结果用
表示);
(3)若对于满足
的所有
能取到的最小值为8,求实数
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91174b2336306191ba275a87864172b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e1dbb4c7999d45d14b499e433a09137.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fde64d21aa8cf8bd96410f7a0b35a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91174b2336306191ba275a87864172b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8626ed40ac561244d7a7d78fdb24bc.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b7d9f40ce4648c9729f49cc071fe631.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6ba4991da3131c3e0cc5126359338e3.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aee9563b393021b8a23fd706969828b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8626ed40ac561244d7a7d78fdb24bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff17456ebb5651fe67e874c9b438c17a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddfb4014c775cf008fadabc87c95866b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95763e154888a080b3b96ff7fb3b39f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
(3)若对于满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38a158c72b071561459803ae1b950d22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7c9593bf5727941ac14317c5e730e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
您最近一年使用:0次
2023-02-13更新
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399次组卷
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3卷引用:上海市七宝中学2021-2022学年高一下学期5月月考数学试题