1 . “杨辉三角”是二项式系数在三角形中的一种几何排列,中国南宋数学家杨辉在1261年所著的《详解九章算法》一书中就有出现,比欧洲早393年发现.如图所示,在“杨辉三角”中,除每行两边的数都是1外,其余每个数都是其“肩上”的两个数之和,例如第4行的6为第3行中两个3的和.则下列命题中正确的是( )
A.由“在相邻两行中,除1以外的每个数都等于它肩上的两个数字之和”猜想![]() |
B.由“第n行所有数之和为2n”猜想:![]() |
C.第20行中,第10个数最大 |
D.第15行中,第7个数与第8个数的比为7:8 |
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2 . 将杨辉三角中的每一个数
都换成分数
,可得到如图所示的分数三角形,成为“莱布尼茨三角形”,从莱布尼茨三角形可以看出,存在x使得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/474f222dbcfcb5b5acf440bc88c9555c.png)
,则x的值是_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afb4fb20d3a3a67baa8505623e0bd9de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/796de9f6d9d237548371658bd8f124a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/474f222dbcfcb5b5acf440bc88c9555c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76cd2ad78cee377a4b74f79aa79f7210.png)
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解题方法
3 . 我国古代数学家僧一行应用“九服晷影算法”在《大衍历》中建立了晷影长l与太阳天顶距θ(
)的对应数表,这是世界数学史上较早的正切函数表.根据三角学知识可知,晷影长l等于表高h与太阳天顶距θ正切值的乘积,即
.对同一“表高”测量两次,第一次和第二次太阳天顶距分别为
,
,第二次的“晷影长”是“表高”的2倍,且
,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b13a855bec1e9c264742bbed4685ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7ba1eba7fba0b56e4e9b4d032e24da.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/5ffb487cd5f8d97cedc263d542caf5bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d3a6ba2ae3161cf7c209a30ea464c1.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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4 . 如果方程
能确定
是
的函数,那么称这种方式表示的函数为隐函数.隐函数的求导方法如下:在方程
中,把
看成
的函数
,则方程可看成关于
的恒等式
,在等式两边同时对
求导,然后解出
即可.例如,求由方程
所确定的隐函数的导数
,将方程
的两边同时对
求导,则有
(
是
的函数,需要用复合函数的求导法则求导),得
.利用隐函数求导方法可求得曲线
在点
处的切线方程为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bbbf52d1f9d61b41bdd4acfc9fac268.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bbbf52d1f9d61b41bdd4acfc9fac268.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d318c5ba5c7154bbcc95f8ec8e3d4a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63fe744d4980ba5e09c4074e0643e635.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8e70497c791fd23d1f37a544f2f73f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f9bf4c61dd281154e8cc5771a5f6e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b4d2174f411d9db6ab7b2aea47818cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f9bf4c61dd281154e8cc5771a5f6e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3731ec916f7866d93c5d22f7f70f202f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/186e696074c13fecc9bf8853d65c3776.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46b3b2e2820c19241b0e5770445f1d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad20e2bc6576fc461419f8f138d26e7.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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5 . 早在公元5世纪,我国数学家祖暅就提出:“幂势既同,则积不容异”.如图,抛物线C的方程为
,过点(1,0)作抛物线C的切线l(l的斜率不为0),将抛物线C、直线l及x轴围成的阴影部分绕y轴旋转一周,所得的几何体记作
,利用祖暅原理,可得出几何体
的体积为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
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6 . “四平方和定理”最早由欧拉提出,后被拉格朗日等数学家证明.“四平方和定理”的内容是:任意正整数都可以表示为不超过四个自然数的平方和,例如正整数
.设
,其中
均为自然数,则满足条件的有序数组
的个数是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19e9edd49b95d101473211fa54acfcdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6f5f4751622b599216b655a679cdd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5e4be004a34cfce346c12feea0a696.png)
A.26 | B.28 | C.29 | D.30 |
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7 . 中国南北朝时期的著作《孙子算经》中,对同余除法有较深的研究.设
为整数,若
和
被
除得的余数相同,则称
和
对模
同余,记为
.若
,
,则
的值可以是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5924004836cc5973c0a701a67c50d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73aeb67aa5fa6797d0a68cfbf1a3d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f82e83349efc625e006bb5636141d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a4cfa5382e39e85e6acc1a98dcdac55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
A.2022 | B.2023 | C.2024 | D.2025 |
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7日内更新
|
380次组卷
|
3卷引用:重庆市杨家坪中学2023-2024学年高二下学期第二次月考数学试题
8 . 如图形状出现在南宋数学家杨辉所著的《详解九章算法商功》中,后人称为“三角垛”.“三角垛”的最上层有1个球,第二层有3个球,第三层有6个球……,设各层球数构成一个数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
的通项公式;
(2)若数列
的前
项和
,数列
满足
,求数列
的前
项和
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2da0f1d689270c2c9cad0c1c9da2a18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45ad1f261d1ca999f8dbd5b1a0305ddf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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9 . “杨辉三角”是中国古代数学文化的瑰宝之一,最早出现在南宋数学家杨辉于1261年所著的《详解九章算法》一书中.“杨辉三角”揭示了二项式系数在三角形数表中的一种几何排列规律(如图所示),则“杨辉三角”中第30行中第12个数与第13个数之比为__________ .
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10 . 帕德近似是法国数学家亨利
帕德发明的用有理多项式近似特定函数的方法.给定两个正整数
,函数
在
处的
阶帕德近似定义为:
,且满足:
,
,
,
,
.(注:
,
,
,
,
为
的导数)已知
在
处的
阶帕德近似为
.
(1)求实数
的值;
(2)证明:当
时,
;
(3)设
为实数,讨论函数
的单调性.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c97ec04a1aa7ac6fce72d589864940a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b85a97933a1d984f6e484b4021c800.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d8688bb9fed24a8dc9f53f8b82a7469.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adcb8c6a69df1a0deaba265e204d5f99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047a8c1ed551fccee1c1848746c5f282.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72029562177dfc99a171c9013eb90227.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4573475f70860a3d99b92a329d0d07f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca214aa6276b96d67a451c3fdbc59b3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cba6d8d56270fc72edd1af793542c036.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030c5fc27fb5c07e4d6c913653af07ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa160e70abb25d476bbd7d720815f4f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a33cfe27fd2276a7c542f062c17b4d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eea7fa65b493fc1bdf84e16d39ae07d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d40624fc4d5a669a76185052ee6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb45f673c56a289ea78831c9237e8d20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40765d09390381658d5b4dc0160366cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8de781718020ed3f99538b8e25d6186.png)
(1)求实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
(2)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/447d6f62c09c1d05346fd16a24159f6e.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b00d47ef1d331094530990ffe38e1d77.png)
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