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1 . 《九章算术》中将底面为矩形且有一条侧棱与底面垂直的四棱锥称为阳马.已知四棱锥
为阳马,底面
是边长为2的正方形,其中两条侧棱长都为3,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.该阳马的体积为![]() | B.该阳马的表面积为![]() |
C.该阳马外接球的半径为![]() | D.该阳马内切球的半径为![]() |
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2 . 十七世纪法国数学家、被誉为业余数学家之王的皮埃尔・德・费马提出的一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,意大利数学家托里拆利给出了解答,当
的三个内角均小于
时,使得
的点
即为费马点;当
有一个内角大于或等于
时,最大内角的顶点为费马点.已知
,
,
分别是
三个内角
,
,
的对边,且
,若点
为
的费马点,
,则实数
的取值范围为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e8036a881da6a4eef036529028a11d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.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/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e3ff728c68ba198cc1d7dcd12b2cfed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73669a238770ba2989e71ec2d1468738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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3 . 二进制是计算机技术中广泛采用的一种数制,二进制数据是用0和1两个数码来表示的数,它的基数为2,进位规则是“逢二进一”,借位规则是“借一当二”,由18世纪德国数理哲学大师莱布尼兹发现.当前的计算机系统使用的基本都是二进制系统,数据在计算机中主要以补码的形式存储,我们用
表示十进制数n在二进制下的数字各项之和(例如:
,则十进制数5的二进制数为101,
),则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d610325871be3bb5da8e9ffa71d9cef8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ff90c2652820fd6d5740e67767f2348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211a8accc4989b5a25d9260ee39dafcc.png)
A.十进制数25的二进制数为1101 | B.![]() |
C.![]() | D.![]() |
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4 . 一般地,对任意角
,在平面直角坐标系中,设
的终边上异于原点的任意一点P的坐标为
,它与原点的距离是
.我们规定:比值
,
,
分别叫做角
的余切、余割、正割,分别记作
,
,
,即
,
,
,把
,
,
分别叫做余切函数、余割函数、正割函数.
(1)已知
,则
的最大值为_______ ;
(2)设
,则
的最小值为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a79a33a83a7ba57a34b5093d1d1d02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4e7bf9200b351a259ddfc6c0266129d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa2d7c084731df9cdabf1f0af121e3e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fee1e0f6c44b3027d0d6f8d9396f209.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d494c34104f679bdbea537164f1907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e609ecb22257c1ca2fe78b1dc2e62141.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f48bd75362790c061d70f80de8febc3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b57070a05279ad5e576d13fb9c1bef2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/851b7eec8ee522611f6b96a60ab9fc63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/147f65043356b475c5c2bba102958807.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd5cac6f59b3e1405a3b64d13c88e8a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/175c64c2a2393743bde92b3e46df42cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7688d35e68414fa995babd7437e678b.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1cf8cc0ca8fbbc8863fb416e25730f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bde963bde77dedd5e9859b5a4f5e49e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
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5 . 米斗是我国古代称量粮食的量器,是官仓、粮栈、米行及地主家里必备的用具,其外形近似一个正四棱台.米斗有着吉祥的寓意,是丰饶富足的象征,带有浓郁的民间文化的味,如今也成为了一种颇具意趣的藏品.已知一个米斗上下底面边长分别为
和
,侧棱长为
,则其外接球的体积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b947152b672cf56e5a8cde6800d71c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b91d650c2fc1a741fabdb333b09aeb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffa37aefb6d45efe4e20ba48c2e7dfa8.png)
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6 . 古希腊数学家阿波罗尼奥斯用不同的平面截同一圆锥,得到了圆锥曲线,其中的一种如图所示.用过
点且垂直于圆锥底面的平面截两个全等的对顶圆锥得到双曲线的一部分,已知高
,底面圆的半径为4,
为母线
的中点,平面与底面的交线
,则双曲线的两条渐近线所成角的余弦值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae890f9e8b32aa53a54158f24f4a87bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab00e0cff0876c4183a47f1272cf9928.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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7 . 我国元代数学家朱世杰在他的《四元玉鉴》一书中对高阶等差数列求和有精深的研究,即“垛积术”.对于数列
,①,从第二项起,每一项与它前面相邻一项的差构成数列
,②,称该数列②为数列①的一阶差分数列,其中
;对于数列②,从第二项起,每一项与它前面相邻一项的差构成数列
,③,称该数列③为数列①的二阶差分数列,其中
按照上述办法,第
次得到数列
,④,则称数列④为数列①的
阶差分数列,其中
,若数列
的
阶差分数列是非零常数列,则称数列
为
阶等差数列(或高阶等差数列).
(1)若高阶等差数列
为
,求数列
的通项公式;
(2)若
阶等差数列
的通项公式
.
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2506385d68e133523a24a5f5770adb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f230a901381bb98bd400c14317e0da8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/780007875adc41be137fd9ff68c255b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b21e02ca8a3d50e257ddc00ca87a0406.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e1f74af47ea73ea7fa4e19a51166244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b397162165301246a7616800610ea6d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/648eaff4ff716932fdbab7ee616b914d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e024470c4aa889689aefdf14fafddb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
(1)若高阶等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a628a7921f38cb09c818b3135aea1a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84f8542101a1cbbf29cfc7a7358a552c.png)
(ⅰ)求的值;
(ⅱ)求数列的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c698639cdba709641d3c91ea1798abfa.png)
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8 . 数列
:1,1,2,3,5,8,13,21,34,……称为斐波那契数列,该数列是由意大利数学家莱昂纳多·斐波那契(Leonardo Fibonacci)以兔子繁殖为例子而引入,故又称为“兔子数列”,
满足
,
(
,
),则
是斐波那契数列的第______________ 项.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8323901a49cac29afd7d62864f088077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a404164c8d199f60d183a59b3647cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/209591cfb9f8271f5ad48d89f214f22e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f4b291192a27a2a49075931fb9bba06.png)
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9 . 北宋科学家沈括在《梦溪笔谈》中记载了“隙积术”,提出长方台形垛积的一般求和公式.如图,由大小相同的小球堆成的一个长方台形垛积的第一层有
个小球,第二层有
个小球,第三层有
个小球……依此类推,最底层有
个小球,共有
层,由“隙积术”可得 这 些 小 球 的 总 个 数 为
若由小球堆成的某个长方台形垛积共8层,小球总个数为240,则该垛积的第一层的小球个数为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/9/08b5b114-291b-48fd-96e1-14b11208b7d0.png?resizew=167)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d993cf0a090de3b01f1dda52c6fdc9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28b4f6054ca644083ecfa85ed1bb7592.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b939f6c148a4e5fae83668aaa627836.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/023fc87942a6821ca78d8cae08917352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d34cf59fcb503ce674a118159af9244c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/9/08b5b114-291b-48fd-96e1-14b11208b7d0.png?resizew=167)
A.1 | B.2 | C.3 | D.4 |
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10 . 已知勒洛四面体是一个非常神奇的“四面体”,它能在两个平行平面间自由转动,并且始终与两平面都接触,因此它能像球一样来回滚动(如图甲),利用这一原理,科技人员发明了转子发动机.勒洛四面体是以正四面体的四个顶点为球心,以正四面体的棱长为半径的四个球的相交部分围成的几何体(如图乙),若勒洛四面体ABCD能够容纳的最大球的表面积为
,则正四面体ABCD的内切球的半径为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ebb9a166d0ce92751a2a1823ea24f13.png)
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