1 . 对于正整数n,
是小于或等于n的正整数中与n互质的数的数目.函数
以其首名研究者欧拉命名,称为欧拉函数,例如
(
与
互质),则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce7cc0ad7521b5771950aea983f0c1c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce7cc0ad7521b5771950aea983f0c1c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4c9e69c7d5a3d7a5633a373a8a39544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/786c6406780167f9744d0f9e9682e471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d02ea8c4988c5c28ab93f0d70fb55a.png)
A.若n为质数,则![]() | B.数列![]() |
C.数列![]() | D.数列![]() |
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2 . 已知椭圆:
的左、右焦点分别为
,点
是
轴正半轴上一点,
交椭圆于点A,若
,且
的内切圆半径为1,则该椭圆的离心率是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b24214f111f7c6d2b64e53ad970438b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c38928a92bc4b44ed3c9b89769f5372.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ac86e1c253297a377e14fb9a1689be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1290d0c55649a0fb9608007d5db5b047.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc19b3c838f96ad50f4450d0e76eb0fc.png)
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解题方法
3 . 已知正数a,b,c满足
为自然对数的底数,则下列不等式一定成立的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14795f0a41bc144cc1f99d948dfc30b0.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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4 . 记集合
无穷数列
中存在有限项不为零,
,对任意
,设
.定义运算
若
,则
,且
.
(1)设
,用
表示
;
(2)若
,证明:
:
(3)若数列
满足
,数列
满足
,设
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/100d76814e366c60298ea21aad6ddea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b21aa3e2f0c8de96d08195e5f66b725a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ba8cfb33f75f570c4d9cab8b522be30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e8b70b7fbc19242014383f0ee8621dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff83cd0e7e7b17d6f90cd29b3fe7a19b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a083253cd5a7df93f553e5e71b4aa7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87adb7b83f14cc809c1b7161e83c171f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77c5afa350510e7a8b3b27b5fa7803ad.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557d484154f5ff1194d22e1b02fff5dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/202bc33cd714c241671d6d4457c5637f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342d0252c1b2f7d2a84b5c985d19d547.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9660167870a1eed0a0d19edc430c8180.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1196e9280fbc7cbd6a01694af1dd42c.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64ed1a5374a245c7cc789dd17c2f9be0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c103c41bc5f744916b1aa6e0b38c23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557d484154f5ff1194d22e1b02fff5dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b597616902954c408ef4d86b25016c98.png)
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5 . 对于任意给定的四个实数
,
,
,
,我们定义方阵
,方阵
对应的行列式记为
,且
,方阵
与任意方阵
的乘法运算定义如下:
,其中方阵
,且
.设
,
,
.
(1)证明:
.
(2)若方阵
,
满足
,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e84c30444f13d37ada78285dc4f83b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e76d1d8e50dda4d50229a8a20c57e58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc29ee719feeedfbc8c529cf11348abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33e11a5b70e1e2e685d1783a4707872e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ec97af19b15cd584710a3faf30c716.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f44b167b4e75af29a18637f71f3ebfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b39fcc210ec89dbc7d684a70a34542c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d17ebf9f595cdb9dab841dec703b512.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16a4ed514630bd37fab9765b3fb5f2cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/709d09c76c222f156df31a1bba5f2ce9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2e4a35eca00ea2f4580d62515d54d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95035eeae686e910be45f08093e406c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93e7d309cb178b71c6e56f5b7f610413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/109b4ece615b08a89a7f69d436f448b0.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/addb109c49695bce8c5b5cf4fad95772.png)
(2)若方阵
![](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/2221c60bc15c59fa1b3ac74a23b57cdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90fa9bfe3bf3e3b7265da3c49d31f1bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35536fb98d8b24cead230c8df95fd9d3.png)
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解题方法
6 . 已知抛物线C:
的焦点为F,过点F的直线l与抛物线C交于A,B两点,O为坐标原点,直线
过点A且与OA垂直,直线
过点B且与OB垂直,直线
与
相交于点Q,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745de5ef1fd897d16e37464172d5e8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
A.设AB的中点为H,则![]() |
B.点Q的轨迹为抛物线 |
C.点Q到直线l距离的最小值为![]() |
D.![]() ![]() |
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7 . 已知数列
的前n项和为
,且满足
,则
( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/234605545a58197b26d52799abbb17b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06fbdd2e3efbef1ff014df55b242eced.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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8 . 在三棱锥
中,已知
是边长为2的正三角形,且
.若
和
的面积之积为
,且二面角
的余弦值为
,则该三棱锥外接球的表面积为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63e4d19bf237a6fca67e0d01a9ddb726.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d294d69caac577339f11f477b2047e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee14db57f0c762aad845cf5b4a243c0.png)
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9 . 如图,曲线
是以原点O为中心,
,
为焦点的椭圆的一部分,曲线
是以O为顶点,
为焦点的抛物线的一部分,
是
和
的交点,我们把
和
合成的曲线W称为“月蚀圆”.
所在椭圆和
所在抛物线的标准方程;
(2)过
作与y轴不垂直的直线l,l与W依次交于B,C,D,E四点,P,Q为
所在抛物线的准线上两点,M,N分别为CD,BE的中点.设
,
,
,
分别表示
,
,
,
的面积,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd38dfea1366aaf54ae7d33e6d8a0a90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
(2)过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.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/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f30f56664446f32dbbc2c5f12a99374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177678001b2ccde1db8f57fa5e017002.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3b52af1d93cf91437881f823ad19623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb9bcf5b109569e6f047c31a7ca9d72e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/845ff4183b0ea6c866247eefc6ea4be8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/add9278c504368ce1edca9d7ab1ac751.png)
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解题方法
10 . 已知正方体
边长为2,动点
满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34b93364f416d0ceadcc84387ae12e83.png)
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34b93364f416d0ceadcc84387ae12e83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e7d4399eaa40f83220ae809bdfbddff.png)
A.当![]() ![]() ![]() |
B.当![]() ![]() ![]() |
C.当![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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吉林省长春市东北师范大学附属中学2023-2024学年高三下学期第七次模拟考试数学试卷江西省重点中学协作体2024届高三第二次联考数学试卷(已下线)模块5 三模重组卷 第2套 复盘卷湖北省武汉市汉铁高级中学2024届高考数学考前临门一脚试卷(已下线)立体几何与空间向量-综合测试卷B卷