1 . 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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解题方法
2 . 在圆锥
中,C是母线
上靠近点S的三等分点,
,底面圆的半径为r,圆锥
的侧面积为
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddaa2c8b030ac1dd8029a2413b166c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9c018281fcaaf52863e1f83d9dad0.png)
A.当![]() ![]() |
B.当![]() ![]() |
C.当![]() ![]() ![]() |
D.当![]() ![]() ![]() |
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3 . 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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4 . 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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5 . 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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2024·全国·模拟预测
名校
解题方法
6 . 已知椭圆
的离心率为
,且过点
.若斜率为
的直线
与椭圆
相切于点
,过直线
上异于点
的一点
,作斜率为
的直线
与椭圆
交于
两点,定义
为点
处的切割比,记为
.
(1)求
的方程;
(2)证明:
与点
的坐标无关;
(3)若
,且
(
为坐标原点),则当
时,求直线
的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a200ca2c4af794f4d1c6a5443830b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1174142f3bba761585b6bc2653009b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7cda9ce6d633bc1f3a249fb0fc458a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f6f17bc385bafb37e8f964e5eb99cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17bcf83ec075e94f9c6a543afcad7a66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e44884f3ac15e7dd933e044aad80d678.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e44884f3ac15e7dd933e044aad80d678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8dea40504f399f4e364f2f7219bf61a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05e0d700800ec8282168f975dcb7f273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce662cd099fa16b8b4f900cd1f90177e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
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吉林省通化市梅河口市第五中学2024届高三三模数学试题(已下线)高三数学考前押题卷2(已下线)安徽省合肥市第一中学2024届高三下学期三模数学试题2024届普通高招全国统一考试临考预测押题密卷数学试题(A卷)
名校
7 . 如图,在五边形
中,四边形
为正方形,
,
,F为AB中点,现将
沿
折起到面
位置,使得
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01ff27eea7545bb06f9472f91290c54e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddbb52f9b226b1db3f6f9f055948bd38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66a6eb75c2bc5a47ec8c8d83d79fd431.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/923189afc198d153c79059a827f63c87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6db6ff4159947ed2dc47d82fa3bcab9a.png)
A.平面![]() ![]() |
B.若![]() ![]() ![]() ![]() |
C.折起过程中,![]() ![]() |
D.三棱锥![]() ![]() |
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解题方法
8 . 已知函数
,且满足
,则实数
的值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d54554cce9ab9d1ae528d4ae30dbb05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9192e24be965dccb0ecf7f945c9bae81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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9 . 如图所示,已知点
是
的重心,过点
作直线分别交
两边于
两点,且
,
,则
的最小值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dec2ca6438c82b43f746057d8129885.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9545f42dc3bb78dabdb73891f2e4a69c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/301af152850e2c795bd385d0d10836f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fc838e1477179b36ca7481ee2cc1e8.png)
A.![]() | B.![]() | C.4 | D.2 |
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10 . 记表
示
在区间
上的最大值,则
取得最小值时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4e9bb42376c12d7d21702ae8062b25a.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2838fbaec7d05f460c677eea011bac5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f030c36bb8786df88d401792062a4100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/316be14489ddd109633e87fef03a5b01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4e9bb42376c12d7d21702ae8062b25a.png)
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吉林省长春市东北师范大学附属中学2024届高三下学期第五次模拟考试数学试题(已下线)山东省淄博实验中学2024届高三下学期第三次模拟考试数学试题(已下线)模型6 分段函数与复合问题模型(已下线)模型7 绝对值函数模型甘肃省兰州市西北师大附中2024届高三第五次诊断考试(三模)数学试题