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1 . 如图,平行四边形
中,
,
.现将
沿
起,使二面角
大小为120°,则折起后得到的三棱锥
外接球的表面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cc201327a8ee3fd646948d3f0c5d9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05d35d8d8bb0dc17f2f86fe5b230a2b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/282d4a8c3476b2b81e3fd73898e64539.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3931333820859378ea6723ff3075189.png)
A.![]() | B.![]() | C.![]() | D.![]() |
您最近一年使用:0次
7日内更新
|
705次组卷
|
4卷引用:江苏省南京市东山高级中学南站校区2023-2024学年高一下学期期末考试数学试卷
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解题方法
2 . 已知椭圆
:
的左、右焦点别为
,
,离心率为
,过点
的动直线
交
于
,
两点,点
在
轴上方,且
不与
轴垂直,
的周长为
,直线
与
交于另一点
,直线
与
交于另一点
,点
为椭圆
的下顶点,如图.
的方程:
(2)若过
作
,垂足为
.
(i)证明:直线
过定点;
(ii)求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347b68f42934c74e0d759a67613a1da9.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/8d5989c84e320b504511f23eeb6e7357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5076289823db419f94e9c0c8f4aafd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5eb2485f90dbfd0dfd6e7d179a856f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e2031d209711b058f3d278ede3c1d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e86dbcf83cd5d3421b3eed7be7dab32d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3656055f5256cd06e636ea96e9f89c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)若过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fb78c5f885034612c0e030b920143d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed2dee66e18a91bf4bdf327c68136a20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(i)证明:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(ii)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00c6e75f0b71f175e27f6b3389353ca0.png)
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解题方法
3 . 在三维空间中,单位立方体的顶点坐标可用三维坐标
表示,其中
.而在
维空间中
,以单位立方体的顶点坐标可表示为
维坐标
,其中
.现有如下定义:在
维空间中,
,
两点的曼哈顿距离为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd16a6bce68922e270867b93251aa45b.png)
(1)在3维单位立方体中任取两个不同顶点,试求所取两点的曼哈顿距离为1的概率;
(2)在
维单位立方体中任取两个不同顶点,记随机变量
为所取两点间的曼哈顿距离
(i)求出
的分布列与期望;
(ii)证明:随机变量
的方差小于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c2a29087dbd2e7635da13f7d288c1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489ac1b21a2d8f1cf07dc4aaca39a2bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677fd74842cbce34aed7073cebbd9c58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3183165f26bf33a2f3e7b6354937524e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/778f11ad6b2139da11259859c06e868c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f173dd2b7c86ba2ec88c614ad334bb37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd16f86170db5efa19732f33aad277b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd16a6bce68922e270867b93251aa45b.png)
(1)在3维单位立方体中任取两个不同顶点,试求所取两点的曼哈顿距离为1的概率;
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4f27f84764f1cca89ce3d93fc1cf603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(i)求出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(ii)证明:随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fe33278e4c69efacf81defce3045cec.png)
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4 . 已知四面体
的各个面均为全等的等腰三角形,且
.设E为空间内任一点,且A,B,C,D,E五点在同一个球面上,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da71b0b81d1f86b85b52ab064eebabab.png)
A.四面体![]() ![]() |
B.四面体![]() ![]() |
C.当![]() ![]() |
D.当三棱锥![]() ![]() ![]() ![]() |
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5 . 已知函数
.
(1)若
,求方程
的实数解;
(2)若关于
的方程
在区间
上有且只有一个解,求实数
的范围;
(3)若
,是否存在实数
,使不等式
在区间
上恒成立?若存在,求出
的最小值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b04e386c4e4cfb4ef0f622b8a3d7b650.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
(2)若关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e792b4df90196152b9ab9ab04abec10c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89aab08e9252668f35b70c1d2a8ee9ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
6 . 定义非零向量
的“相伴函数”为
,向量
称为函数
的“相伴向量”(其中O为坐标原点).记平面内所有向量的“相伴函数”构成的集合为S.
(1)设
,求证:
;
(2)求(1)中函数
的“相伴向量”模的取值范围;
(3)已知点
满足:
,向量
的“相伴函数”
在
处取得最大值.当点M运动时,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b84dac41f87e939f6cc39f38dc59b78d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b0e4e35cf9b9f97c19e4b72cc2a1b09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b84dac41f87e939f6cc39f38dc59b78d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abc54bc56c16baa3643686b85a6130e4.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/314bf3721381f67a49fa6a8068f465b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bfc9270aef191b473d38ffe9108b339.png)
(2)求(1)中函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
(3)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2d6bb01f1044358cc5fee441bc62489.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bec0705d808bfdd465aa1b585acb628.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e723e57753f0a4fe1ef8ca1aee0e2117.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11abb76da45ffa52b47c3a6b9a03ac7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63e0e24323fe73e5d9fc6136219306da.png)
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7 . 已知
,
.设p:
,q:
,则p是q的( )条件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d33da711e50e96568facb18cef27165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/731bdc8d2686a05f12a2ba8a7e3b01be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efdc0e0ca559f0f1af6127545f356fa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3af884e11dbf6b215c8bf893e993c3c.png)
A.充分不必要 | B.必要不充分 | C.充要 | D.既不充分又不必要 |
您最近一年使用:0次
2024-06-11更新
|
161次组卷
|
2卷引用:江苏省南京市金陵中学2023-2024学年高二下学期期中考试试卷
名校
8 . 棱长为2的正方体
中,点
,
,
分别是棱
,
,
的中点,则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
A.![]() ![]() |
B.![]() ![]() ![]() |
C.平面![]() ![]() |
D.点![]() ![]() ![]() ![]() ![]() ![]() |
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解题方法
9 . 若
,
,且
,则
的最小值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fcbdcb52d7b9e7c54d8382f42e278fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7dc87628c486529f615d51675515d98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dedab547d3f1d4824d6b68f88ef4e80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee4c25755e08df6c44fa0d2be441f709.png)
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10 . 已知函数
.
(1)当
时,求
的单调区间;
(2)当
时,
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1996957bc581e5d03c0f733233e9956.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b269c11e6f9dbbf1a1efcda572f13ea8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1c8abdaf1a883ca6d2fed14f8341696.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次