1 . 下列给出的命题正确的是( )
A.若![]() ![]() |
B.点![]() ![]() ![]() ![]() |
C.若直线![]() ![]() ![]() ![]() ![]() ![]() ![]() |
D.两个不重合的平面![]() ![]() ![]() |
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2024高三·全国·专题练习
2 . 正四棱柱
中,底面
是边长为4的正方形,
与
交于点
与
交于点
,且
.
(1)用向量方法求
的长;
(2)对于
个向量
,如果存在不全为零的
个实数
,
,使得
,则称
个向量
叫做线性相关,否则称为线性无关.试判断
是否线性相关.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cfbc0b5a8fbde804bd8425a4b76d207.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6d912c3d63e7566f63a5d6820f9bb5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c7f786fc33d0506d64047034e12fd7a.png)
(1)用向量方法求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
(2)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a27bb33ccdad573e2b2b0e7facbcca07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75b5dc876d7dcd3c971b36d26668b1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a5eb204cdd638a40eded9c9d8453388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65237e0feff86664a9315cb175d30cac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a27bb33ccdad573e2b2b0e7facbcca07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7e84a3d14f243496cf60c7260cc1b42.png)
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解题方法
3 . 如图,在圆台
中,截面
分别交圆台的上下底面于点
,
,
,
四点.点
为劣弧
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/b1a293cc-0531-4fb5-a1dc-c34eaf1d8ca5.png?resizew=149)
(1)求过点
作平面
垂直于截面
,请说明作法,并说明理由;
(2)若圆台上底面的半径为1,下底面的半径为3,母线长为3,
,求平面
与平面
所成夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abbe2aba242716238b79c46bb1f40e88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d1a8ed65b138016acff8c465165337.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/a0ed1ec316bc54c37c4286c208f55667.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/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/b1a293cc-0531-4fb5-a1dc-c34eaf1d8ca5.png?resizew=149)
(1)求过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d1a8ed65b138016acff8c465165337.png)
(2)若圆台上底面的半径为1,下底面的半径为3,母线长为3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73926ba72492b6a6a935dae64ed59e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/105f1100148bbf6d789b9048281755a1.png)
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4 . 已知
是平面
的一个法向量,
是平面
的一个法向量,且平面
平面
,则向量
在
上的投影向量为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edaaa35e08248eebefd781cfd2597d8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92a260f0d6364e0f81d29b48c6ff1983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a60be170a52db82cf37b30db0cde26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c45c7469ee6e876c4ffa4d87cd67d3e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c37564ec4e9e92485f1769e8ffaac31d.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
5 . 下列选项正确的是( )
A.空间向量![]() ![]() |
B.已知向量![]() ![]() ![]() ![]() ![]() ![]() ![]() |
C.已知空间向量![]() ![]() ![]() ![]() ![]() |
D.点![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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6 . 古代城池中的“瓮城”,又叫“曲池”,是加装在城门前面或里面的又一层门,若敌人攻入瓮城中,可形成“瓮中捉鳖”之势.如下图的“曲池”是上、下底面均为半圆形的柱体,
平面
为
的中点,则直线
与平面
所成角的正弦值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e4271abd1d57bf573558879f30e130f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8a3d5d669ac76a2ffb07da81d949adf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec353880a1e680a37ecce3b6fb4896f3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/21/a5612c4d-08b1-418f-922a-ed1d34f6ad66.png?resizew=498)
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解题方法
7 . 平面
两两互相垂直且有一个公共点
,
,
,
,直线
过点
,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61ba63ad02b1d5af2982fac3d91eb15c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ad734d66ce0bc4fbbac7cb052a85a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58143850496a20be45d79b4bad3d451d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab5483e2254688b4cd18fff7aeaf4c21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
A.若![]() ![]() ![]() ![]() ![]() ![]() |
B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() ![]() ![]() ![]() |
D.若点![]() ![]() ![]() ![]() ![]() |
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解题方法
8 . 已知正四面体
的棱长为a,,N为
的重心,P为线段CN上一点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
A.正四面体的体积为![]() |
B.正四面体的外接球的体积为![]() |
C.若![]() |
D.P点到各个面的距离之和为定值,且定值为![]() |
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9 . 已知平面
的一个法向量为
,平面
的一个法向量为
,直线
的方向向量为
,直线
的方向向量为
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8264299603ddbdda4f25d7c630a24877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb7a3d29fd0a1007f74d26ec5027c25a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f916000aed0824c4db0a18a2ba0e3b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/933402f1b0de6cd8ede886529abcd75c.png)
A.![]() |
B.![]() |
C.![]() ![]() |
D.![]() ![]() ![]() |
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福建省南安市华侨中学2023-2024学年高二上学期8月月考数学试题山东省潍坊市安丘市国开中学2022-2023学年高二下学期6月月考数学试题(已下线)1.4.1 用空间向量研究直线、平面的位置关系(重难点突破)-【冲刺满分】2023-2024学年高二数学重难点突破+分层训练同步精讲练(人教A版2019选择性必修第一册)山东省济南第一中学2023-2024学年高二上学期10月月考数学试题内蒙古赤峰市第二实验中学2023-2024学年高二上学期期中数学试题山东省青岛第五十八中学2023-2024学年高二上学期9月月考检测数学试题江苏省无锡市天一中学2023届高三考前最后一模数学试题(已下线)江苏省金陵中学、海安中学、南京外国语学校2024届高三三模数学试题
名校
解题方法
10 . 如图,四棱锥
的底面为正方形,
底面
,
平面
,垂足为
,
为
上的点,
,以
为坐标原点,分别以
,
,
为
,
,
轴的正方向,并均以1为单位长度,建立空间直角坐标系,设
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9df740160690029ac1e730c85f20347.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/218054144a13435580cd132b9459546c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b624742fe28db114e0554c6c87bff05c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a754ad0537577221e7be168127d7cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c085dbb9d78aef7d81c3f4d6855f067b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e0214e5f5b15dbbfa80b0335c2f0740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95296f53585ee03c52f0f94bea8b94b6.png)
A.![]() |
B.平面![]() ![]() |
C.当![]() ![]() ![]() ![]() |
D.当![]() ![]() ![]() ![]() |
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