解题方法
1 . 如图,PA,PB,PC,他们之间每两条的夹角都是
,则直线PC与平面PAB所成角的大小为 __________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/71dd45ec-e413-4394-9266-a12414dfde33.png?resizew=192)
您最近一年使用:0次
解题方法
2 . 在正方体
中,异面直线
与
所成的角的余弦值为___
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/30/379beef1-613e-4ac9-8bfa-4085fbf8947d.png?resizew=160)
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解题方法
3 . 如图,棱长为1的正方体
中,点
为
的中点,则下列说法正确的是____________ .
与
为异面直线
②
与平面
所成角的正切值为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/821309f088a175c00dc0f4828334503d.png)
③过
三点的平面截正方体所得两部分的体积相等
④线段
在底面
的射影长为
![](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/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/821309f088a175c00dc0f4828334503d.png)
③过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc05a73194804cfd0465d00c91721f8c.png)
④线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
您最近一年使用:0次
2023-12-29更新
|
210次组卷
|
2卷引用:四川省乐山市犍为外国语实验学校2023-2024学年高二上学期12月月考数学试题
名校
解题方法
4 . 在三棱锥
中,
,
,
,平面
平面
,则三棱锥
的外接球体积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13d28cb7181257cf732af4b615fc47d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2e70e6efce668b8f5b3e54763d71ec2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07160f14b3b453bebb64cb2bf96dc85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcf6dc837ae85207789b94d109c5c2eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
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2023高三·全国·专题练习
5 . 阅读下面题目及其解答过程.
如图,在直三棱柱
中,
,
,
分别为
,
的中点.
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/STEM/925fc86dad1c48f09cf7adf52bb8d990.png?resizew=138)
(1)求证:
;
(2)求证:
.
解:(1)取
的中点
,连接
,
,如图所示.
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/EXPLANATION/d8115a60a30b48ac99ac18fb6deb2ed9.png?resizew=139)
在
中,
,
分别为
,
的中点,
,
.
由题意知,四边形
为_ .
为
的中点,
,
.
,
.
四边形
为平行四边形,
.又_ ,
平面
,
.
(2)
为直三棱柱,
平面
.
又
平面
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
_ .
,且
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
_ .
又
平面
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
_ ,
.
以上题目的解答过程中,设置了①~⑤五个空格,如下的表格中为每个空格给出了两个选项,其中只有一个符合逻辑推理.请选出符合逻辑推理的选项(只需填写“A”或“B”).
如图,在直三棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/STEM/925fc86dad1c48f09cf7adf52bb8d990.png?resizew=138)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33cce492aefef0c3a24fffcae3a3ccba.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9874eca4abea481fa84eb772a920f9c7.png)
解:(1)取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/735056c174e8dd7906257a2a50a962a7.png)
![](https://img.xkw.com/dksih/QBM/2023/12/1/3379858988621824/3380140028157952/EXPLANATION/d8115a60a30b48ac99ac18fb6deb2ed9.png?resizew=139)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.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/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb1038fe742b2121709231eed48fcb11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcfbf154e19cbd0580d58ccc9bac077c.png)
由题意知,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f96c673a2381f118ea2d3efc0bca1f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c56afd59592dbb194c87cdd725b7dc8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adec13cc3d4b82438803ac7bfa18d61b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38d0463b6e3d27b5cfc1df0e6c14fbef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/758d6c28ad9f09ae4c5dbe5649cdf9f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeeadcae4a2964c73187962918724ae7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d5a164bf56f8fb92527ad78bc10ccf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28e425314ef91a4b7d9522ac79fbed6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e3ffd599e4fb57893b141bad96c66b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e53b212640dadf751ef7f65a78a209.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/625899f6b0246330b5ac95b6538f5ca6.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/215eb19188ab59c8ec06776d0aee2085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a753598c7dafac4e9f2841b8b9a7132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be509ef5101aae24609ff9941cb246fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bab896c46e21eade473ddabf245263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83499936f532ddce9068dd1ff8eb2b01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e3ffd599e4fb57893b141bad96c66b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e53b212640dadf751ef7f65a78a209.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0375c6c592f61ee820127b9261e96d5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/691da5f9c37b146ea9abbc50b8560c51.png)
以上题目的解答过程中,设置了①~⑤五个空格,如下的表格中为每个空格给出了两个选项,其中只有一个符合逻辑推理.请选出符合逻辑推理的选项(只需填写“A”或“B”).
空格序号 | 选项 |
① | A.矩形 B.梯形 |
② | A.![]() ![]() ![]() ![]() |
③ | A.![]() ![]() |
④ | A.![]() ![]() ![]() ![]() |
⑤ | A.![]() ![]() |
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6 . 在正方体
中,
与平面
所成角的大小为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26d9636ad77369535852c6e4493446a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b4cd2b33bd983a9ed6575b9de04a46a.png)
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名校
解题方法
7 . 在棱长为2的正四面体
中,点
是
所在平面内以
为左、右顶点,
为半短轴长的椭圆上的一动点(异于
两点).取
的中点为坐标原点
,以直线
为
轴,直线
为
轴建立平面直角坐标系
,若直线
和
的斜率分别为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d82f6055983b8e9bb5a9cd8e09ebe5.png)
_____ ;
的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4357d5744046d4d44abb09e1ee35fcb.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/0ae6f48b9a53c0155a692509cf31f7e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/827ccf0c04aa941ba20d5f4c6068b46b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ae6f48b9a53c0155a692509cf31f7e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c407500c3f3d395fdfdc366851ef3fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92d82f6055983b8e9bb5a9cd8e09ebe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
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8 . 已知P,A,B,C四点不共面,若
,直线
与平面
所成的角为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4961a7dd688b8a12db7d1d3480c6b87f.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/341170b2f67a1b7c85c8efba0e451c99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4961a7dd688b8a12db7d1d3480c6b87f.png)
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解题方法
9 . 如图,二面角
的棱上有两个点A,B,线段BD与AC分别在这个二面角
的两个面内,并且都垂直于棱
.若
,
,
,
,则平面
与
的夹角是 __________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49ac5c414260b82119691c62f54fa027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc14778010a33f90902ff17b1ec0ac73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70dc2c20619a4fc12a0cfda59af5b69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7eb3d1070981fed5ca65a34bb2282e6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3abbaa130997ad1bed362edb6c2eaef9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
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10 . 如图,在棱长为1的正方体
中,
是棱
上的一个动点,给出下列四个结论:
①三棱锥
的体积为定值;
②存在点
使得
平面
;
③
的最小值为
;
④对每一个点
,在棱
上总存在一点
,使得
平面
;
⑤
是线段
上的一个动点,过点
的截面
垂直于
,则截面
的面积的最小值为
.
其中正确的命题的序号是________ .
![](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/2777840758e70e7dbbc18cef8f3d6d2b.png)
①三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8bcd4d16a1f2e89bb43fd1731a05ab1.png)
②存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/565133e91e3ace2b2187cfc6f1db5be6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a83c0b8db2205a6815811aa4ff5390f.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5a2fee2a78208f17068351405599b0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1f190b17530d81d927c358ac84757a4.png)
④对每一个点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80aa75fd1351786abd39fa546f3c7833.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a83c0b8db2205a6815811aa4ff5390f.png)
⑤
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8772aa893a9c1d40f714cb25701701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/839c7616cd0d90265f4b2c9c021254fe.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/5/68789707-893b-4b2a-b900-6f451484d747.png?resizew=165)
其中正确的命题的序号是
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