解题方法
1 . 如图,四棱锥
中,底面
是矩形,
,
平面
,下列叙述中错误的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b929269c53a44907dba8ee298a0a522.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.![]() ![]() | B.![]() |
C.![]() | D.平面![]() ![]() |
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2 . 在三棱锥
中,平面
平面
,
为等腰直角三角形,
,
,
,
为
的中点.
;
(2)求二面角
的余弦值;
(3)在线段
上是否存在点
使得平面
平面
?若存在,求出
的值,若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fbfcae2cecc98e2d6c16dde6d3ec1c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/036de574712cad14bddadf6653c7e714.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615fc8790237a1b09af51d6bcad6b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0028211551dd418eaaf51dde450f8b73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a7ebf74ae4daefad4350f9d1103a891.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74ec3d90e5f12cd8946d4dc638c1a357.png)
(3)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9f9bab9ec616f69811e860d0f0dca5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07b1e52dfd144ab4afda4d4aa5a92c1f.png)
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3 . 如图,三棱锥
中,
,平面
平面
,点
是棱
的中点,再从条件①、条件②这两个条件中选择一个作为已知.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/5/476a976a-fce8-4985-a541-ee7ee108bdc2.png?resizew=159)
(1)求证:
;
(2)求二面角
的余弦值.
条件①:
;
条件②:直线
与平面
所成角为
.
注:如果选择条件①和条件②分别解答,按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/752cd7bd1a9918e43586f5c4786d3955.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/5/476a976a-fce8-4985-a541-ee7ee108bdc2.png?resizew=159)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bafa8c14100a4f847b41b9148954116c.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5617a404c5a3356753136e5a6b6d51e5.png)
条件①:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d77ce3f557c59114eab8be453b86282.png)
条件②:直线
![](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/f1e5fa72f2878b476bc57f0df12d6555.png)
注:如果选择条件①和条件②分别解答,按第一个解答计分.
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4 . 如图1所示,平面多边形
中,
,
,
,且
,现沿直线
将
折起,得到四棱锥
,如图2所示.
(1)求证:
;
(2)在图(2)中,若直线
与平面
所成角的正弦值为
,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6723f8b9d907981aa735cd96386bee36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/373585f2c60ca4e176e695b360b7ba6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e7990d7f75ab69abc8a3a08365c9a72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/840d92478eeda81ed76c9c945a27a544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a664bf10399f679b60e7e36cdf0fb08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefe4a3e7a7fa195ed6a6712447639b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d02bd5cfe804460846423e77f72db10f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/1/f46b94fe-5db2-41dc-947d-1fbf73071055.png?resizew=341)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3edc1be5141e54264eb9ade93bb8bb33.png)
(2)在图(2)中,若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3520ee9cc97a075e889e1625dba1157c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa42621cd6793e7f3673fdb49bc3123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/840798a31aba0783f96584e0ad7c0d2e.png)
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5 . 如图, 在三棱柱
中,
为等边三角形,四边形
是边长为2的正方形, D为AB中点, 且 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50b193e9a6e49d78f89a479fd04f97a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/22/219733bf-a7ed-4c58-b52e-08bc2dd0019f.png?resizew=171)
(1)求证: CD⊥平面
;
(2)已知点 P 在线段
上,且直线AP 与平面
所成角的正弦值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/770abdd10660689c605577f9cb6d9db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50b193e9a6e49d78f89a479fd04f97a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/22/219733bf-a7ed-4c58-b52e-08bc2dd0019f.png?resizew=171)
(1)求证: CD⊥平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
(2)已知点 P 在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74bca84ad86c648d3bb20c8909c8da3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9868f77d5ab5073b6145f1c6d272122e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ca3bdb7a6f14db41bbc013000c510a.png)
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6 . 如图,棱柱
的所有棱长都为2,
,侧棱
与底面
的所成角为
平面
为
的中点.
;
(2)证明:
平面
;
(3)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a23f01af749100e1888bba06268843db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baa1d0a713c6fe6a3e70c2deb8fdd432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2caef76fe98904ad7d8c395b8036c610.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2eb89294b31ffdd2680b4361e8994d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e735911ba4cd7f8fca6b3f65d705b573.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a78b6fb582468bdd5c3afa5461aefce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
(3)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fb606548eca84c3b64e1b1f17fd2999.png)
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解题方法
7 . 如图,在边长为2的正方体
中,点
是该正方体对角线
上的动点,给出下列四个结论:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/1/c46dae4e-d1ce-4f90-944e-210aad65078e.png?resizew=161)
①
;
②
面积的最小值是
;
③只存在唯一的点
,使
平面
;
④当
时,平面
平面
.
其中正确结论的个数是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe734023d4e70010a6b2cc3267cb86e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/1/c46dae4e-d1ce-4f90-944e-210aad65078e.png?resizew=161)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c17c529214eb10f3b29856db1969b5.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9524e3810e06dc781285f1289e75d653.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
③只存在唯一的点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4645450a006f2c20087486d0833afbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf4c26f3f4d96117f087400a0f32ece8.png)
④当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df34dcb774c6b51eed4f26dbb1167d0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4df8bcc1e5df6bbe6db0a7c39585940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62a52848aff08399a36f217356007a4b.png)
其中正确结论的个数是( )
A.1个 | B.2个 | C.3个 | D.4个 |
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8 . 如图所示的几何体是由正方形
沿直线
旋转
得到的,设
是圆弧
的中点,
是圆弧
上的动点(含端点),给出下列四个结论:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/6c4c8853-4a80-4c84-add3-ce8c3e4a5be5.png?resizew=151)
①存在点
,使得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4514b6d7e2c6fb9e8a0d83768db7ffe8.png)
②不存在点
,使得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fab349a4b810b3b0bbed7b4cb5a1a008.png)
③存在点
,使得
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bb10d645970e5860afd3430957fab6c.png)
④不存在点
,使得直线
与平面
的所成角为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
其中,所有正确结论的序号为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8192018a58bb1fe23769a48a4d9042ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbdb53e0fdf3ebeb96e4f69feacbd80e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/27/6c4c8853-4a80-4c84-add3-ce8c3e4a5be5.png?resizew=151)
①存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4514b6d7e2c6fb9e8a0d83768db7ffe8.png)
②不存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fab349a4b810b3b0bbed7b4cb5a1a008.png)
③存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8801e1441e6dff0f8880179db56cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bb10d645970e5860afd3430957fab6c.png)
④不存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589786dd7c3a2679c3230b671cd232d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bb10d645970e5860afd3430957fab6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
其中,所有正确结论的序号为
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解题方法
9 . 如图,已知平面
平面
,四边形
是矩形,
,点
,
分别是
,
的中点.
(1)若点
为线段
中点,求证:
平面
;
(2)求证:
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/829f9180ddd9aa1a0ee0dc520f4e0b5f.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/24/e121ec03-39f8-4a97-b585-ab88d104cef5.png?resizew=161)
(1)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c19129982fd8389238b303e091bd94c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a22d6b860f06fe23618b0d3de6768fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
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2023-11-23更新
|
858次组卷
|
4卷引用:北京市汇文中学2023-2024学年高三上学期期中考试数学试题
北京市汇文中学2023-2024学年高三上学期期中考试数学试题江西省上饶市婺源县天佑中学2024届高三上学期期中数学试题(已下线)考点9 垂直的判定与性质 2024届高考数学考点总动员四川省内江市第六中学2023-2024学年高二上学期第2次月考数学(创新班)试题
名校
解题方法
10 . 如图,直四棱柱
中,底面
是边长为1的正方形,点
在棱
上.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/7/0daaa5d5-1d2e-483c-b1b8-d28103505dee.png?resizew=142)
(1)求证:
;
(2)从条件①、条件②、条件③这三个条件中选择两个作为已知,使得
平面
,并给出证明.
条件①:
为
的中点;
条件②:
平面
;
条件③:
.
(3)若
为
的中点,且点
到平面
的距离为1,求
的长度.
![](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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/7/0daaa5d5-1d2e-483c-b1b8-d28103505dee.png?resizew=142)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0af20277067fdd1bf5b03cea567c0a84.png)
(2)从条件①、条件②、条件③这三个条件中选择两个作为已知,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88b90810b206fa24f92d84504169e02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81bf7a859123936a07193592e089340a.png)
条件①:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
条件②:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7542b49ab149f2be8ba6b48392bef1f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81bf7a859123936a07193592e089340a.png)
条件③:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a948b9a017c589727827d944ef1224c.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81bf7a859123936a07193592e089340a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
您最近一年使用:0次
2023-11-14更新
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239次组卷
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2卷引用:北京市西城区北京师范大学附属实验中学2023-2024学年高二上学期期中考试数学试题