名校
1 . 如图,矩形ABCD中,AB=2AD,E为边AB的中点.将
ADE沿直线DE翻折成
A1DE(A1
平面BCDE).若M在线段A1C上(点M与A1,C不重合),则在
ADE翻折过程中,给出下列判断:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/6c74a8b1-9cce-4664-af4e-7f1442648b97.png?resizew=223)
①当M为线段A1C中点时,|BM|为定值;
②存在某个位置,使DE
A1C;
③当四棱锥A1—BCDE体积最大时,点A1到平面BCDE的距离为|A1H|(DE的中点为H);
④当二面角A1—DE—B的大小为
时,异面直线A1D与BE所成角的余弦值为
.
其中判断正确的个数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e581739cffb5676d997a58ab10d58880.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/6c74a8b1-9cce-4664-af4e-7f1442648b97.png?resizew=223)
①当M为线段A1C中点时,|BM|为定值;
②存在某个位置,使DE
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1633988fd62a652de726ee92a917b52d.png)
③当四棱锥A1—BCDE体积最大时,点A1到平面BCDE的距离为|A1H|(DE的中点为H);
④当二面角A1—DE—B的大小为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/457eed9a9809a61c38d9143c00d311b5.png)
其中判断正确的个数为( )
A.1 | B.2 | C.3 | D.4 |
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2021-07-04更新
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1197次组卷
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2卷引用:天津市滨海新区2020-2021学年高一下学期期末数学试题
名校
2 . 2021年6月17日,神舟十二号载人飞船顺利升空并于6.5小时后与天和核心舱成功对接,这是中国航天史上的又一里程碑,我校南苍穹同学既是航天迷,又热爱数学,于是他为正在参加期末检测的你们编就了这道题目,如图,是神舟十二号飞船推进舱及其推进器的简化示意图,半径相等的圆
与圆柱
底面相切于
四点,且圆
与
与
与
与
分别外切,线段
为圆柱
的母线.点
为线段
中点,点
在线段
上,且
.已知圆柱
,底面半径为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/3/78e25f34-db50-45eb-82da-472097cfab1d.png?resizew=153)
![](https://img.xkw.com/dksih/QBM/2021/6/29/2753530368434176/2754982265675776/STEM/095b1cdbb79d4b85b5e8d416ecefceb9.png?resizew=257)
(1)求证:
平面
;
(2)线段
上是否存在一点
,使得
平面
若存在,请求出
的长,若不存在,请说明理由;
(3)求二面角
的余弦值;
(4)如图,是飞船推进舱与即将对接的天和核心舱的相对位置的简化示意图.天和核心舱为底面半径为2的圆柱
,它与飞船推进舱共轴,即
共线.核心舱体两侧伸展出太阳翼,其中三角形
为以
为斜边的等腰直角三角形,四边形
为矩形.已知推进舱与核心舱的距离为4,即
,且
,
.在对接过程中,核心舱相对于推进舱可能会相对作出逆时针旋转的运动,请你求出在舱体相对距离保持不变的情况下,在舱体相对旋转过程中,直线
与平面
所成角的正弦值的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cedae9bc8fe49c3c98914f89a0670158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f1ac49b4139636fb1809fe970b23a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46336351200396508fcc05fef6a26625.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba671abb53449fe1012b8f5e017f5c05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e430821d59f79d5aa0fc9934e240603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f1ac49b4139636fb1809fe970b23a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a696a182fff038a86b2bbe8ca099442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edf3e262887cd9bfad78cad43b979b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c4c2157cf374ebe6352715ef100471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47d2403519f528c80887ad7045b630c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50b993fdfc49cb68be99ff1f04f71388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e980772d05dd94fafd77f69c20e6e882.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/3/78e25f34-db50-45eb-82da-472097cfab1d.png?resizew=153)
![](https://img.xkw.com/dksih/QBM/2021/6/29/2753530368434176/2754982265675776/STEM/095b1cdbb79d4b85b5e8d416ecefceb9.png?resizew=257)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ac480d8d9d7821b62a603cf5cfda236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfaf581b4f42a25087f7eee23a7d66b6.png)
(2)线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9de7ea432599108b34a0ccaa0f2c75e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b131bc54bb464a496e79ada0c6a7cabd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
(3)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7f3fd3258a508a38c7616464cfb86e4.png)
(4)如图,是飞船推进舱与即将对接的天和核心舱的相对位置的简化示意图.天和核心舱为底面半径为2的圆柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3711953485a76de370a04756009a644a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aeaf090a8dc4e381e3001c24473a0ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e046095eefc95b26511f64d1cb3bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08cce6cac0fdd4b1a434af8bcaec8fef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134c3d2c318a33a82da4134dd17fa57e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cbf49977b753e293bdf415fccd91abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23be45176cc25e19752dc551147b02eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19d2a5b65d9119ddd8649164ecde37ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134c3d2c318a33a82da4134dd17fa57e.png)
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