名校
解题方法
1 . 先判断下列命题的真假,如果是假命题,就括号部分的结果进行改正
(1)若
,
共线,则(
)
(2)经过两点
的直线方程为(
)
(3)经过点
,倾斜角为
的直线方程为(
)
(4)截距相等的直线都可以用方程(
)表示
(5)在空间中,直线
的方向向量
,平面
的一个法向量
,若
则(
平面
)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9795e7f5cb9b366776c41d8f3f43942.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ed9c2e9fabbfc63733bae8fa079d00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
(2)经过两点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/771a1341eed7e8c04fa8432aa9d77517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d719138091c843fed7ec6fd6f14efac.png)
(3)经过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abc095361d36bbfb29a38fbdf5f21d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c8ecfb8dfad67ca567f8bb1a1ca8a09.png)
(4)截距相等的直线都可以用方程(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/087af7a99d3e752994c8081970f5050d.png)
(5)在空间中,直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb80eb942aafb194fadc473776f35b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36f1a8e551cba7ec9f451749f60e628d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/516c9750b593ab00666ea4ec2730df2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93a4afe69e9ee3c701f1f109c3a0a7d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
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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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解题方法
3 . 在滨海文化中心有天津滨海科技馆,其建筑有鲜明的后工业风格,如图所示,截取其中一部分抽象出长方体和圆台组合,如图所示,长方体
中,
,圆台下底圆心
为
的中点,直径为2,圆与直线
交于
,圆台上底的圆心
在
上,直径为1.
![](https://img.xkw.com/dksih/QBM/2021/4/23/2705880246927360/2730846585544704/STEM/45b2fb4d77584581b5ffa7988d973437.png?resizew=234)
![](https://img.xkw.com/dksih/QBM/2021/4/23/2705880246927360/2730846585544704/STEM/5d45fccb0ba7472eb5b9f5ed084f1e3a.png?resizew=205)
(1)求
与平面
所成角的正弦值;
(2)求二面角
的余弦值;
(3)圆台上底圆周上是否存在一点
使得
,若存在,求点
到直线
的距离,若不存在则说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2089110ce065e01dc57e38edb840dceb.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://img.xkw.com/dksih/QBM/2021/4/23/2705880246927360/2730846585544704/STEM/45b2fb4d77584581b5ffa7988d973437.png?resizew=234)
![](https://img.xkw.com/dksih/QBM/2021/4/23/2705880246927360/2730846585544704/STEM/5d45fccb0ba7472eb5b9f5ed084f1e3a.png?resizew=205)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88c9f260496ba23993238601a89eca5c.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae190120755c4dd7070df21dd0a90a01.png)
(3)圆台上底圆周上是否存在一点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44b72acd3da8b22454e5d461379f5bde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
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