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1 . 如图,在四棱锥PABCD中,底面ABCD为正方形,∠PAB=90°,PBPDPAABE为线段PB的中点,F为线段BC上的动点.
说明: figure
(1)求证:AE⊥平面PBC
(2)是否存在点F,使平面AEF与平面PCD所成的锐二面角为30°?若存在,试确定点F的位置;若不存在,请说明理由.
2 . 如图,在直三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,eqIdab1b2b01ce8e4594b15830e6b67e1079eqId1a59f675e3ea4309bfb83ff3b4953b43eqId9da5ed85dcc642b6a2ee462907d9a821,点eqId2381423d4cd146cab95f55527681a766是棱eqIdea82c30b37894b1bb742dee61200db07的中点,则下列说法中正确的是(   )
说明: figure
A.eqIdd4e49fdecf234767af214da91f5369b9
B.平面eqId3027f6c378fc4eb4a272a1d38f64b9ee平面eqId9b26d52d6cbc46fda65515d6f10c26e1
C.异面直线eqId0627be51821d4be7b3e024a354815d64eqIddaf8c155668d4bc28935bf064d7c5f7d所与成的角为直角
D.过点eqId346946f1395a41008d3b3b56c9ae4fbd作平面eqId180eb7168a7544f7995aeb0ffdac6b2a的平行平面eqIdc13953f2514e4c8f9c8aaaf5241c33ac,若eqIdc13953f2514e4c8f9c8aaaf5241c33ac交侧面eqId50dcbbdb425943e8b5a0e6b8b508d09f于线段eqId0743582db60f420c9cae978529e721bb,则eqIdefc5feeedd884cf386ca7ac33cf08b60
3 . 棱长为2的正方体eqId588284d93dc5489295f8f224f8e30d13中,eqIdbedf755e0fdb4d078d6859360706b163为正方体表面上一个动点,且总有eqId04b9bf478cee4a53a1bc7ee8d8e96c7c,则动点eqIdbedf755e0fdb4d078d6859360706b163的轨迹所围成的面积为(   )
A.eqIdefe7549d43a142158aee9b47aa9a1354B.eqIdf35f1e45d1d840c2ae8371573f781126C.6D.eqId7bcd573cee334383ad72cc4b179fcfb3
4 . 在四棱锥eqIddd5f333edd7c4a01b271754239943f8d中,侧面eqId42ab127176d64d389237593e8dd9c7bf底面eqId5ce7a06ae7a34393b92b5277978ac014,底面eqId5ce7a06ae7a34393b92b5277978ac014为矩形,eqIdead78591f6944789b224097f5e9043b2为等边三角形,eqId790707287e604e1a97e1484c1457c054eqId922285b1cf32406693c10c09849dafcf,点eqId93cbffaa5ae045d6ac45d1e979991c3aeqId8a76bbe21fb549e3a9c2038d58c7a3d8上,eqId234cb036ed814cec8b8fa1de3fa56fcb
说明: figure
(1)求证:eqIdb16183aa581b4939914cd894a1d7b214
(2)求二面角eqId18306063f55b4eb79b8f30acdb1a23a5的正切值.
5 . 如图,在四棱锥P-ABCD中,eqId61306e0b1eaa49b6847d04985aca3fb4平面eqId6854fcf0032e4a459902daced9d7503feqIdcaf648c9c5bb43898723088f4ba0c16beqIdf26d139651814796af967d52319d3910 eqId513b964714c8461baad486d3ad4ead9cEPD中点,点F在线段PC上,且DF//平面PAB.
说明: figure
(1)求证:eqIdc2d64d1363c445c9b89c744f60500a64平面eqIde62a05cd0c654eaf9aa3b0adc0bee259;
(2)求二面角F-AE-P的正弦值.
6 . 如图所示,正三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43各棱的长度均相等,eqId0cd8063abf2b458f80091bc51b75a904eqIdea82c30b37894b1bb742dee61200db07的中点,eqId2381423d4cd146cab95f55527681a766eqId517584fed25c413ba8b7bb33ffa2d5c6分别是线段eqIdfe9cbfb067114ce8857e621d77cb5feb和线段eqId8f39d706a505485987ccbb3f6177d72d上的动点(含端点),且满足eqIdccf4acf49de7476b83c9541e0a427a92,当eqId2381423d4cd146cab95f55527681a766eqId517584fed25c413ba8b7bb33ffa2d5c6运动时,下列结论中正确的是(   )
说明: figure
A.eqIdee228a02cb55436fb230451a6e1c2d2a是等腰三角形
B.在eqIdee228a02cb55436fb230451a6e1c2d2a内总存在与平面eqId50dcbbdb425943e8b5a0e6b8b508d09f垂直的线段
C.三棱锥eqId37a09c6f98a24e279f6a1ffa9d5e930a的体积是三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43的体积的eqId231e56cfb91a489890b9c771c5aad8e9
D.eqId8dac7508d6a74a68b6392b4d24eaed1c
7 . 如图1,eqId0b4671382ad6491cb648fae82fdffac1中,eqId97840d2105de460387d5d0989b35a28aeqIdbfafc9c818504a68969d0160bf2b7a65eqIddad31040fd02475ea52d192682cf59c6DE分别是eqId99a3187c2b8f4bcc9703c74c3b72f1f3eqIdcf2da96900c948a1b3ce7cbfd420c080的中点.把eqId1ba8a6a849e04b31abd4a64a759061b5沿eqId66d409c343f04cfda647f9bfc7da6b2a折至eqId0c25b35a4d0d4a42864c79c88b69652b的位置,eqId3f47e43b97c2420f88283510f4eaa2d0平面eqId8cd0bdb8c4664be490944f0c7c4d6c57,连接eqId0fd9ac73d66042c6bbd74f8ccdcbd9fceqIdb53b60c33a254a27b9bfd58e818cc7e6F为线段eqId0fd9ac73d66042c6bbd74f8ccdcbd9fc的中点,如图2.
说明: figure
(1)求证:eqIdddc5b665c41c4b1faaf27015f3d97e36平面eqId0902deb68f0b430f932caadd41cd761a
(2)当三棱锥eqIdbc4ee36eb33b44249b77b80bebf978a4的体积为eqId49b7b111d23b44a9990c2312dc3b7ed9时,求直线eqId1b51efe7c2fa42748ac5a6be262e2fa4eqIdb53b60c33a254a27b9bfd58e818cc7e6所成角的正切值.
8 . 如图,平面eqIda1d171b61b794cc8b8e4c283d1f43965平面eqIda670baf5d73f4d7689915a00c51820f2eqIdebd11f5d4e374e6380fbd2c317bfe068eqIdf6e90d22eba747b9baafdd19320e196aeqIda9747524f0914207af1c78e0d6939818eqId351908e4a8ea4f3bbb1b9f9709af6c5a.
说明: figure
(1)求证:eqId96c91195de72444194ce914ce0899821平面eqIda670baf5d73f4d7689915a00c51820f2
(2)求证:eqId7781ea7b749c4a4c928e5c730a35d197.
9 . 如图,在三棱锥eqId12175c05cfe240a0b8270ee03df1962f中,eqIde78cce043a6c43329b55788351ff3484平面eqIda670baf5d73f4d7689915a00c51820f2eqIdfb4e495ba1384cb8895c57c4ec5236e5eqId0ec95099d64646bcb6f32da71f3681caeqIdbfafc9c818504a68969d0160bf2b7a65EeqIdb53b60c33a254a27b9bfd58e818cc7e6的中点,过点AeqId0e68c58823d5409cb9a55963d56da3c0,垂足为点F.
说明: figure
(1)求证:eqId49b07c005155438286e7ee699acdca49平面eqId0902deb68f0b430f932caadd41cd761a
(2)求eqId7111d4972b7c401c903ff4927c13b5ec与平面eqId0902deb68f0b430f932caadd41cd761a所成角的正弦值.
10 . 如图,棱长为1的正方体eqId588284d93dc5489295f8f224f8e30d13中,eqIdbedf755e0fdb4d078d6859360706b163为线段eqId30ace6b4ce264294a0534221c1b01e26上的动点,则下列结论正确的是(   )
说明: figure
A.eqId8cbeedd6eb164e17b3f9254b4d9bede6B.平面eqIda14254ede29c495187f41263ab21581b平面eqIdc660d75ecea84461b455241cd2c0a0e6
C.eqId490313b5b87242b3851cfe1e7d96d2d4的最小值为eqId66393f4f7f004632a900c09a1ea18033D.eqIdee6e28ede4324abb841a28f9bb8839b7与平面eqId227f5b5400174e55ad063cfd2f95d019所成角正弦值的取值范围是eqId25f381554ef04784a96e1ebfe70abef4
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