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1 . 如图,已知在四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,底面eqId5ce7a06ae7a34393b92b5277978ac014是矩形,eqIde78cce043a6c43329b55788351ff3484平面eqId5ce7a06ae7a34393b92b5277978ac014eqId0f1f33dbfc2e4961a3ef0973f38fc86beqIdb60bb1649c8b41cb935771acc097167eeqId93cbffaa5ae045d6ac45d1e979991c3aeqId63db14a5b4334f3ea583c8fb12b0d175分别是eqId99a3187c2b8f4bcc9703c74c3b72f1f3eqIdd293923b221045898dcc5442f88aa67a的中点.
说明: figure
(Ⅰ)求证:eqIdfec789eef46d41f085a9535585488fc5平面eqId6bc768e544bb4cc683314bfcee6add55
(Ⅱ)求eqIdb53b60c33a254a27b9bfd58e818cc7e6与平面eqId5ce7a06ae7a34393b92b5277978ac014所成角的正弦值;
(Ⅲ)求二面角eqIdee697987e8dd469bac9e26109dddb1ac的正切值.
2 . 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.
3 . 在四棱锥eqIddd5f333edd7c4a01b271754239943f8d中,平面eqId3f27fe03c88f4272858390eb2f5ff7ba平面eqIde499a0d7181a4bf4bfcd70c68255bab7,底面eqId5ce7a06ae7a34393b92b5277978ac014是矩形,eqIdebdd941186214c56922309cef10fe0a8eqId0627be51821d4be7b3e024a354815d64的中点eqIda5437c1f6bc94bbd816a1d878423d345eqId32d6ae8f3e024b05a7c614ef2b17abc7
说明: figure
(1)求证:eqId22440a0a1e694463be65ea5f94cdecd0
(2)若eqId437378f7f9f94ed8ba9420ca83c7e74f,求二面角eqId15f449a0fabe43f998618a2717078f57的余弦值.
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,已知平行四边形eqId5ce7a06ae7a34393b92b5277978ac014中,eqId2807fcad45ad4b3b8490dba6af5d22e6eqId7ff34b6deb764c09a59b578c206ea8cceqId2381423d4cd146cab95f55527681a766为边eqId0627be51821d4be7b3e024a354815d64的中点,将eqIde9b428a8af374c1ea1c3d923e0f79510沿直线eqId10381e9fca8b4095b66c934255c907b4翻折成eqIdf1127c924cc34f6aa1eb4366d7b98265,若eqId517584fed25c413ba8b7bb33ffa2d5c6为是eqId6e9e3153fca343779f77e9d76e94d641的中点,则在eqIde9b428a8af374c1ea1c3d923e0f79510的翻折过程中,下列命题正确的是(   )
说明: figure
A.线段eqId322e25d340bc4a49bcc4951064148f48的长为定值
B.异面直线eqId10381e9fca8b4095b66c934255c907b4eqId6e9e3153fca343779f77e9d76e94d641所成角为eqIdcbf8f098b4634cf49e9e651e10b7cf1d
C.直线eqId322e25d340bc4a49bcc4951064148f48与平面eqIddbbcdc1769324f6a9bae763d6573e39a所成角为定值
D.二面角eqId0fccdc2f486645c3b5d6524f89fb30b9可以为直二面角
6 . 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.
7 . 如图,在直三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,点eqId0cd8063abf2b458f80091bc51b75a904eqId93cbffaa5ae045d6ac45d1e979991c3aeqId63db14a5b4334f3ea583c8fb12b0d175eqId3f5a5d0fed7d4dc1bcce73cf1b7a92d3分别为棱eqId99a3187c2b8f4bcc9703c74c3b72f1f3eqIdea82c30b37894b1bb742dee61200db07eqIdfe9cbfb067114ce8857e621d77cb5febeqId31581e8b21734ef88d2a14232de7e7a6的中点,点eqId2381423d4cd146cab95f55527681a766eqId34c6b6e427744abfb88dc90b8c7c2111上.
说明: figure
(1)证明:eqId85e441f9128241ff8557a92523934ed6平面eqIdb5ec8343fcc14a308ebbd14f60948c48
(2)若eqId485e0ed66f8e483995b667bfb1bb3fa4eqId70263cdda99d47efb3c3418d1a811ee0,求二面角eqIda144616b1469430fb025cba0252c1bcc的大小.
8 . 如图,矩形eqId5ce7a06ae7a34393b92b5277978ac014中,eqIdb60bb1649c8b41cb935771acc097167eeqIdfe737ff044dd42c1bba7c0bd71ecb93beqId93cbffaa5ae045d6ac45d1e979991c3aeqId34c6b6e427744abfb88dc90b8c7c2111的中点,把eqId2120aba3a51d4273b0a51bf9edc287f3沿eqId7111d4972b7c401c903ff4927c13b5ec翻折,满足eqId59a84e7a9d6542319fd4ffd6bb029797.
说明: figure说明: figure
(1)求证:平面eqId152c68569180470696ea37ab76b3d7bd平面eqId32b0a8afc9814c2f8a154deafdcf7192
(2)求二面角eqId26f21a634aff46f5a3943d6f18dd99bb的余弦值.
9 . 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.
10 . 如图,在棱长为2的正方体eqIdb36ac1a6ce704ae182b2eabdfef2c664中,eqId93cbffaa5ae045d6ac45d1e979991c3aeqId63db14a5b4334f3ea583c8fb12b0d175分别为棱eqId5c3140c2a22d4e188020a38ef7665ae7eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,则eqId1a0830891ff846de8422ea80714ecab4与平面eqId5ce7a06ae7a34393b92b5277978ac014所成角的正切值是()
说明: figure
A.eqId49b7b111d23b44a9990c2312dc3b7ed9B.eqId5f816d038c584dc7a249120298677ef3C.eqIdbf4d8b310ca74c22a7eb63a406f4674aD.eqId5222c2ea3b544c58bafb590c9238c493
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