数学试卷
一、选择题
1.设集合72ef49ec0a5f452814fee47ffb1673a6.png,7b5697a3f2c24688c2a5fd2eb3894e65.png,则( )
A.c12878237139b7ba1eeac7dda64dbfe6.png B.6cffc3111ab421aec39d72f19d96d8d0.png C.b796ec591d40c77f88b4a0c740960b60.png D.74a4c729f8bd652cb0153b7fc577d9cc.png
2.已知复数fbade9e36a3f36d3d676c1b808451dd7.png满足:5407665e9b5dbaef689aa988af08cdec.png其中i为虚数单位.,则复数fbade9e36a3f36d3d676c1b808451dd7.png的虚部等于( )
A.6c8000f03d933531c86f3487fd602aea.png B. 10cfc0dd6f03ce6a747d565f6e35fd9b.png C. 27abf3c3c0ceec6fce6416dc3fcf1951.png D. 463e10b4289d71d8f76004d317ee77b5.png
3.“9d6180535d9a630651400a641470c8e7.png”是“直线155307be2cf4893944351e234660dbe0.png的倾斜角大于cd3ba7dbe6650fbf0b092d3d3c833d5d.png”的( )
A.充分而不必要条件
B.必要而不充分条件
C.充分必要条件
D.既不充分也不必要条件
4.已知31de0d07f5ecf3e5ef4d6dd2f44e0557.png,则07b103119ce2f2894a934e516ace9d04.png( )
A.1 B.-1 C.93b05c90d14a117ba52da1d743a43ab1.png D.0
5.若0779458e5cb9f831cdfa2b3b7e9553e2.png是两条不同的直线,5f2abeeac98cd1096d5658f9f306e651.png是两个不同的平面,则下列命题正确的是( )
A.若4b3aa87a8f511636d1f8dda243783d8f.png,则8c2b9ccc9d1561f905733e45009887b4.png
B.若c626c36474eec1e7445fb700d1450c38.png,则fa77724abe2f213f8c1bc64cb96df1e2.png
C.若b845c6579016698b89e0ef4807163d47.png,则dac87ce23f58f8d3aaabc2785ff234ea.png
D.若0d195323d3c80f7ad6510b75f83b1f0f.png,则3a3c174dd0b6e631ebac1d9dbe08548f.png
6.下列命题正确的个数是( )
a9ede36fb169b63a29b4a4d160c98e4e.png已知点db7e83b803c05fdb00aead61b16d96f4.png在圆6f01700fbd354149d7a2d44fb27cccb1.png外,则直线e9994df7545f18ac84dea5b4988605fe.png与圆f186217753c37b9b9f958d906208506e.png没有公共点.
5055008c66be740e074a66b3a1960d53.png命题“dd1a780be8018c532f8efb9639903110.png”的否定是“f4fb1fec3c3dba8c2376f5f895ba970e.png” .
a38dcd9b7af969d544d58708ec8ce60d.png已知随机变量02129bb861061d1a052c592e2dc6b383.png服从正态分布d157416b01500df09fb6bdc4875512fe.png,33d7e70ab0ee1da491011fa8cadbb634.png,则65ec8aeee451f73a85ef4d53d26d89a1.png.
44189a991ff9acb07663ecccddcac6d0.png实数f10bc3c94b77e1d6b9f98106daf335c1.png满足约束条件d0f408839e5bb35235d14214b540945f.png,则目标函数7fff789024649743e7be6ffd40edb218.png的最小值为1.
A.1个 B.2个 C.3个 D.4个
7.函数aa1750e003511d2661225fbfee1b4bab.png的图象大致为( )
A. B. C. D.
8.等比数列02731f49cef7ec135770140b199cf7cb.png的首项c9cbc41caa2e11a695d71be3e1ea90c4.png,前7b8b965ad4bca0e41ab51de7b31363a1.png项和为44d853a7808a331d95220fcb38095649.png,若7a58168037d6c1cdd51adca19e9ef43e.png,则数列cd7736b5ab5c98e4b6dc49749c61ebca.png的前d3d9446802a44259755d38e6d163e820.png项和为( )
A.65 B.75 C.90 D.110
9.高斯是德国著名的数学家,近代数学奠基者之一,享有“数学王子”的称号,用其名字命名的“高斯函数”为:设b65d5152fd3b93a2f4bf4c94f5fc8cc3.png,用ef2388922dbd928d592f5da09c1c9eeb.png表示不超过9dd4e461268c8034f5c8564e155c67a6.png的最大整数,则ee6a958f736a6b6021fc9cd524e0ead7.png称为高斯函数,例如:484276e142d22bde9d299c3397e9e417.png,d37e385c474441ed238f592b8f286b07.png,已知函数f3834f40d1eef1b47a124beed070eefc.png,则函数fef1332d5189f3fb2327ec3ed6bd8046.png的值域是( )
A.2f22877729561346fb3f82681f530340.png B.85151f02c0ad6308a892c2adbfe6353a.png C.62bec01e6036f25b8dd29bc69142078d.png D.a0c1248d6bdaa9bacf00ab991f2551c7.png
10.某多面体的三视图如图所示,则该几何体的体积与其外接球的表面积的数值之比为( )
A.c0a66beb835d045fdedee309979f9feb.png B.010395aa7f5447d10c87ed515f90b8ff.png C.b4854d59fdbeb52bc3e81ff9ab2200b5.png D.04d837f837ba84139c1c01757f9e9270.png
11.已知点b10bb9cc61fe30f42c2f367d69038e88.png是双曲线45b71004573972dd1ed36b0340f53450.png的左焦点,过800618943025315f869e4e1f09471012.png且平行于双曲线渐近线的直线与圆222041b22f9bed1bdd29fae79ecd9958.png交于点800618943025315f869e4e1f09471012.png和另一个点44c29edb103a2872f519ad0c9a0fdaaa.png,且点44c29edb103a2872f519ad0c9a0fdaaa.png在抛物线2cbaa22399be501b5e277dd60ab44377.png上,则该双曲线的离心率是( )
A.aa4e3cfb024c7ff30a8846913966dfb1.png B.e091b8fa7233f8fad57977e3a64289fd.png C.a13c27c8b637f6a6f639760d8272b7ca.png D.a82f4247f85daecf06487906d965c226.png
12.已知函数0aaa7790b59eef832fb8cfab3f3d93fc.png在区间b6dbc33006b907f2db1855810abfce98.png内任取两个实数c1753c36ab4eb582f1420d5178cb4bc5.png,且fd070ae3cb267feef69234e1461307bc.png,不等式b4bd9c83f27aadcf87ecad12a9885a33.png恒成立,则实数0cc175b9c0f1b6a831c399e269772661.png的取值范围是 ( )
A.8bb3c7edfa89d385bc6e576646ee43a2.png B.b2618e73c5f2ab1ef3e2ad348184245d.png C.4266685f01df9ad0a9591024cb2e3491.png D.e0d76bedf61f74701879d69fda845ae1.png
二、填空题
13.一个盒子中装有6张卡片,上面分别写着如下六个定义域为e1e1d3d40573127e9ee0480caf1283d6.png的函数:a53435592b15d45111cf5a5f77e31478.png, d00a5ce74add4fa4b2cbaf7d034602d1.png,6546295215fc92c739d88ca9f5293bcf.png,2c703e9222912add2a4831f9c658132a.png,65109f50ea7c909049e7d1caffc8d8ae.png,3deb89edf30d70ce0ed75062e5f1e936.png从盒子中任取2张卡片,将卡片上的函数相乘得到一个新函数,所得新函数为奇函数的概率是 .
14.二项式4284f13b8477a8353eec4ee7a81674cb.png的展开式中7a47687226b8a6b097ae16e6de59da37.png的系数为91a24814efa2661939c57367281c819c.png,则cbd473412676b374b8b64c9bbb665e33.png_______________。
15.在533c003242f935720a3ff6d1bc2c631e.png中,f623e75af30e62bbd73d6df5b50bb7b5.png是f85b7b377112c272bc87f3e73f10508d.png的中点,c1d9f50f86825a1a2302ec2449c17196.png是e182ebbc166d73366e7986813a7fc5f1.png的中点,过点c1d9f50f86825a1a2302ec2449c17196.png作一直线943afaf25ac17fe7bc39fdaae916e3a4.png分别与边1c82f1f08f1e0404dda6a37e41292e65.png交于c307df251313e30dbffc899da627b0d6.png,若702d86c57c80def8d251886525d24fba.png,74727a1ada880dddfdd10d3247e9a875.png,则172909ae4f6140793a7cd7b5978d8b38.png的最小值是________.
16.不等式bbd5098bf976d51239b97af53cc82f16.png对b9e9d4f269543a3b4fe2f566494af63e.png恒成立,则实数0cc175b9c0f1b6a831c399e269772661.png的取值范围是________.
三、解答题
17.在533c003242f935720a3ff6d1bc2c631e.png中,角ce04be1226e56f48da55b6c130d45b94.png的对边分别为a44c56c8177e32d3613988f4dba7962e.png,且efb9b01280d747417281357cb29774a8.png.
1.求角7fc56270e7a70fa81a5935b72eacbe29.png的大小;
2.若等差数列02731f49cef7ec135770140b199cf7cb.png的公差不为零,4e7de42dc1f2baa2d92d87f0ece9e9d6.png,且0f768ac5d5dea8d93716a27da05871de.png、3245e1e5ae22ab11774bb424bcc68e53.png、b7a13d1e0b257580b5e7518b53acb172.png成等比数列,求74168c36e4a6b2b6bc22cb15fd5e0353.png的前7b8b965ad4bca0e41ab51de7b31363a1.png项和44d853a7808a331d95220fcb38095649.png.
18.如图,在空间四面体cb08ca4a7bb5f9683c19133a84872ca7.png中, e182ebbc166d73366e7986813a7fc5f1.png⊥平面902fbdd2b1df0c4f70b4a5d23525e932.png,6c4d7b8dd23cf9fa2c5133d9295c1cf6.png,且61f9a8753334a2310b55ada40e3fc88a.png.
1.证明:平面75b85826a15607f238debae369a5571c.png⊥平面8539ef1fba74a70f5a77fcc3f25c1659.png;
2.求四面体cb08ca4a7bb5f9683c19133a84872ca7.png体积的最大值,并求此时二面角1cb568f1ded1415cd0575f2b83d0af27.png的余弦值.
19.2018年7月24日,长春长生生物科技有限责任公司先被查出狂犬病疫苗生产记录造假,后又被测出百白破疫苗“效价测定”项不符合规定, 由此引发的疫苗事件牵动了无数中国人的心.疫苗直接用于健康人群,尤其是新生儿和青少年,与人民的健康联系紧密.因此,疫苗在上市前必须经过严格的检测,并通过临床实验获得相关数据,以保证疫苗使用的安全和有效.某生物制品研究所将某一型号疫苗用在动物小白鼠身上进行科研和临床实验,得到统计数据如下:
现从所有试验小白鼠中任取一只,取到“注射疫苗”小白鼠的概率为add2b5c8b974155f65e931df2054a985.png.
1.求249543bbeb4e6a832f228d60a680ff60.png列联表中的数据ffd1b2a14ec9bd9750e3132e8c07ef1d.png的值;
2.能否有fcd81b95cf4c09d8f5c9c7d075edefce.png把握认为注射此种疫苗有效?
3.现从感染病毒的小白鼠中任意抽取三只进行病理分析,记已注射疫苗的小白鼠只数为8b8bf6c426eb40b0d06df646f36a4ae3.png,求8b8bf6c426eb40b0d06df646f36a4ae3.png的分布列和数学期望.
附:6fa5cdd8e37181a1f57462e7d7a0fa35.png,d45902a6f8071534eeee4791053bad49.png.
20.已知椭圆0eca03d989d4eba51a30ceddfbe7e542.png的左、右焦点分别为8b73bdbf452fd7cf6840411396bfb043.png,离心率d9493e3354a648fea41f4bbdc8adb50c.png,点P是椭圆上的一个动点,678e38aa49efc26f0daff91fa7b273ff.png面积的最大值是9f525fcad8e41db98394ae9281e4da10.png.
1.求椭圆的方程;
2.若93f9dc4db1b82845990c5124e8b1c45a.png是椭圆上不重合的四点,22ae662cbeefd95571ca1be581c0141b.png相交于点39a427e0b250982dd0fab7c404b4e2c2.png,6d291b0f2686c7b2126da6ba2af4aa65.png,且59da07bc514f9b21aed1a640ca2d29eb.png,求此时直线4144e097d2fa7a491cec2a7a4322f2bc.png的方程.
21.已知函数9a8e418dfab81c1cb3e3d4e71c890fd1.png
1.若曲线7c1c9491ba7c6e8d6d2cfa82e39b22ca.png在点a255512f9d61a6777bd5a304235bd26d.png处的切线方程是9b55e58c93c101cb9349682c1eb816a6.png,求实数0779458e5cb9f831cdfa2b3b7e9553e2.png的值;
2.若f9db00e998cf022870915dba8d1b9815.png,对任意c1262207fb8672dcf0f1ed34383816c0.png,不等式3f8568771d9be0a9e1acf61fd051c85e.png恒成立,求实数e358efa489f58062f10dd7316b65649e.png的取值范围.
22.在极坐标系中,曲线d41d8cd98f00b204e9800998ecf8427e.png的极坐标方程为73726975f084131de5b71110170adbbc.png,曲线0d61f8370cad1d412f80b84d143e1257.png与曲线f623e75af30e62bbd73d6df5b50bb7b5.png关于极点对称.
1.以极点为坐标原点,极轴为9dd4e461268c8034f5c8564e155c67a6.png轴的正半轴建立直角坐标系,求曲线f623e75af30e62bbd73d6df5b50bb7b5.png的极坐标方程;
2.设44c29edb103a2872f519ad0c9a0fdaaa.png为曲线f623e75af30e62bbd73d6df5b50bb7b5.png上一动点,记44c29edb103a2872f519ad0c9a0fdaaa.png到直线8b6445c5ec8df8909e1c9df0bbb4608d.png与直线bcca71a08303595e8bc1c411187cbf24.png的距离分别为d508ce714ad200eecf7c39eb5e85fc32.png,求6872bf6c24b8886c2b6defc03a73ff15.png的最小值.
23.已知函数d1f045d668bd5f41db7387e21b0c4327.png
1.当3872c9ae3f427af0be0ead09d07ae2cf.png时,解不等式3941ed61a34fefa31d44be9f8c273332.png;
2.若存在0b21a666a81629962ade8afd967826ed.png满足1e077bb2fecff1afc3f62968772e9868.png,求实数0cc175b9c0f1b6a831c399e269772661.png的取值范围.
参考答案
1.答案:B
解析:
2.答案:C
解析:
3.答案:A
解析:
4.答案:D
解析:
5.答案:D
解析:考查空间中线面的位置关系
6.答案:A
解析:
7.答案:D
解析:
8.答案:A
解析:
9.答案:D
解析:1a136b218d8ce2fd2fb63cad6d226a34.png
∴当5c77023c5630081535d65c257628da48.png时,9f7fc9615d98f1be118a07256c954ab6.png
当6aa1e6948d09d634566855e433a2b2be.png时,026217f412d72ad7a9c52c85c2a8f9f9.png;
当4a81d4e66bf4b3e3f3aa420ba51da0a3.png时,01ffd34989f5efe136d660a0e792a335.png
∴函数19ff4550cdd3459c1b809e87c2f199de.png的值域是a0c1248d6bdaa9bacf00ab991f2551c7.png
故答案为a0c1248d6bdaa9bacf00ab991f2551c7.png
10.答案:B
解析:
11.答案:C
解析:
12.答案:C
解析:
13.答案:add2b5c8b974155f65e931df2054a985.png
解析:
14.答案:6ca8c824c79dbb80005f071431350618.png
解析:
15.答案:b26e03cebe297985d1c61ae12b9348de.png
解析:
16.答案:9a28e54adb9ae0f831e3e4ac767b8ec1.png
解析:
17.答案:1.由efb9b01280d747417281357cb29774a8.png
得393ea13edce6642ce23f861b6c0b859a.png,658333c031c77898516fee736e2e5640.png,
所以a52e7bbd752732c5e87c409f165ab2c7.png
又df4ab748303ae74343f4a5632909404d.png
defc044f9c81f9ba22a30a86e76352b6.png
2.设3d0299a906f22a56ae7e72f5cb3590bf.png的公差为8277e0910d750195b448797616e091ad.png,由得4178829a8c36cd8fceb2b713b302671a.png,且01ff45f66fbed86b5fbc4d89259c25ac.png,
∴bfd4355784f89619e2fd7688ebf8774c.png.
又70c3b0045afdc51be2540c9d3191da7b.png,∴87dd9e833d58394690e907aed4c2e01c.png,∴9087857e7f19f29d4614931ea339ac72.png
∴426b1ed78b44279712541839d80660e3.png
∴4200e7c9811522a7c503af20c1371b1e.png
e87b9f765d6f0a282a741d70c087566c.png
解析:
18.答案:1.8c6ad34ef784091d0a954a0276252dfa.png,
f54b3dc0469ffc4eb3eb602d3ce3dcf4.png
故bc7c405cec747c5fe9d8496d68881ec1.png
即42ba7705b75c8d7aa17b5c83cabfa19b.png
又c0aa34183c3b9b2188c32ed9fe9a98f6.png面902fbdd2b1df0c4f70b4a5d23525e932.png
478e57688cc55f89fa7cb1ef4bcf01ac.png
由42ba7705b75c8d7aa17b5c83cabfa19b.png、02a62e29d36eeb07f1b68af39e76281d.png得6badd10cbda3dbe6a0586c220097d64a.png面75b85826a15607f238debae369a5571c.png
故有平面a7cdf8675192a55cc0f673f4dbc11859.png平面8539ef1fba74a70f5a77fcc3f25c1659.png
2.设6e950f863356ca8b4db83a7d723c33a4.png,则0681132ca86cf8434a94b3e8a3a48c24.png
四面体cb08ca4a7bb5f9683c19133a84872ca7.png的体积d432266fc53da3b684a99c96ce401c70.png
bfcf9ef5f75990dbd6dee815dbbdb7dc.png,
故7da3f9417e349bc472da3b128413cd70.png在ab53a4f568e6c71f30aeda1497f8bfcd.png单增,在edb9a54e6bc851c55113bb76be692614.png单调递减
易知440bcb2225cd249b09bb29454f83249d.png时四面体cb08ca4a7bb5f9683c19133a84872ca7.png的体积5206560a306a2e085a437fd258eb57ce.png最大,且最大值是4f68aed6db5d75011aff3ac823640679.png
以9d5ed678fe57bcca610140957afab571.png为原点,f85b7b377112c272bc87f3e73f10508d.png所在直线为9dd4e461268c8034f5c8564e155c67a6.png轴,5fc810cf62601df84b7923b9964c53e6.png所在直线为415290769594460e2e485922904f345d.png轴,
建立空间直角坐标系则36aa8d970e7036a843b4f0eb75bccbd4.png
设平面8539ef1fba74a70f5a77fcc3f25c1659.png的法向量为1fabdc2edb871cd374d9900db3c06b84.png
则由bb94f580df5fe3089c0177c84cedd3b5.png
取6a267007228f9f654a0d28dec6932c31.png,得平面8539ef1fba74a70f5a77fcc3f25c1659.png的一个法向量为a508cd411fd419228c9205c4f270fb2b.png
同理可得平面72c5cc0e2586935d16539f31a2a4fec4.png的一个法向量d5639b9166446ca432e8732b7af1dad6.png
adc5ff64937cdc221204b2eba6db17dd.png
由于1cb568f1ded1415cd0575f2b83d0af27.png是锐二面角,故所求二面角的余弦值为b78f19034bdedb175c5790f1f19c63e1.png
解析:
19.答案:1.设“从所有试验小白鼠中任取一只,取到‘注射疫苗’小白鼠”为事件A,
由已知得33ecd841eb14e6ac007201148cdf97b8.png,
所以ab5fff50cf8118de0be2b8abd258a968.png
2.80e58a1d017a41428ef7f16cceb442da.png
所以至少有fcd81b95cf4c09d8f5c9c7d075edefce.png的把握认为疫苗有效.
3.由已知8b8bf6c426eb40b0d06df646f36a4ae3.png的取值为37770ad1bcf26046c97ffab80fefb809.png
dc34d65018c6227a52d42b6b6a38e3bc.png
6f8ee83a4165f36e6129b886fe2daaf3.png
3160f5bfc6a1fd8764cd9cadd008572f.png
b271ede9d2d87853f8f18a769476b62b.png
8b8bf6c426eb40b0d06df646f36a4ae3.png的分布列为:
数学期望55eb72ec41eeb7d864fa07ece7dd3adb.png
解析:
20.答案:1.由题意知,当点P是椭圆上、下顶点时,678e38aa49efc26f0daff91fa7b273ff.png面积取得最大值
此时,是2b6bd31ba1673dda26a7643cad91de8b.png,又01520e2940bc4ac941d465485cc5c10c.png
解得7db2f38b6531097336c807ac43ceeb4d.png,所求椭圆的方程为1400d71ef707997adf38ed49ffc1c267.png
2.由1知3502cfb6729f78543237ff697b5b6f63.png,由6d291b0f2686c7b2126da6ba2af4aa65.png得500e6cfbb23c16776c395e671acb327e.png,
①当直线4144e097d2fa7a491cec2a7a4322f2bc.png与87a47565be4714701a8bc2354cbaea36.png有一条直线的斜率不存在时,9d5d74d7b59ab0b85efd2b60b6eef74d.png,不合题意
②当直线的斜率为4336ea2e6db6f7115128b06bd5e6deeb.png存在且不为50bd4711a422a838a8a585e77a1b3fd8.png时,其方程为551a7cd67a8ac6676a41baeb86ccc473.png
由a4d745b57175a661d09435d22fa77420.png消去y得2013651ca41444af1d32afe5df6d6e0c.png
设778feebb10e13e1680584eaba84c595a.png
则913ecae45943bf35dd16d25f4ad1a319.png
所以22958f6a22876ac57050caebe5f2e123.png
直线87a47565be4714701a8bc2354cbaea36.png的方程为8af3e137f8d872f95f5daf80703d6e09.png,同理可得9edf7db820acc5ae97169e3e3c2dd34b.png
由f60e2ff8883258e57b2dbaa56b4c4b70.png解得e9ae5470ec06ba7dab8906f83b2f3d06.png
故所求直线4144e097d2fa7a491cec2a7a4322f2bc.png的方程为32d620c507f9dac7d7d6250cd3f52fa1.png
解析:
21.答案:1.因为9a8e418dfab81c1cb3e3d4e71c890fd1.png,所以f219f3327105ac7ecd804c3cc63eda8f.png
因曲线7c1c9491ba7c6e8d6d2cfa82e39b22ca.png在点a255512f9d61a6777bd5a304235bd26d.png处的切线方程是9b55e58c93c101cb9349682c1eb816a6.png
0d665264876f3e2b8ec2c350bfce4d05.png,又切点为ec6014ffc3d92709aa6a4fea11bb3788.png,得13aefef79524eb1b231abbd479e50fff.png
所以541bb80b59d2823b6f0f5337d66f28b0.png
2. f219f3327105ac7ecd804c3cc63eda8f.png,f9db00e998cf022870915dba8d1b9815.png,
所以844e017ca3b5b705b04a04dc654bf16e.png时,08c640c4f51d912be5fee1e5f1e8ed35.png恒成立
故函数50bbd36e1fd2333108437a2ca378be62.png在eb1351ddf84d6ca6e06bbb319592766d.png上单调递增
不妨设5260b620b443afa17fbda5c943dabb89.png,则3f8568771d9be0a9e1acf61fd051c85e.png
可化为71069832276d9c02bb920df49deb7ddb.png
设54333558d55fde266ec5d8e01f1e6771.png
则37b9520dde59fda5ecbab5de2d110f36.png,即ca8e608169b20a94570ac837e8ba0833.png在eb1351ddf84d6ca6e06bbb319592766d.png是减函数
即f69344cbcd06fc93a7e4c96adf9f9b54.png在eb1351ddf84d6ca6e06bbb319592766d.png上恒成立,
等价于4a223c1854b95803b649b357afe69835.png在eb1351ddf84d6ca6e06bbb319592766d.png上恒成立
即324665482e0103433196e4b7755f01ac.png对任意844e017ca3b5b705b04a04dc654bf16e.png恒成立
由于d26beab3529b76f2503d2f5112f9cba8.png在844e017ca3b5b705b04a04dc654bf16e.png是增函数,
故d26beab3529b76f2503d2f5112f9cba8.png最大值是82712d720499b86477d59b5224eacac5.png
故782852e8e90db6d844d1a77272032957.png即实数e358efa489f58062f10dd7316b65649e.png的取值范围是cbd104e5097437c74c89eb9406735052.png
解析:
22.答案:1.设d861aea4410c246600eebd69c2d5d6ff.png是曲线f623e75af30e62bbd73d6df5b50bb7b5.png上任意一点,则44c29edb103a2872f519ad0c9a0fdaaa.png关于原点的对称点4af9cb7140eff406f917e3b6d2ad8bb3.png在曲线0d61f8370cad1d412f80b84d143e1257.png上,
且512e6f81b2db5df0defa8033496de030.png,将512e6f81b2db5df0defa8033496de030.png代入73726975f084131de5b71110170adbbc.png得a2c3208f2e458ab6bb8f28f730de3356.png,
则75b0201349fd1df4e6dad94024d91c96.png,即曲线f623e75af30e62bbd73d6df5b50bb7b5.png的极坐标方程为75b0201349fd1df4e6dad94024d91c96.png。
2.由曲线f623e75af30e62bbd73d6df5b50bb7b5.png的极坐标方程为75b0201349fd1df4e6dad94024d91c96.png得直角坐标方程为8af4cf5b1e59edff62d14598f9f847c4.png,
设5e76e38c02ee046d40b6e9d2aa885def.png,
直线8b6445c5ec8df8909e1c9df0bbb4608d.png与直线f337883ca8c50fbb604aef1091d222aa.png的直角坐标方程分别为b1b71c4dadec856441a9071646dd6f1f.png,
从而e86c94df7d9e2befb3f33290d9c4bd2e.png
3e08260b4a8f0f213c4160b01a594e72.png,
故6872bf6c24b8886c2b6defc03a73ff15.png的最小值为c5b67deff9fbc5667ba8ef41b8f0a130.png
解析:
23.答案:1.当3872c9ae3f427af0be0ead09d07ae2cf.png时,58bbb7d6055f2e1286385cf3a9fe2e68.png,
由3941ed61a34fefa31d44be9f8c273332.png,得b4026537a752e578d0c28bd2a4c4d8f5.png。
当961493342dc3a50043fbcf08176ac65b.png时,不等式等价于6b1fb6f5a3c69aef52731cb50dfbaa1f.png,解得961493342dc3a50043fbcf08176ac65b.png,所以961493342dc3a50043fbcf08176ac65b.png;
当46d46146e05d3efe96d9ada7fc25d614.png时,等价于95432003ae12a53023d2314b3efd33ad.png,解得961493342dc3a50043fbcf08176ac65b.png,所以9dd4e461268c8034f5c8564e155c67a6.png无解;
当56d3b110b0ef773f673064e0dcbdfe77.png 时,不等式等价于8717064b9819a52f34ae1597660d11ea.png,
解得0e1564adaf624eb2ff2c1a729c36a204.png,所以0e1564adaf624eb2ff2c1a729c36a204.png。
故原不等式的解集为9bf57b218580a25305ae230c0f9fb167.png
2.0bbeeb3e60e36017f9d64ec19bfc7113.png
89b08a5740c500e416e5090b030c37b3.png
由题意09806835c0e8fab0478f61de827474f7.png,所以d1cbf7841e975e94dd7249a303b8ef7a.png,
解得e984a47bd6b9a733ef84687c96427b53.png。
解析:
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