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2020年普通高等学校招生全国统一考试压轴试题(一)理科科数学试卷
河南 高三 月考 2020-06-02 245次

一、单选题添加题型下试题

1. 已知集合eqIddcfe3befcdbc4ca78464f60fb7fe1445eqIda20a1ba6bb144664882b4e422357b4cf,则eqId170a03e31a0e4656b66452088e998e61(   )
A.eqId7080e98f665346d1899fef3a00732af8B.eqIdcb892a350d064235bab7142e12a18afaC.eqIdc4ede4fc7df041e3a07e3f97318bdd84D.eqIdcb79cfbd0e8a43b49fc80ab6ff77de1c
2. 欧拉公式eqId39a8f1b2f21642aca0b9b6dbf0b0c1f5eqIdb6e15e51d29a47a7aff44f98b1aafaee为虚数单位)是由瑞士著名数学家欧拉发明的,它将指数函数的定义域扩大到复数,建立了三角函数和指数函数的关系,它在复变函数论里非常重要,被誉为“数学中的天桥”.根据欧拉公式,设复数eqId64bbe399ad2e47619bd963a2b2442263,则eqIdff0418f825be4a969cc22934df7cdb95等于(   )
A.eqId00b44129cf064d1eafb1a6235bc2e32dB.eqIdfca343da1efc413487359c1551da9dbeC.eqId1b88108496aa40f880467a8a169e3fd9D.eqId20fc9915bf574cc0bc4fe6bb8248b72a
3. 月形是一种特殊的平面图形,指有相同的底,且在底的同一侧的两个弓形所围成的图形.月形中的一种特殊的情形是镰刀形,即由半圆和弓形所围成的图形(如下图),若半圆的半径与弓形所在圆的半径之比为eqId4ddbd9c96b16408ab74a0f572988df3b,现向半圆内随机取一点,则取到镰刀形中的一点的概率为(   )
figure
A.eqId25c36e7fadac4d7bbdf795df59bd117eB.eqIddacd6db7c776440690c87a14277501dcC.eqId62eec4ab779444d2b6756163c8ba3cd3D.eqId3dc4f75e77264a7ca6d82bfc7056c0d8
4. 数列eqId93e38ecd74a24cb59da79181b95bfd3a的前几项是:0、2、4、8、12、18、24、32、49、50eqId26d0a7425a7845e499195ee6177ac033其规律是:偶数项是序号平方再除2;奇数项是序号平方减1再除2.如图所示的程序框图是为了得到该数列的前100项而设计的,那么在两个判断框中,可以先后填入(   )
说明: figure
A.eqIdf458db68122849abb588feb2e682d9ac是偶数?,eqIdd2d6117191874770be6fb6f04885686e
B.eqIdf458db68122849abb588feb2e682d9ac是奇数?,eqIdd2d6117191874770be6fb6f04885686e
C.eqIdf458db68122849abb588feb2e682d9ac是偶数?,eqId4e9bce5673754586af32820443318875
D.eqIdf458db68122849abb588feb2e682d9ac是奇数?,eqId4e9bce5673754586af32820443318875
5. 已知数列eqId93e38ecd74a24cb59da79181b95bfd3a的前eqIdf458db68122849abb588feb2e682d9ac项和为eqIddb5481de79c946c0a760143297d5eade,且对任意eqId075d44d1e421488b950d298144815f07都有eqId218f1c7b4af849dc92e3bef859d9d9ad,设eqIde1a58f7520014f09b4e3ee800bbdc54b,则数列eqIdeeb4ca98872f4e6d91cea28f43fc0b63的前6项之和为(   )
A.11B.16C.10D.15
6. 声音中包含着正弦函数.音的四要素:音调、响度、音长和音色都与正弦函数的参数有关.我们平时听到的音乐不只是一个音在响,是由基音和许多个谐音的结合,其函数可以是eqId65f7d8395ff147e48839e92ac90c7d36,则eqId4837c94ef0ff4dcf9b1dda4df363275a的图象可以是(   )
A.figureB.figure
C.figureD.figure
7. 过双曲线eqId2381423d4cd146cab95f55527681a766eqIda3dc8ed52c4a473694b8116e30ac1b7c的左顶点eqIdcc614bd3390c4d028df189b234dcc351作斜率为1的直线eqId417e80f1349244878d01fe90e0891f5f,若eqId417e80f1349244878d01fe90e0891f5f与双曲线的渐近线分别交于eqId8754ce8cf7f34f04abb9a0c041f57f5ceqId19a4eb16029e4550a14f2afe4741a3c3两点,且eqIde39425cec1d74681ae3d1b197802b3d4,则双曲线的离心率是(   )
A.eqId5e30b3f3682042e3bf61665ea4945f08B.eqId611d188c5f51433484033ec4dba90dadC.eqIdd1f30dbc126c4c019fb6297ec9249e7fD.eqIdda6144fc77744175b7255caf26a0c4b3
8. 已知定义在eqId7dcd2312d0bf4bb1befdfcc170e45791上的连续可导函数eqId4837c94ef0ff4dcf9b1dda4df363275a无极值,且eqId6709283711bf4ae39fde99d031d3bb28eqId7708a7e45a184cc69e7a46b7332606b4.若eqId6747ec6b3fc244f4a8cb4c291a16c0f6eqId89b891c8d1004611b92492332072b804上与函数eqId4837c94ef0ff4dcf9b1dda4df363275a的单调性相同,则实数eqId3a4208e0040c42ed8b09de2cae9d562e的取值范围是(   )
A.eqId2ab02efa03c2467c8bc857883cfd7e05B.eqIda5ac802ac5064c38bc7ed727929e4152C.eqIdf24f2087da7d41928f03984990f3c5c8D.eqId77c0d933bf1f4361a26d0747602918bf
9. 在平面四边形eqId5ce7a06ae7a34393b92b5277978ac014中,eqIdd4a44957091544c48937a2acb70631fceqId3a3f8c3527304c3ebbf05d194722aa5aeqId29f60da2619b49628cd86f1620a2fe51,若将eqIdc1606811a17a43ab89872b189e4eecc5沿eqId1b51efe7c2fa42748ac5a6be262e2fa4折成直二面角eqIdd6b45960c6954316943730fe84c281a8,则三棱锥eqId3802a92c6e594ffba940fdc2b72520dc外接球的表面积是(   )
A.eqId7681784440c04c469bb134c3ea9fb53aB.eqId754b0f61a97c4c8d9e32c2aca27cb1d3C.eqId9d1a6fc5b77c428fa86d41d660efa327D.eqIdf02c7a6cd286450983fa0c3df26806cd
10. 若eqIdb00a12d1a9764515897eed31b72591f9eqId81f9c1b78c2c472ba1c8e285a3f89598eqIdd1078b83ab8942d990e4f2d675209985,则eqId70a27b6ddf6b478285353abb3b1f3741eqIdaea992e70d4943e49e893817eb885ed7eqIdad7cefefaba947e1a168bca491c4fca1的大小关系为(   )
A.eqId98df60d61acf40a38ea6e88af8316379B.eqIdde23c9bee5f2497a98f967afa9b73a7aC.eqId0d0cc8a6003c4e21bacd608a53c711cfD.eqId2cf6c501fcaf451c89bbbe8a7b5da7ad
11. 已知eqId63db14a5b4334f3ea583c8fb12b0d175为抛物线eqId19a4eb16029e4550a14f2afe4741a3c3eqIda807245456b04e3397b6665c0a779287的焦点,过eqId63db14a5b4334f3ea583c8fb12b0d175作两条互相垂直的直线eqId74c7e0292b094e54ae596d37798a05edeqId439fb93f7745417b8bb1747d963c6e60,直线eqId74c7e0292b094e54ae596d37798a05edeqId19a4eb16029e4550a14f2afe4741a3c3交于eqIdcc614bd3390c4d028df189b234dcc351eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,直线eqId439fb93f7745417b8bb1747d963c6e60eqId19a4eb16029e4550a14f2afe4741a3c3交于eqId0cd8063abf2b458f80091bc51b75a904eqId93cbffaa5ae045d6ac45d1e979991c3a两点,则eqId4ad31cb143594f1d863e392de35e3045的最小值为(   )
A.60B.62C.64D.66
12. 已知函数eqId4837c94ef0ff4dcf9b1dda4df363275aeqId28c4c21271884dfeac29ce0222782376定义域为eqId7dcd2312d0bf4bb1befdfcc170e45791eqId1b77ff0956c44f0dbc07c8204e87a450.若eqId09061a8026224a4c80fc6c37982c06a8eqId17a5a7a42b014ee5b6aea440fceb899a,则关于eqId8e15963708e9415da25069fea5906e24的方程eqIdd32cbf9bc737485ea89f259f1a6641ef有两解时,eqId70a27b6ddf6b478285353abb3b1f3741的取值范围为(   )
A.eqId4b2c9db50b2e4842b1a2f2be6a34f274B.eqIdf059509053ef4f828c95083f0880f903C.eqIda9d7234c30024932ad5a4a1faafcbf06D.eqIdb8c3a22bf4834263a533744973d167f3

二、填空题添加题型下试题

13. 变量eqId8e15963708e9415da25069fea5906e24eqId072d7d6b911b42bc89207e72515ebf5f满足约束条件eqId9d86a53c2a724113ab60683e33967f7e则目标函数eqId6d5f26f8b7d9461fa8404321bfa4b989的取值范围是______.
14. 设eqId71d6a2ff828b41caa8cf9562454f4794eqId32de792adcf94b1285482ab77748208a为单位向量,非零向量eqIdc128e891b46248798a4e8ac0d421a099,若eqId71d6a2ff828b41caa8cf9562454f4794eqId32de792adcf94b1285482ab77748208a的夹角为eqId87a9bbb6dd564186905f2064b8c22182,则eqId25f8cde4953b49628671dfc9e5aa97c9的最大值等于______.
15. 在数列eqId93e38ecd74a24cb59da79181b95bfd3aeqIdeeb4ca98872f4e6d91cea28f43fc0b63中,eqId46a97e855da04987b2a1599fe90202c6eqId51c9c62421b74733a9802a1c5e75fdd7eqId77b3f6b9955e499dbdab033a38779496eqId46d68f4e66cf4136891c86d837a74e34.设eqId7354555c8dab40b596ee3ef934be6656,则数列eqId74ef5fe5183946caaee41d8974816a20的通项公式eqId51e2551e6fb9477eb69913f8890696dc______.
16. 已知eqIde9edd995034840cfb557ea415074294a,函数eqIdaa28def486cb491eb370d4e794c6efe5在区间eqId2ae988b00be3499786a3510849d92ff2上的最大值为eqId49b7b111d23b44a9990c2312dc3b7ed9,则eqId70a27b6ddf6b478285353abb3b1f3741的取值范围为______.

三、解答题添加题型下试题

17. 已知eqId64b9b599fdb6481e9f8d9a94c102300b的内角为eqIdcc614bd3390c4d028df189b234dcc351eqId8754ce8cf7f34f04abb9a0c041f57f5ceqId19a4eb16029e4550a14f2afe4741a3c3,它们的对边分别为eqId70a27b6ddf6b478285353abb3b1f3741eqIdaea992e70d4943e49e893817eb885ed7eqIdad7cefefaba947e1a168bca491c4fca1,已知eqId975db053026d4bf0ab32a09c2db70c4e.
(1)求角eqId8754ce8cf7f34f04abb9a0c041f57f5c的大小;
(2)若eqId9a766568cce54b6683aa197d86ba6c44eqIdb2a33d2535254431ba668d6e7c18f3af,求eqId64b9b599fdb6481e9f8d9a94c102300b的面积.
18. 如图,三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,eqIda469016de944432093211431e595c191eqId70e40cad473d412bb44fc2cd7d64475beqId3cde7181a4424dc0be079b0ff8d411d5.
figure
(1)求证:平面eqId5d815026ad8640a3b58ec658e7c6c457平面eqIde34726e130e14183a5fc4272dc576bfc
(2)若eqId285b8ae3f04f4b01a9d4f90b7c3b3624,直线eqIdbd70ef61616a454bae30f5691884094b与平面eqId0bd8d6a420744c698e9cb156a89b8b6f所成角为45°,eqId0cd8063abf2b458f80091bc51b75a904eqId8f39d706a505485987ccbb3f6177d72d的中点,求二面角eqId447fa317a1de4287bbec37db4e4f7db0的余弦值.
19. 某工厂质检部门要对该厂流水线生产出的一批产品进行检验,如果检查到第eqId0af3cc7bf89043019788e3e371917a68件仍未发现不合格品,则此次检查通过且认为这批产品合格,如果在尚未抽到第eqId0af3cc7bf89043019788e3e371917a68件时已检查到不合格品则拒绝通过且认为这批产品不合格.设这批产品的数量足够大,可以认为每次检查查到不合格品的概率都为eqId0c243e5dded24989821daf9bef2676a6,即每次抽查的产品是相互独立的.
(1)若eqIdb3fff4d8190f472583ee733f93e7b3df,求这批产品能够通过检查的概率;
(2)已知每件产品质检费用为50元,若eqId78a81a6715a94650b13a5a836f5109f1,设对这批产品的质检个数记作eqId86c70780eab347ccb2043d1d7bc9ed96,求eqId86c70780eab347ccb2043d1d7bc9ed96的分布列;
(3)在(2)的条件下,已知1000批此类产品,若eqId329ab47201e14604bd49d8cdf373b5cf,则总平均检查费用至少需要多少元?(总平均检查费用eqId1f2c92dfd3d8438ca50bd7b1c0ed8a80每批次平均检查费用eqId125def3606544bc58ccdd097b4c3556a批数)
20. 平面内与两定点eqId20c08a62962443a28784dca4c3b7e03eeqIddfc529a2af954e87a592451070ca3442连线的斜率之积等于eqId62db1d1f989e4c4da063e07a8ae90dd7的点的轨迹,加上eqId346946f1395a41008d3b3b56c9ae4fbdeqId894f8d53415741dea8f7a841c68384d1两点所成的曲线为eqId19a4eb16029e4550a14f2afe4741a3c3.若曲线eqId19a4eb16029e4550a14f2afe4741a3c3eqId072d7d6b911b42bc89207e72515ebf5f轴的正半轴的交点为eqId2381423d4cd146cab95f55527681a766,且曲线eqId19a4eb16029e4550a14f2afe4741a3c3上的相异两点eqIdcc614bd3390c4d028df189b234dcc351eqId8754ce8cf7f34f04abb9a0c041f57f5c满足eqIdbf88b556e4ea43a7960cfefdf19389d0.
(1)求曲线eqId19a4eb16029e4550a14f2afe4741a3c3的轨迹方程;
(2)求eqIde9b428a8af374c1ea1c3d923e0f79510面积eqId1b7d8b87b50e4cfdbf89d48980209a52的最大值.
21. 已知函数eqId3f0319a6a83c4db78cc0efa7e98219c8的最小值为0,其中eqId982da3835b7946d999cc8604a2ece266.
(1)求eqId70a27b6ddf6b478285353abb3b1f3741的值;
(2)若对任意的eqIdcb9cb8feb01f448397391534bb1367d5,有eqId5cfd692c87b24977af367edf04dfd083恒成立,求实数eqId4d2187284c5d4de29906363f7d21f60f的最小值;
(3)记eqIddf8f853d201549ddb485a08d62e77b76eqId7b5559fa52604169ac0cb99796b61142为不超过eqId8e15963708e9415da25069fea5906e24的最大整数,求eqIda2789a0c782641b5837564f6eb7ba80b的值.
(参考数据:eqIdda4f2992b10647a1922f43efc5815dfaeqIdee3ec2a7f7dc489981f3e7106616c098eqId8679ed14e4b14b6f8fb8af8d0a571f6e
22. 已知在极坐系中,点eqIdae2f33d2e3f74f048755b42562934171绕极点eqId2efda802d5534f6d92d9f8af7aaec28b顺时针旋转角eqIdc13953f2514e4c8f9c8aaaf5241c33ac得到点eqId9135159fbbd34d8f80d69a925eda69b8.以eqId2efda802d5534f6d92d9f8af7aaec28b为原点,极轴为eqId8e15963708e9415da25069fea5906e24轴非负半轴,并取相同的单位长度建立平面直角坐标系,曲线eqId93cbffaa5ae045d6ac45d1e979991c3aeqId7d4e23b3494e4d6a98d2f7cfc75a947ceqId2efda802d5534f6d92d9f8af7aaec28b逆时针旋转eqIdb47bf79338b1478a89eac53d16970cdf得到曲线eqId19a4eb16029e4550a14f2afe4741a3c3.
(1)求曲线eqId93cbffaa5ae045d6ac45d1e979991c3a的极坐标方程和曲线eqId19a4eb16029e4550a14f2afe4741a3c3的直角坐标方程;
(2)点eqId2381423d4cd146cab95f55527681a766的极坐标为eqId302f6af78e9e429789312a2f53d75c27,直线eqId417e80f1349244878d01fe90e0891f5f过点eqId2381423d4cd146cab95f55527681a766且与曲线eqId93cbffaa5ae045d6ac45d1e979991c3a交于eqIdcc614bd3390c4d028df189b234dcc351eqId8754ce8cf7f34f04abb9a0c041f57f5c两点,求eqIdf98c39e854a74584bf7045a26ec1db9e的最小值.
23. 已知函数eqIdfdf5c17ad9ad4f888596830664713f0a的最小值为eqId2381423d4cd146cab95f55527681a766.
(1)求eqId2381423d4cd146cab95f55527681a766
(2)若正实数eqId70a27b6ddf6b478285353abb3b1f3741eqIdaea992e70d4943e49e893817eb885ed7eqIdad7cefefaba947e1a168bca491c4fca1满足eqId25fdd70846e74deaa63aed1ad0355783,求证:eqId7bff1cd761484bcab4b00e2eb82c8ead.