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上海市川沙中学2018—2019学年高二上学期期末数学试题
上海 高二 期末 2019-11-11 95次

一、填空题添加题型下试题

1. 设抛物线的焦点坐标为(1,0),则此抛物线的标准方程为_________.
2. 已知复数eqId76a724d4b69b47c799ae65e317817cb3eqIdb6e15e51d29a47a7aff44f98b1aafaee为虚数单位),则它的虚部为_________.
3. 若复数eqId2cd9e2fe3d6540df9399dbefe49ada54满足eqId78736a7a5300479dbdca8a9654e34d43eqIdb6e15e51d29a47a7aff44f98b1aafaee是虚数单位),则eqId0b4fc8a120af4fab80df6ffdcf7c46f8_________.
5. 将椭圆的参数方程eqId559f84f4d725413c8ed54d4d07a465a3eqIdf9db0aee08264a238b9741675591065e为参数)转化为普通方程_________.
6. 已知抛物线eqIda807245456b04e3397b6665c0a779287的焦点与双曲线eqId9aca4748e10b4a46802f26823a3e28c6的右焦点重合,则双曲线的渐近线方程为_________.
典型
7. 已知圆eqId688ef50da5d24dd2969d28f9e4547a8e和点eqId2f4ad4d58c454b2890517481598f7690,则过点eqIdcc614bd3390c4d028df189b234dcc351的圆的切线方程为._________
典型
8. 若复数eqIdb1fed5e7c89b428d944e438da1566f03复平面上对应的点在直线eqIdd47194db52584b01bf9b35de491b53ca上,则eqId71b066f4ef034bf6b75c671bd9877701的最小值是_________.
9. 已知抛物线型拱桥的顶点距水面2米时,量得水面宽为8米,当水面下降1米后,水面的宽为_________米.
典型
10. 已知椭圆eqId2b0c2340657f4135ba6b1dd71e3559c2上的一点eqIdd7be5abad1e34f6db17a56660de8d54f也在抛物线eqId19a2a3e0d6404c9c8948d2ea5a37d7e1上,设抛物线焦点为eqIdd5b10b2a13f148b78e0d5a1b820538fd,若eqId5dd68526f6c04987b7e8eebb87d8ef49,则eqId783c5f01aa1e49b29f88c27d2eca4354_________.
12. 点eqId0aa72756ed0c4c7a89f26aa94f43e47deqIdde963a24921f4ee7a991bf9d4f43a42e分别是椭圆eqIde5a7720992084ff68ee34362c1638331的左、右两焦点,点eqId517584fed25c413ba8b7bb33ffa2d5c6为椭圆eqId19a4eb16029e4550a14f2afe4741a3c3的上顶点,若动点eqId2381423d4cd146cab95f55527681a766满足:eqId8b1a53867fde43069e510f8c184785ea,则eqId0835fd5903f844e5abeb78b92f41ef2a的最大值为__________.

二、单选题添加题型下试题

典型
13. 在复数范围内,下列命题中,假命题的是(   )
A.若eqId2cd9e2fe3d6540df9399dbefe49ada54为实数,则eqId3fbacf76b8f94fc1869edca7cd0df433B.若eqId3fbacf76b8f94fc1869edca7cd0df433,则eqId2cd9e2fe3d6540df9399dbefe49ada54为实数
C.若eqId969c9e9418ec4187acf06f198e501bb6为实数,则eqId2cd9e2fe3d6540df9399dbefe49ada54为实数D.若eqId2cd9e2fe3d6540df9399dbefe49ada54为实数,则eqId969c9e9418ec4187acf06f198e501bb6为实数
14. 当eqId3bc166368ff1453996dab191171f0950时,方程eqIdef923a1879424cbdbd932e8a9d2581ae所表示的曲线是(    )
A.焦点在eqIdc12aac8c2448479687a40735b3572a23轴的椭圆B.焦点在eqIdc12aac8c2448479687a40735b3572a23轴的双曲线
C.焦点在eqId072d7d6b911b42bc89207e72515ebf5f轴的椭圆D.焦点在eqId072d7d6b911b42bc89207e72515ebf5f轴的双曲线
15. 若实系数一元二次方程eqId1ccf1352c7974556849406a4a6bfbee7有两虚数根eqId053a2a1403874a859cbcaf8e354928c9,且eqIdcd38f6fbccbe4000ac164b4908ab3188,那么实数eqId3a4208e0040c42ed8b09de2cae9d562e的值是(   )
A.eqId2411e237a6024ba5812575d3840e3781B.eqId37705e1ef6a84bbdbe88433e75932cdfC.eqIdfca343da1efc413487359c1551da9dbeD.eqIdab7cb7cf6f374dca93e01c743e2fffba

三、解答题添加题型下试题

典型
17. 设eqId2cd9e2fe3d6540df9399dbefe49ada54为关于eqId8e15963708e9415da25069fea5906e24的方程eqId5cf138e8de424da3954546e81bfb76e8的虚根,eqIdb6e15e51d29a47a7aff44f98b1aafaee为虚数单位.
(1)当eqIdc57a73522f2f4fecb1b1f056b498150b时,求eqId864d6cde05884f65ad6037d69a37db41的值;
(2)在(1)的条件下,若eqId5d3d31151f0649bcaee0eb188a99242ceqId1af0f44f5eeb49a282da88e153aebf8d,求eqId70a27b6ddf6b478285353abb3b1f3741的取值范围.
18. 已知动点eqId5675989e12e2426db6140e258ba34b55到点eqIda9a6893a743340ba93c1a9b5bdfd5309的距离为eqIdcb9b2b8bd99c447bbccaaa8a411aa71b,动点eqId5675989e12e2426db6140e258ba34b55到直线eqId158110ba4d5942a3b554d4b3cf51f5b6的距离为eqIdef8988307d4c45ce94db1755a94bb661,且eqId5b3f3387fedb43cdad42cdfd639bae12.
(1)求动点eqId5675989e12e2426db6140e258ba34b55的轨迹eqId19a4eb16029e4550a14f2afe4741a3c3的方程;
(2)若直线eqId42b4056c845e4f6499ef2baf88f9dd1d交曲线eqId19a4eb16029e4550a14f2afe4741a3c3eqIdad5a3fe6c747453488b22964c985d62c两点,求eqId81232f29758e4449908f62031953893f的面积.
19. 已知双曲线eqId3891caf29e5d4c8099c4e8ebea653bc4.
(1)若经过点eqId9cda99dba8d74312be9e06ed7f7299fc的直线eqId417e80f1349244878d01fe90e0891f5f与双曲线eqId19a4eb16029e4550a14f2afe4741a3c3的右支交于不同两点eqId1f64331cf3034c38a68946ea0fe1ef14,求直线eqId417e80f1349244878d01fe90e0891f5f的斜率的取值范围;
(2)在(1)的条件下,求线段eqId996e2048847f41d5a5fe0e2282909254的中垂线eqIda20558bac276438caa0c85b6224a1e5ceqId072d7d6b911b42bc89207e72515ebf5f轴上的截距eqId32454977564e4ee18da36bd51156c509的取值范围.
20. 如图,已知满足条件eqId049e68ae25034bb7afe3e2faacacec97(其中eqIdb6e15e51d29a47a7aff44f98b1aafaee为虚数单位)的复数eqId2cd9e2fe3d6540df9399dbefe49ada54在复平面eqId2272a344734c4fb088737b84294f7219上的对应点eqId1ef00b79fdeb46e1b2869c91759ed19d的轨迹为圆eqId19a4eb16029e4550a14f2afe4741a3c3(圆心为eqId19a4eb16029e4550a14f2afe4741a3c3),定直线eqId3a4208e0040c42ed8b09de2cae9d562e的方程为eqId17d1574eb04d4d929c95ce98a1db1de4,过eqId998a5d5a212c46efa96c585d6ac13318斜率为eqId4d2187284c5d4de29906363f7d21f60f的直线eqId417e80f1349244878d01fe90e0891f5f与直线eqId3a4208e0040c42ed8b09de2cae9d562e相交于eqId517584fed25c413ba8b7bb33ffa2d5c6点,与圆eqId19a4eb16029e4550a14f2afe4741a3c3相交于eqIdad5a3fe6c747453488b22964c985d62c两点,eqId2381423d4cd146cab95f55527681a766是弦eqId4ea1af672e334fb1a6a3ab675d494e72中点.
(1)若直线eqId417e80f1349244878d01fe90e0891f5f经过圆心eqId19a4eb16029e4550a14f2afe4741a3c3,求证:eqId417e80f1349244878d01fe90e0891f5feqId3a4208e0040c42ed8b09de2cae9d562e垂直;
(2)当eqId2d1ccba965854b3186aceeb7011a1f29时,求直线eqId417e80f1349244878d01fe90e0891f5f的方程;
(3)设eqIde4ad32d097e64689a81ab7eeb433936d,试问eqId32454977564e4ee18da36bd51156c509是否为定值?若为定值,请求出eqId32454977564e4ee18da36bd51156c509的值,若eqId32454977564e4ee18da36bd51156c509不为定值,请说明理由.figure
压轴
21. 给出定理:在圆锥曲线中,eqId99a3187c2b8f4bcc9703c74c3b72f1f3是抛物线eqId31a730a2c4884736880f7e84cde89e34的一条弦,eqId19a4eb16029e4550a14f2afe4741a3c3eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,过点eqId19a4eb16029e4550a14f2afe4741a3c3且平行于eqId8e15963708e9415da25069fea5906e24轴的直线与抛物线的交点为eqId0cd8063abf2b458f80091bc51b75a904.若eqIdeb53cb7cb0274f7d8f111c60f824c243两点纵坐标之差的绝对值eqIdceb2f1336a1147cea4695308d3dc0c69,则eqId0859a3ea0e614cf8aec9b9d35def4203的面积eqId5911d3b446dc4b6eae279af6a33983c3,试运用上述定理求解以下各题:
(1)若eqId10733093f7274b47abdf247b093a0d5beqId99a3187c2b8f4bcc9703c74c3b72f1f3所在直线的方程为eqId762500ba8ef1409c9131f0f2fd5091aceqId19a4eb16029e4550a14f2afe4741a3c3eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,过eqId19a4eb16029e4550a14f2afe4741a3c3且平行于eqId8e15963708e9415da25069fea5906e24轴的直线与抛物线eqIdbf31c16f10784d76851cf83e57c2eeed的交点为eqId0cd8063abf2b458f80091bc51b75a904,求eqIda26c3a9be6d4443db09979e9bf9fc1b8
(2)已知eqId99a3187c2b8f4bcc9703c74c3b72f1f3是抛物线eqId31a730a2c4884736880f7e84cde89e34的一条弦,eqId19a4eb16029e4550a14f2afe4741a3c3eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,过点eqId19a4eb16029e4550a14f2afe4741a3c3且平行于eqId8e15963708e9415da25069fea5906e24轴的直线与抛物线的交点为eqId0cd8063abf2b458f80091bc51b75a904eqId80024c560b2a4efc8e3275bcf0f79ebb分别为eqIdbf4db7d066034b139ff80a09ab139d45eqId1b51efe7c2fa42748ac5a6be262e2fa4的中点,过eqId80024c560b2a4efc8e3275bcf0f79ebb且平行于eqId8e15963708e9415da25069fea5906e24轴的直线与抛物线eqId31a730a2c4884736880f7e84cde89e34分别交于点eqId1f64331cf3034c38a68946ea0fe1ef14,若eqIdeb53cb7cb0274f7d8f111c60f824c243两点纵坐标之差的绝对值eqIdceb2f1336a1147cea4695308d3dc0c69,求eqIddd1272ba95524f3c85fba0a12f3ea751eqId42008372275a484d9e12b4393677ff7d
(3)请你在上述问题的启发下,设计一种方法求抛物线:eqId092b9364b6f44556b0af9d3cc851e476与弦eqId99a3187c2b8f4bcc9703c74c3b72f1f3围成成的“弓形”的面积,并求出相应面积.