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2019年普通高等学校招生全国统一考试(上海卷)
数 学
一、填空题(本大题共12题,满分54分,第1-6题每题4分,第7-12题每题5分)
1.(4分)已知集合word/media/image2_1.png,2,3,4,word/media/image3_1.png,word/media/image4_1.png,5,word/media/image5_1.png,则1e49ca292213f572d50c447279071024.png
2.(4分)计算word/media/image7_1.png .
3.(4分)不等式word/media/image8_1.png的解集为 .
4.(4分)函数word/media/image9_1.png的反函数为 .
5.(4分)设word/media/image10_1.png为虚数单位,word/media/image11_1.png,则word/media/image12_1.png的值为 .
6.(4分)已知word/media/image13_1.png,当方程有无穷多解时,word/media/image14_1.png的值为 .
7.(5分)在word/media/image15_1.png的展开式中,常数项等于 .
8.(5分)在word/media/image16_1.png中,word/media/image17_1.png,word/media/image18_1.png,且word/media/image19_1.png,则word/media/image20_1.png .
9.(5分)首届中国国际进口博览会在上海举行,某高校拟派4人参加连续5天的志愿者活动,其中甲连续参加2天,其他人各参加1天,则不同的安排方法有 种(结果用数值表示)
10.(5分)如图,已知正方形word/media/image21_1.png,其中word/media/image22_1.png,函数word/media/image23_1.png交word/media/image24_1.png于点word/media/image25_1.png,函数word/media/image26_1.png交word/media/image27_1.png于点word/media/image28_1.png,当word/media/image29_1.png最小时,则word/media/image30_1.png的值为 .
11.(5分)在椭圆word/media/image32_1.png上任意一点word/media/image33_1.png,word/media/image34_1.png与word/media/image35_1.png关于word/media/image36_1.png轴对称,若有word/media/image37_1.png,则word/media/image38_1.png与word/media/image39_1.png的夹角范围为 .
12.(5分)已知集合word/media/image40_1.png,word/media/image41_1.png,word/media/image42_1.png,word/media/image43_1.png,存在正数word/media/image44_1.png,使得对任意word/media/image45_1.png,都有word/media/image46_1.png,则word/media/image47_1.png的值是 .
二、选择题(本大题共4题,每题5分,共20分)
13.(5分)下列函数中,值域为word/media/image48_1.png,word/media/image49_1.png的是( )
A.word/media/image50_1.png B.word/media/image51_1.png C.word/media/image52_1.png D.word/media/image53_1.png
14.(5分)已知word/media/image54_1.png、word/media/image55_1.png,则“word/media/image56_1.png”是“word/media/image57_1.png”的( )
A.充分非必要条件 B.必要非充分条件
C.充要条件 D.既非充分又非必要条件
15.(5分)已知平面word/media/image58_1.png、word/media/image59_1.png、word/media/image60_1.png两两垂直,直线word/media/image61_1.png、word/media/image62_1.png、word/media/image63_1.png满足:word/media/image64_1.png,word/media/image65_1.png,word/media/image66_1.png,则直线word/media/image67_1.png、word/media/image68_1.png、word/media/image69_1.png不可能满足以下哪种关系( )
A.两两垂直 B.两两平行 C.两两相交 D.两两异面
16.(5分)以word/media/image70_1.png,word/media/image71_1.png,word/media/image72_1.png,word/media/image73_1.png为圆心的两圆均过word/media/image74_1.png,与word/media/image75_1.png轴正半轴分别交于word/media/image76_1.png,word/media/image77_1.png,word/media/image78_1.png,word/media/image79_1.png,且满足word/media/image80_1.png,则点word/media/image81_1.png的轨迹是( )
A.直线 B.圆 C.椭圆 D.双曲线
三、解答题(本大题共5题,共14+14+14+16+18=76分)
17.(14分)如图,在正三棱锥word/media/image82_1.png中,word/media/image83_1.png.
(1)若word/media/image84_1.png的中点为word/media/image85_1.png,word/media/image86_1.png的中点为word/media/image87_1.png,求word/media/image88_1.png与word/media/image89_1.png的夹角;
(2)求word/media/image90_1.png的体积.
word/media/image91_1.png
18.(14分)已知数列word/media/image92_1.png,word/media/image93_1.png,前word/media/image94_1.png项和为word/media/image95_1.png.
(1)若word/media/image96_1.png为等差数列,且word/media/image97_1.png,求word/media/image98_1.png;
(2)若word/media/image99_1.png为等比数列,且word/media/image100_1.png,求公比word/media/image101_1.png的取值范围.
19.(14分)改革开放40年,我国卫生事业取得巨大成就,卫生总费用增长了数十倍.卫生总费用包括个人现在支出、社会支出、政府支出,如表为2012年word/media/image102_1.png年我国卫生货用中个人现金支出、社会支出和政府支出的费用(单位:亿元)和在卫生总费用中的占比.
(数据来源于国家统计年鉴)
(1)指出2012年到2015年之间我国卫生总费用中个人现金支出占比和社会支出占比的变化趋势:
(2)设word/media/image106_1.png表示1978年,第word/media/image107_1.png年卫生总费用与年份word/media/image108_1.png之间拟合函数word/media/image109_1.png研究函数word/media/image110_1.png的单调性,并预测我国卫生总费用首次超过12万亿的年份.
20.(16分)已知抛物线方程word/media/image111_1.png,word/media/image112_1.png为焦点,word/media/image113_1.png为抛物线准线上一点,word/media/image114_1.png为线段word/media/image115_1.png与抛物线的交点,定义:word/media/image116_1.png.
(1)当word/media/image117_1.png时,求word/media/image118_1.png;
(2)证明:存在常数word/media/image119_1.png,使得word/media/image120_1.png;
(3)word/media/image121_1.png,word/media/image122_1.png,word/media/image123_1.png为抛物线准线上三点,且word/media/image124_1.png,判断word/media/image125_1.png与word/media/image126_1.png的关系.
21.(18分)已知等差数列word/media/image127_1.png的公差word/media/image128_1.png,word/media/image129_1.png,数列word/media/image130_1.png满足word/media/image131_1.png,集合word/media/image132_1.png.
(1)若word/media/image133_1.png,求集合word/media/image134_1.png;
(2)若word/media/image135_1.png,求word/media/image136_1.png使得集合word/media/image137_1.png恰好有两个元素;
(3)若集合word/media/image138_1.png恰好有三个元素:word/media/image139_1.png,word/media/image140_1.png是不超过7的正整数,求word/media/image141_1.png的所有可能的值.
2019年普通高等学校招生全国统一考试(上海卷)
数 学 答 案
一、填空题(本大题共12题,满分54分,第1-6题每题4分,第7-12题每题5分)
1.【解答】解:word/media/image142_1.png集合word/media/image143_1.png,2,3,4,word/media/image144_1.png,word/media/image145_1.png,5,word/media/image146_1.png,word/media/image147_1.png,word/media/image148_1.png.
故答案为:word/media/image149_1.png,word/media/image150_1.png.
2.【解答】解:word/media/image151_1.png.故答案为:2.
3.【解答】解:由word/media/image152_1.png得word/media/image153_1.png,即word/media/image154_1.png,故答案为:word/media/image155_1.png,word/media/image156_1.png.
4.【解答】解:由word/media/image157_1.png解得word/media/image158_1.png,word/media/image159_1.png,
故答案为word/media/image160_1.png word/media/image161_1.png.
5.【解答】解:由word/media/image162_1.png,得word/media/image163_1.png,即word/media/image164_1.png,
word/media/image165_1.png.故答案为:word/media/image166_1.png.
6.【解答】解:由题意,可知:word/media/image167_1.png方程有无穷多解,word/media/image168_1.png可对①word/media/image169_1.png,得:word/media/image170_1.png.
再与②式比较,可得:word/media/image171_1.png.故答案为:word/media/image172_1.png.
7.【解答】解:word/media/image173_1.png展开式的通项为word/media/image174_1.png令word/media/image175_1.png得word/media/image176_1.png,
故展开式的常数项为第3项:word/media/image177_1.png.故答案为:15.
8.【解答】解:word/media/image178_1.png,word/media/image179_1.png由正弦定理可得:word/media/image180_1.png,word/media/image181_1.png由word/media/image182_1.png,可得:word/media/image183_1.png,
word/media/image184_1.pngword/media/image185_1.png,word/media/image186_1.png由余弦定理可得:word/media/image187_1.png,word/media/image188_1.png解得:word/media/image189_1.png.故答案为:word/media/image190_1.png.
9.【解答】解:在五天里,连续的2天,一共有4种,剩下的3人排列,故有word/media/image191_1.png种,
故答案为:24.
10.【解答】解:由题意得:word/media/image192_1.png点坐标为word/media/image193_1.png,word/media/image194_1.png,word/media/image195_1.png点坐标为word/media/image196_1.png,
word/media/image197_1.png,当且仅当word/media/image198_1.png时,取最小值,故答案为:word/media/image199_1.png.
11.【解答】解:设word/media/image200_1.png,则word/media/image201_1.png点word/media/image202_1.png,椭圆word/media/image203_1.png的焦点坐标为word/media/image204_1.png,word/media/image205_1.png,word/media/image206_1.png,word/media/image207_1.png,word/media/image208_1.pngword/media/image209_1.png,word/media/image210_1.png,结合word/media/image211_1.png,可得:word/media/image212_1.png,word/media/image213_1.png,
故word/media/image214_1.png与word/media/image215_1.png的夹角word/media/image216_1.png满足:
word/media/image217_1.png,word/media/image218_1.png,
故word/media/image219_1.png,word/media/image220_1.png,故答案为:word/media/image221_1.png,word/media/image222_1.png.
12.【解答】解:当word/media/image223_1.png时,当word/media/image224_1.png,word/media/image225_1.png时,则word/media/image226_1.png,word/media/image227_1.png,
当word/media/image228_1.png,word/media/image229_1.png时,则word/media/image230_1.png,word/media/image231_1.png,
即当word/media/image232_1.png时,word/media/image233_1.png;当word/media/image234_1.png时,word/media/image235_1.png,即word/media/image236_1.png;
当word/media/image237_1.png时,word/media/image238_1.png,当word/media/image239_1.png时,word/media/image240_1.png,即word/media/image241_1.png,
word/media/image242_1.png,解得word/media/image243_1.png.
当word/media/image245_1.png时,当word/media/image246_1.png,word/media/image247_1.png时,则word/media/image248_1.png,word/media/image249_1.png.
当word/media/image250_1.png,word/media/image251_1.png,则word/media/image252_1.png,word/media/image253_1.png,
即当word/media/image254_1.png时,word/media/image255_1.png,当word/media/image256_1.png时,word/media/image257_1.png,即word/media/image258_1.png,
即当word/media/image259_1.png时,word/media/image260_1.png,当word/media/image261_1.png时,word/media/image262_1.png,即word/media/image263_1.png,
word/media/image264_1.png,解得word/media/image265_1.png.
当word/media/image267_1.png时,同理可得无解.
综上,word/media/image268_1.png的值为1或word/media/image269_1.png.故答案为:1或word/media/image270_1.png.
二、选择题(本大题共4题,每题5分,共20分)
13.【解答】解:word/media/image271_1.png,word/media/image272_1.png的值域为word/media/image273_1.png,故word/media/image274_1.png错,
word/media/image275_1.png,word/media/image276_1.png的定义域为word/media/image277_1.png,word/media/image278_1.png,值域也是word/media/image279_1.png,word/media/image280_1.png,故word/media/image281_1.png正确,
word/media/image282_1.png,word/media/image283_1.png的值域为word/media/image284_1.png,故word/media/image285_1.png错,
word/media/image286_1.png,word/media/image287_1.png的值域为word/media/image288_1.png,word/media/image289_1.png,故word/media/image290_1.png错.故选:word/media/image291_1.png.
14.【解答】解:word/media/image292_1.png等价,word/media/image293_1.png,得“word/media/image294_1.png”,
word/media/image295_1.png “word/media/image296_1.png”是“word/media/image297_1.png”的充要条件,故选:word/media/image298_1.png.
15.【解答】解:如图1,可得word/media/image299_1.png、word/media/image300_1.png、word/media/image301_1.png可能两两垂直;
如图2,可得word/media/image302_1.png、word/media/image303_1.png、word/media/image304_1.png可能两两相交;如图3,可得word/media/image305_1.png、word/media/image306_1.png、word/media/image307_1.png可能两两异面;故选:word/media/image308_1.png.
16.【解答】解:因为word/media/image310_1.png,则word/media/image311_1.png,
同理可得word/media/image312_1.png,又因为word/media/image313_1.png,所以word/media/image314_1.png,则word/media/image315_1.png,
即word/media/image316_1.png,则word/media/image317_1.png,设word/media/image318_1.png,则word/media/image319_1.png为直线,故选:word/media/image320_1.png.
三、解答题(本大题共5题,共14+14+14+16+18=76分)
17.【解答】解:(1)word/media/image321_1.png,word/media/image322_1.png分别为word/media/image323_1.png,word/media/image324_1.png的中点,word/media/image325_1.png,
则word/media/image326_1.png为word/media/image327_1.png与word/media/image328_1.png所成角,在word/media/image329_1.png中,由word/media/image330_1.png,word/media/image331_1.png,
可得word/media/image332_1.png,word/media/image333_1.png与word/media/image334_1.png的夹角为word/media/image335_1.png;
(2)过word/media/image336_1.png作底面垂线,垂直为word/media/image337_1.png,则word/media/image338_1.png为底面三角形的中心,
连接word/media/image339_1.png并延长,交word/media/image340_1.png于word/media/image341_1.png,则word/media/image342_1.png,word/media/image343_1.png.word/media/image344_1.png.
word/media/image345_1.pngword/media/image346_1.png.
18.【解答】解:(1)word/media/image348_1.png,word/media/image349_1.png,word/media/image350_1.png;
(2)word/media/image351_1.png,word/media/image352_1.pngword/media/image353_1.png存在,word/media/image354_1.png,
word/media/image355_1.pngword/media/image356_1.png存在,word/media/image357_1.png且word/media/image358_1.png,word/media/image359_1.pngword/media/image360_1.png,
word/media/image361_1.pngword/media/image362_1.png,word/media/image363_1.png,word/media/image364_1.png或word/media/image365_1.png,
word/media/image366_1.png公比word/media/image367_1.png的取值范围为word/media/image368_1.png,word/media/image369_1.png,word/media/image370_1.png.
19.【解答】解:(1)由表格数据可知个人现金支出占比逐渐减少,社会支出占比逐渐增多.
(2)word/media/image371_1.png是减函数,且word/media/image372_1.png,
word/media/image373_1.png在word/media/image374_1.png上单调递增,
令word/media/image375_1.png,解得word/media/image376_1.png,
word/media/image377_1.png当word/media/image378_1.png时,我国卫生总费用超过12万亿,
word/media/image379_1.png预测我国到2028年我国卫生总费用首次超过12万亿.
20.【解答】解:(1)抛物线方程word/media/image380_1.png的焦点word/media/image381_1.png,word/media/image382_1.png,
word/media/image383_1.png,word/media/image384_1.png的方程为word/media/image385_1.png,代入抛物线的方程,解得word/media/image386_1.png,
抛物线的准线方程为word/media/image387_1.png,可得word/media/image388_1.png,
word/media/image389_1.png,word/media/image390_1.png;
(2)证明:当word/media/image391_1.png时,word/media/image392_1.png,
设word/media/image393_1.png,word/media/image394_1.png,word/media/image395_1.png,则word/media/image396_1.png,
联立word/media/image397_1.png和word/media/image398_1.png,可得word/media/image399_1.png,word/media/image400_1.png,
word/media/image401_1.png
word/media/image402_1.png,则存在常数word/media/image403_1.png,使得word/media/image404_1.png;
(3)设word/media/image405_1.png,word/media/image406_1.png,word/media/image407_1.png,则
word/media/image408_1.png
word/media/image409_1.png,
由word/media/image410_1.png,
word/media/image411_1.png,
则word/media/image412_1.png.
21.【解答】解:(1)word/media/image413_1.png等差数列word/media/image414_1.png的公差word/media/image415_1.png,word/media/image416_1.png,数列word/media/image417_1.png满足word/media/image418_1.png,集合word/media/image419_1.png.word/media/image420_1.png当word/media/image421_1.png,集合word/media/image422_1.png,0,word/media/image423_1.png.
(2)word/media/image424_1.pngword/media/image425_1.png,数列word/media/image426_1.png满足word/media/image427_1.png,集合word/media/image428_1.png恰好有两个元素,如图:
根据三角函数线,①等差数列word/media/image429_1.png的终边落在word/media/image430_1.png轴的正负半轴上时,集合word/media/image431_1.png恰好有两个元素,此时word/media/image432_1.png.
②word/media/image433_1.png终边落在word/media/image434_1.png上,要使得集合word/media/image435_1.png恰好有两个元素,可以使word/media/image436_1.png,word/media/image437_1.png的终边关于word/media/image438_1.png轴对称,如图word/media/image439_1.png,word/media/image440_1.png,此时word/media/image441_1.png.
综上,word/media/image442_1.png或者word/media/image443_1.png.
(3)①当word/media/image444_1.png时,word/media/image445_1.png,集合word/media/image446_1.png,word/media/image447_1.png,word/media/image448_1.png,符合题意.
②当word/media/image449_1.png时,word/media/image450_1.png,word/media/image451_1.png,word/media/image452_1.png,或者word/media/image453_1.png,
等差数列word/media/image455_1.png的公差word/media/image456_1.png,word/media/image457_1.png,故word/media/image458_1.png,word/media/image459_1.png,又word/media/image460_1.png,2
当word/media/image461_1.png时满足条件,此时word/media/image462_1.png,1,word/media/image463_1.png.
③当word/media/image464_1.png时,word/media/image465_1.png,word/media/image466_1.png,word/media/image467_1.png,或者word/media/image468_1.png,因为word/media/image469_1.png,word/media/image470_1.png,故word/media/image471_1.png,2.
当word/media/image472_1.png时,word/media/image473_1.png,1,word/media/image474_1.png满足题意.
④当word/media/image475_1.png时,word/media/image476_1.png,word/media/image477_1.png,
所以word/media/image478_1.png或者word/media/image479_1.png,word/media/image480_1.png,word/media/image481_1.png,故word/media/image482_1.png,2,3.
当word/media/image483_1.png时,word/media/image484_1.png,满足题意.
⑤当word/media/image485_1.png时,word/media/image486_1.png,word/media/image487_1.png,所以word/media/image488_1.png,或者word/media/image489_1.png,word/media/image490_1.png,word/media/image491_1.png,故word/media/image492_1.png,2,3
当word/media/image493_1.png时,因为word/media/image494_1.png对应着3个正弦值,故必有一个正弦值对应着3个点,必然有word/media/image495_1.png,word/media/image496_1.png,word/media/image497_1.png,word/media/image498_1.png,不符合条件.
当word/media/image499_1.png时,因为word/media/image500_1.png对应着3个正弦值,故必有一个正弦值对应着3个点,必然有word/media/image501_1.png,word/media/image502_1.png,word/media/image503_1.png不是整数,不符合条件.
当word/media/image504_1.png时,因为word/media/image505_1.png对应着3个正弦值,故必有一个正弦值对应着3个点,必然有word/media/image506_1.png或者word/media/image507_1.png,word/media/image508_1.png,或者word/media/image509_1.png,此时,word/media/image510_1.png均不是整数,不符合题意.
综上,word/media/image511_1.png,4,5,6.
¥29.8
¥9.9
¥59.8