真题
1 . 若函数
、
满足
,则称
、
在区间
上的一组正交函数,给出三组函数:①
;②
;③
.
其中为区间
的正交函数的组数是
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/6e871961286d484bbc3797582d727839.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/ccffdada8a7b477f8f855803e955d1ab.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/6c05be5342e34ed68d37f556afd23f16.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/6e871961286d484bbc3797582d727839.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/ccffdada8a7b477f8f855803e955d1ab.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/3d3efcfa35b346f1861057c86054203b.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/b00fc86f38c04334931d45e68eef6ab5.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/d8da35df1a4e440388f1ce6cd75bddbe.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/174ab9b89e694bb598063ca875bb4a2e.png)
其中为区间
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127547904/STEM/3d3efcfa35b346f1861057c86054203b.png)
A.0 | B.1 | C.2 | D.3 |
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真题
2 . 某实验室一天的温度(单位:
)随时间
(单位:
)的变化近似满足函数关系;
.
(1)求实验室这一天的最大温差;
(2)若要求实验室温度不高于11
,则在哪段时间实验室需要降温?
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127851008/STEM/66cc0293d59c4cc198264ee63dd34376.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127851008/STEM/2fc032839979449193da0c8500132f09.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127851008/STEM/4f13372b16074975950a70387ed7058a.png)
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127851008/STEM/cb9fb54162ef4e00ba1997a25e18f72a.png)
(1)求实验室这一天的最大温差;
(2)若要求实验室温度不高于11
![](https://img.xkw.com/dksih/QBM/2014/6/20/1571783122010112/1571783127851008/STEM/66cc0293d59c4cc198264ee63dd34376.png)
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3 . 假设每天从甲地去乙地的旅客人数X是服从正态分布
的随机变量.记一天中从甲地去乙地的旅客人数不超过900的概率为p0.
(1)求p0的值;
(参考数据:若
,有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b06f6f8e7aaf143692274df350e60e.png)
)
(2)某客运公司用A,B两种型号的车辆承担甲、乙两地间的长途客运业务,每车每天往返一次,A,B两种车辆的载客量分别为36人和60人,从甲地去乙地的营运成本分别为1600元/辆和2400元/辆.公司拟组建一个不超过21辆车的客运车队,并要求B型车不多于A型车7辆.若每天要以不小于p0的概率运完从甲地去乙地的旅客,且使公司从甲地去乙地的营运成本最小,那么应配备A型车、B型车各多少辆?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19bd86b4ac74137f706f577a87f1917d.png)
(1)求p0的值;
(参考数据:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e454e3d0f307caa85ca0645d365baba6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b06f6f8e7aaf143692274df350e60e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f555d8d32e43cb7894b8b8d0cc488c.png)
(2)某客运公司用A,B两种型号的车辆承担甲、乙两地间的长途客运业务,每车每天往返一次,A,B两种车辆的载客量分别为36人和60人,从甲地去乙地的营运成本分别为1600元/辆和2400元/辆.公司拟组建一个不超过21辆车的客运车队,并要求B型车不多于A型车7辆.若每天要以不小于p0的概率运完从甲地去乙地的旅客,且使公司从甲地去乙地的营运成本最小,那么应配备A型车、B型车各多少辆?
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4 . 如图,某地质队自水平地面A,B,C三处垂直向地下钻探,自A点向下钻到A1处发现矿藏,再继续下钻到A2处后下面已无矿,从而得到在A处正下方的矿层厚度为A1A2=d1.同样可得在B,C处正下方的矿层厚度分别为B1B2=d2,C1C2=d3,且d1<d2<d3.过AB,AC的中点M,N且与直线AA2平行的平面截多面体A1B1C1﹣A2B2C2所得的截面DEFG为该多面体的一个中截面,其面积记为S中.
(1)证明:中截面DEFG是梯形;
(2)在△ABC中,记BC=a,BC边上的高为h,面积为S.在估测三角形ABC区域内正下方的矿藏储量(即多面体A1B1C1﹣A2B2C2的体积V)时,可用近似公式V估=S中﹣h来估算.已知V=
(d1+d2+d3)S,试判断V估与V的大小关系,并加以证明.
(1)证明:中截面DEFG是梯形;
(2)在△ABC中,记BC=a,BC边上的高为h,面积为S.在估测三角形ABC区域内正下方的矿藏储量(即多面体A1B1C1﹣A2B2C2的体积V)时,可用近似公式V估=S中﹣h来估算.已知V=
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321018368/STEM/ead9b85f037843e0a6bc9c9072abeaf1.png?resizew=10)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321018368/STEM/97bae1f9cf8041fd8235389f44c5c4f1.png?resizew=170)
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5 . 如图,AB是圆O的直径,点C是圆O上异于A,B的点,直线PC⊥平面ABC,E,F分别是PA,PC的中点.
(1)记平面BEF与平面ABC的交线为l,试判断直线l与平面PAC的位置关系,并加以证明;
(2)设(1)中的直线l与圆O的另一个交点为D,且点Q满足
.记直线PQ与平面ABC所成的角为θ,异面直线PQ与EF所成的角为α,二面角E﹣l﹣C的大小为β.求证:sinθ=sinαsinβ.
(1)记平面BEF与平面ABC的交线为l,试判断直线l与平面PAC的位置关系,并加以证明;
(2)设(1)中的直线l与圆O的另一个交点为D,且点Q满足
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735197384704/1571735202594816/STEM/fe360ef4630f4137844de7807a4b52b9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/10/29f8132e-d52b-4bb3-be15-39727acee098.png?resizew=195)
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2016-12-03更新
|
2780次组卷
|
4卷引用:2013年普通高等学校招生全国统一考试理科数学(湖北卷)
真题
6 . 设n是正整数,r为正有理数.
(1)求函数f(x)=(1+x)r+1﹣(r+1)x﹣1(x>﹣1)的最小值;
(2)证明:
;
(3)设x∈R,记[x]为不小于x的最小整数,例如
.令
的值.
(参考数据:
.
(1)求函数f(x)=(1+x)r+1﹣(r+1)x﹣1(x>﹣1)的最小值;
(2)证明:
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735197384704/1571735202652160/STEM/b210b038fbb344ac94a54aec26f9c0b2.png)
(3)设x∈R,记[x]为不小于x的最小整数,例如
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735197384704/1571735202652160/STEM/5941330c7eba43a5b07a8f3fbdc39e10.png)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735197384704/1571735202652160/STEM/108a733b55f84c2e9e222d7a48ca2ca8.png)
(参考数据:
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735197384704/1571735202652160/STEM/2c1b4efceba3424a99b79389879110ff.png)
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7 . (1)已知函数
,其中
为有理数,且
. 求
的最小值;
(2)试用(1)的结果证明如下命题:设
,
为正有理数. 若
,则
;
(3)请将(2)中的命题推广到一般形式,并用数学归纳法证明你所推广的命题.
注:当
为正有理数时,有求导公式
.
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/6f8c59605248485685f77b4c3bc96421.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/463c8fd45c7a46d091d17966d0dde464.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/ec0238437f0648b59824118d5895b004.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/15304a712f2f41ad99057e7d9272da55.png)
(2)试用(1)的结果证明如下命题:设
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/9168a2f310ea47cc83a3465bfda82387.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/d880ea4848af43a4b24c4d5936d36803.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/52faf3dc378d4b879bac8296a50c796b.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/4bfd68028da6483185124093e14fcdcd.png)
(3)请将(2)中的命题推广到一般形式,并用数学归纳法证明你所推广的命题.
注:当
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/0ef3fec917b847cbb8950a0b417f15cd.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924211200/STEM/73ea234624264f66b7949c96edc822b1.png)
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真题
解题方法
8 . 设
是单位圆
上的任意一点,
是过点
与
轴垂直的直线,
是直线
与
轴的交点,点
在直线
上,且满足
. 当点
在圆上运动时,记点M的轨迹为曲线
.
(Ⅰ)求曲线
的方程,判断曲线
为何种圆锥曲线,并求其焦点坐标;
(Ⅱ)过原点且斜率为
的直线交曲线
于
,
两点,其中
在第一象限,它在
轴上的射影为点
,直线
交曲线
于另一点
. 是否存在
,使得对任意的
,都有
?若存在,求
的值;若不存在,请说明理由.
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/001976725cd34c8181378ebdfa21bf25.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/baf71f96bdaa4ee5a9fc78c51ef59c40.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/120a670dd9334a308e03726e079bed76.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/001976725cd34c8181378ebdfa21bf25.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/d491db393b074f0d98151cc22c1eea37.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/1f13a0ec4ce2478ab3e3d9386086ce53.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/120a670dd9334a308e03726e079bed76.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/d491db393b074f0d98151cc22c1eea37.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/693358a568be418a9ffea081f51cf087.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/120a670dd9334a308e03726e079bed76.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/024715c5d57f4f81b227288b65b6f79d.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/001976725cd34c8181378ebdfa21bf25.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
(Ⅰ)求曲线
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
(Ⅱ)过原点且斜率为
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/5ba0ca206a3f495bb7f84502756e0047.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/093d559355dd48989cb219517e38d285.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/df6ac5a967654322841b5b80c2985362.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/093d559355dd48989cb219517e38d285.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/5593c0ed176944f1bc312521594bc146.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/5639d80f631042e38ded909204e53fc7.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/03174559494c4eb89a1efb2fdd228055.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/3f2ae1284ac74accae202b67e1667b19.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/279528b1f450405ab5e9ff1d548fd1a8.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/76c801e2bfc94c39ab48ef1a822b6f1f.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/b3a6e666ab914319a09af470aca75e57.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/cf05650da1fe49f994141f0c6e91f057.png)
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570916918902784/1570916924162048/STEM/76c801e2bfc94c39ab48ef1a822b6f1f.png)
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真题
9 . f(x)=x3+2ax2+bx+a,g(x)=x2﹣3x+2,其中x∈R,a、b为常数,已知曲线y=f(x)与y=g(x)在点(2,0)处有相同的切线l.
(Ⅰ) 求a、b的值,并写出切线l的方程;
(Ⅱ)若方程f(x)+g(x)=mx有三个互不相同的实根0、x1、x2,其中x1<x2,且对任意的x∈[x1,x2],f(x)+g(x)<m(x﹣1)恒成立,求实数m的取值范围.
(Ⅰ) 求a、b的值,并写出切线l的方程;
(Ⅱ)若方程f(x)+g(x)=mx有三个互不相同的实根0、x1、x2,其中x1<x2,且对任意的x∈[x1,x2],f(x)+g(x)<m(x﹣1)恒成立,求实数m的取值范围.
您最近一年使用:0次
2016-12-03更新
|
2038次组卷
|
2卷引用:2011年普通高等学校招生全国统一考试文科数学(湖北卷)
真题
10 . 如图,已知正三棱柱ABC﹣A1B1C1的底面边长为2,侧棱长为3
,点E在侧棱AA1上,点F在侧棱BB1上,且AE=2
,BF=
.
![](https://img.xkw.com/dksih/QBM/2014/6/5/1571758950432768/1571758955798528/STEM/cf5c7152-9660-411b-b1d9-443a4cfd042c.png?resizew=126)
(I) 求证:CF⊥C1E;
(II) 求二面角E﹣CF﹣C1的大小
![](https://img.xkw.com/dksih/QBM/2014/6/5/1571758950432768/1571758955798528/STEM/98b692d740b7403bb6665276a0b11295.png)
![](https://img.xkw.com/dksih/QBM/2014/6/5/1571758950432768/1571758955798528/STEM/98b692d740b7403bb6665276a0b11295.png)
![](https://img.xkw.com/dksih/QBM/2014/6/5/1571758950432768/1571758955798528/STEM/98b692d740b7403bb6665276a0b11295.png)
![](https://img.xkw.com/dksih/QBM/2014/6/5/1571758950432768/1571758955798528/STEM/cf5c7152-9660-411b-b1d9-443a4cfd042c.png?resizew=126)
(I) 求证:CF⊥C1E;
(II) 求二面角E﹣CF﹣C1的大小
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