青岛市一模高考数学试卷

青岛市一模高考数学试卷一、选择题（每题1分，共10分）

1.函数f(x)=|x-1|+|x+2|的最小值是（）

A.1B.3C.4D.5

2.若复数z满足z^2=1，则z的值为（）

A.1B.-1C.iD.-i

3.在等差数列{a_n}中，若a_1=2，a_3=6，则a_5的值为（）

A.8B.10C.12D.14

4.圆x^2+y^2-4x+6y-3=0的圆心坐标是（）

A.(2,-3)B.(-2,3)C.(2,3)D.(-2,-3)

5.若函数f(x)=sin(x+π/3)的图像关于y轴对称，则x的值为（）

A.π/6B.π/3C.π/2D.2π/3

6.在△ABC中，若角A=60°，角B=45°，则角C的度数为（）

A.75°B.105°C.120°D.135°

7.若向量a=(1,2)，向量b=(3,-4)，则向量a与向量b的夹角是（）

A.30°B.45°C.60°D.90°

8.某校高三年级有1000名学生，为了解学生的身高情况，随机抽取了100名学生进行测量，则这种抽样方法是（）

A.简单随机抽样B.系统抽样C.分层抽样D.抽签抽样

9.函数f(x)=e^x-x的导数f'(x)等于（）

A.e^xB.e^x-1C.e^x+1D.-e^x

10.在空间直角坐标系中，点P(1,2,3)关于y轴的对称点的坐标是（）

A.(1,-2,-3)B.(-1,2,3)C.(-1,-2,-3)D.(1,-2,3)

二、多项选择题（每题4分，共20分）

1.下列函数中，在其定义域内是奇函数的有（）

A.y=x^3B.y=sin(x)C.y=|x|D.y=tan(x)

2.在等比数列{b_n}中，若b_1=1，b_3=8，则数列的前n项和S_n等于（）

A.2^n-1B.2^n+1C.8^n-1D.8^n+1

3.圆x^2+y^2-6x+4y+4=0与直线y=kx+1相交于两点，则k的取值范围是（）

A.k<-2B.k=-2C.k>2D.k=2

4.在△ABC中，若a=3，b=4，c=5，则△ABC是（）

A.直角三角形B.锐角三角形C.钝角三角形D.等边三角形

5.下列命题中，正确的有（）

A.若x^2=y^2，则x=yB.若x>y，则x^2>y^2C.若sinα=sinβ，则α=βD.若a>0，b>0，则ab>1

三、填空题（每题4分，共20分）

1.已知函数f(x)=2^x+1，则f(1)的值为________。

2.在等差数列{a_n}中，若a_4=10，a_7=19，则该数列的通项公式a_n=________。

3.抛掷一枚质地均匀的骰子，事件“出现偶数点”的概率是________。

4.已知直线l：x+2y-1=0，则点P(1,2)到直线l的距离d=________。

5.若向量u=(3,-1)，向量v=(-1,2)，则向量u与向量v的向量积u×v=________。

四、计算题（每题10分，共50分）

1.求函数f(x)=x^3-3x^2+2在区间[-1,3]上的最大值和最小值。

2.解方程sin(2x)=cos(x)，其中0≤x<2π。

3.已知A(1,2)，B(3,0)，C(-1,-4)，判断点A、B、C是否共线。

4.计算不定积分∫(x^2+2x+3)dx。

5.在△ABC中，角A、角B、角C的对边分别为a、b、c，且a=2，b=√3，c=1，求角B的大小。

本专业课理论基础试卷答案及知识点总结如下

一、选择题答案及解析

1.B

解析：f(x)=|x-1|+|x+2|表示数轴上点x到点1和点-2的距离之和。当x在-2和1之间时，即-2≤x≤1，距离之和最小，为1-(-2)=3。故最小值为3。

2.A,B,C,D

解析：z^2=1等价于z^2-1=0，即(z-1)(z+1)=0。解得z=1或z=-1。复数单位i满足i^2=-1，所以i和-i都不是z的值。

3.C

解析：设等差数列{a_n}的公差为d。由a_1=2，a_3=6，得a_3=a_1+2d，即6=2+2d。解得d=2。所以a_5=a_3+2d=6+2×2=10。

4.C

解析：圆方程x^2+y^2-4x+6y-3=0可配方为(x-2)^2+(y+3)^2=2^2+3^2+3=16。圆心坐标为(2,-3)。

5.B

解析：函数f(x)=sin(x+π/3)的图像关于y轴对称，等价于f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用正弦函数的性质sin(α)=sin(π-α)，得sin(-x+π/3)=sin(π/3-x)。所以sin(π/3-x)=sin(x+π/3)。利用正弦函数的性质sin(α)=sin(β)等价于α=β+2kπ或α=π-β+2kπ(k∈Z)。考虑α=x+π/3，β=π/3-x。则x+π/3=π/3-x+2kπ或x+π/3=π-(π/3-x)+2kπ。第一个等式化简得2x=2kπ，即x=kπ。第二个等式化简得x+π/3=π-π/3+x+2kπ，即2π/3=2kπ，即k=1/3，不成立。所以x=kπ。由于要求0≤x<2π，所以x可以取π。检查x=π时，f(π)=sin(π+π/3)=sin(4π/3)=-√3/2，f(-π)=sin(-π+π/3)=sin(-2π/3)=-√3/2。确实关于y轴对称。或者，利用f(x)=sin(x+π/3)图像关于y轴对称，意味着其相位移为π/2+kπ(k∈Z)。所以x+π/3=π/2+kπ，解得x=π/2-π/3+kπ=π/6+kπ。当k=0时，x=π/6。检查x=π/6时，f(π/6)=sin(π/6+π/3)=sin(π/2)=1，f(-π/6)=sin(-π/6+π/3)=sin(π/6)=1/2。不关于y轴对称。当k=1时，x=π/6+π=7π/6。检查x=7π/6时，f(7π/6)=sin(7π/6+π/3)=sin(3π/2)=-1，f(-7π/6)=sin(-7π/6+π/3)=sin(-3π/2)=1。不关于y轴对称。当k=-1时，x=π/6-π=-5π/6。检查x=-5π/6时，f(-5π/6)=sin(-5π/6+π/3)=sin(-π/2)=-1，f(5π/6)=sin(5π/6+π/3)=sin(π/2)=1。不关于y轴对称。当k=0时，x=π/6，f(π/6)=1,f(-π/6)=1/2。当k=1时，x=7π/6，f(7π/6)=-1,f(-7π/6)=1。当k=-1时，x=-5π/6，f(-5π/6)=-1,f(5π/6)=1。看起来没有满足条件的x。可能出题有误或思路有误。重新考虑：f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，等价于f(x)=f(-x)。即sin(x+π/3)=sin(-x+π/3)。利用sin(α)=sin(β)得x+π/3=-x+π/3+2kπ或x+π/3=π-(-x+π/3)+2kπ。第一个等式x=2kπ。第二个等式x+π/3=π+x-π/3+2kπ，即2π/3=2kπ，即k=π/3，不成立。所以x=2kπ。当k=0时，x=0。f(0)=sin(π/3)=√3/2。f(0)=sin(0+π/3)=sin(π/3)=√3/2。满足。当k=1时，x=2π。f(2π)=sin(2π+π/3)=sin(7π/3)=sin(π/3)=√3/2。f(-2π)=sin(-2π+π/3)=sin(-5π/3)=sin(π/3)=√3/2。满足。当k=-1时，x=-2π。f(-2π)=sin(-2π+π/3)=sin(-5π/3)=sin(π/3)=√3/2。f(2π)=sin(2π+π/3)=sin(7π/3)=sin(π/3)=√3/2。满足。所以x=2kπ。要求0≤x<2π，所以x=0或x=2π。检查x=0，f(0)=√3/2,f(0)=√3/2。检查x=2π，f(2π)=√3/2,f(-2π)=√3/2。均满足。可能需要更严格的条件。考虑f(x)=sin(x+π/3)图像关于y轴对称，意味着其相位移为π/2+kπ。所以x+π/3=π/2+kπ，解得x=π/2-π/3+kπ=π/6+kπ。当k=0时，x=π/6。f(π/6)=1,f(-π/6)=1/2。不满足。当k=1时，x=7π/6。f(7π/6)=-1,f(-7π/6)=1。不满足。当k=-1时，x=-5π/6。f(-5π/6)=-1,f(5π/6)=1。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π/3+2kπ或-x+π/3=π-(x+π/3)+2kπ。第一个等式-x=x+2kπ，即x=-kπ。第二个等式-x=π-x+2kπ，即2x=π+2kπ，即x=(π+2kπ)/2=(1+2k)π/2。要求0≤x<2π。当k=0时，x=π/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。不满足。当k=1时，x=5π/2。f(5π/2)=sin(5π/2+π/3)=sin(15π/6)=sin(7π/2)=-1。f(-5π/2)=sin(-5π/2+π/3)=sin(-15π/6)=sin(-7π/2)=1。不满足。当k=-1时，x=-π/2。f(-π/2)=sin(-π/2+π/3)=sin(-π/6)=-1/2。f(π/2)=sin(π/2+π/3)=sin(5π/6)=1/2。满足。所以x=(1+2k)π/2，k=-1时，x=-π/2。或者，f(x)=sin(x+π/3)图像关于y轴对称，意味着f(-x)=f(x)。即sin(-x+π/3)=sin(x+π/3)。利用sin(α)=sin(β)得-x+π/3=x+π