大专考试数学题库及答案

大专考试数学题库及答案

一、单项选择题(每题2分,共10题)1.函数\(y=\frac{1}{x-1}\)的定义域是(）A.\(x\neq0\)B.\(x\neq1\)C.\(x\gt1\)D.\(x\lt1\)2.下列函数中,既是奇函数又是增函数的是()A.\(y=x+1\)B.\(y=-x^3\)C.\(y=\frac{1}{x}\)D.\(y=x^3\)3.已知\(\sin\alpha=\frac{3}{5}\),且\(\alpha\)是第二象限角,则\(\cos\alpha=\)()A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)4.等差数列\(\{a_n\}\)中,\(a_1=2\),\(a_3=6\),则公差\(d=\)()A.1B.2C.3D.45.抛物线\(y=2x^2\)的焦点坐标是()A.\((0,\frac{1}{2})\)B.\((0,\frac{1}{8})\)C.\((\frac{1}{8},0)\)D.\((\frac{1}{2},0)\)6.已知向量\(\vec{a}=(1,2)\),\(\vec{b}=(2,3)\),则\(\vec{a}-\vec{b}=\)()A.\((-1,-1)\)B.\((1,1)\)C.\((-3,-1)\)D.\((3,1)\)7.函数\(y=\log_2(x+1)\)的反函数是()A.\(y=2^x-1\)B.\(y=2^x+1\)C.\(y=\log_2(x-1)\)D.\(y=\log_2(x+1)\)8.不等式\(x^2-2x-3\lt0\)的解集是()A.\((-1,3)\)B.\((-3,1)\)C.\((-\infty,-1)\cup(3,+\infty)\)D.\((-\infty,-3)\cup(1,+\infty)\)9.若\(x\gt0\),\(y\gt0\),且\(x+y=1\),则\(\frac{1}{x}+\frac{1}{y}\)的最小值是()A.2B.3C.4D.510.曲线\(y=x^3\)在点\((1,1)\)处的切线方程是()A.\(y=3x-2\)B.\(y=3x+2\)C.\(y=-3x-2\)D.\(y=-3x+2\)答案:1.B2.D3.B4.B5.B6.A7.A8.A9.C10.A二、多项选择题(每题2分,共10题)1.下列函数中,是幂函数的有（）A.\(y=x^2\)B.\(y=2x\)C.\(y=x^{\frac{1}{2}}\)D.\(y=\frac{1}{x^2}\)2.下列三角函数值为正数的是()A.\(\sin120^{\circ}\)B.\(\cos120^{\circ}\)C.\(\sin(-30^{\circ})\)D.\(\cos(-30^{\circ})\)3.已知数列\(\{a_n\}\)是等比数列,公比\(q=2\),\(a_1=1\),则()A.\(a_2=2\)B.\(a_3=4\)C.\(a_4=8\)D.\(a_5=16\)4.下列函数中,在\((0,+\infty)\)上单调递增的是()A.\(y=x^2\)B.\(y=\lnx\)C.\(y=2^x\)D.\(y=\frac{1}{x}\)5.已知向量\(\vec{a}=(1,0)\),\(\vec{b}=(0,1)\),则()A.\(\vec{a}\cdot\vec{b}=0\)B.\(|\vec{a}|=1\)C.\(|\vec{b}|=1\)D.\(\vec{a}\)与\(\vec{b}\)垂直6.下列说法正确的是()A.一个命题的逆命题为真,则它的否命题为假B.一个命题的逆命题为真,则它的逆否命题为真C.一个命题的逆否命题为真,则它的否命题为真D.一个命题的否命题为真,则它的逆命题为真7.已知函数\(y=f(x)\)的定义域为\([-1,1]\),则函数\(y=f(2x-1)\)的定义域为（）A.\([0,1]\)B.\([-1,0]\)C.\([0,2]\)D.\([-1,2]\)8.下列曲线中,关于原点对称的有()A.\(y=x^3\)B.\(y=\sinx\)C.\(y=\lnx\)D.\(y=\frac{1}{x}\)9.已知\(a\gtb\gt0\),则()A.\(a^2\gtb^2\)B.\(\frac{1}{a}\lt\frac{1}{b}\)C.\(a^3\gtb^3\)D.\(\sqrt{a}\gt\sqrt{b}\)10.若直线\(l_1:y=k_1x+b_1\),\(l_2:y=k_2x+b_2\)平行,则()A.\(k_1=k_2\)B.\(b_1=b_2\)C.\(k_1k_2=-1\)D.\(k_1\neqk_2\)答案:1.ACD2.AD3.ABCD4.ABC5.ABCD6.D7.A8.ABD9.ABCD10.A三、判断题(每题2分,共10题)1.空集是任何集合的子集。()2.函数\(y=\sinx\)的最小正周期是\(2\pi\)。()3.若\(a\gtb\),则\(ac^2\gtbc^2\)。(）4.向量\(\vec{a}=(x_1,y_1)\),\(\vec{b}=(x_2,y_2)\),则\(\vec{a}\cdot\vec{b}=x_1x_2+y_1y_2\)。（)5.函数\(y=\sqrt{x-1}\)的定义域是\(x\geq1\)。()6.等比数列的公比不能为0。()7.若\(f(x)\)是奇函数,则\(f(0)=0\)。（)8.直线\(x=1\)的斜率不存在。()9.椭圆的离心率\(e\)的范围是\((0,1)\)。()10.若\(A\capB=A\),则\(A\subseteqB\)。（）答案:1.√2.√3.×4.√5.√6.√7.×8.√9.√10.√四、简答题(每题5分,共4题)1.求函数\(y=2x^2-4x+3\)的最小值。配方得\(y=2(x-1)^2+1\),所以当\(x=1\)时,\(y\)取最小值\(1\)。2.已知\(\tan\alpha=2\),求\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值。分子分母同除以\(\cos\alpha\),得\(\frac{\tan\alpha+1}{\tan\alpha-1}=\frac{2+1}{2-1}=3\)。3.求过点\((1,2)\)且与直线\(y=3x+1\)平行的直线方程。设所求直线方程为\(y=3x+b\),代入点\((1,2)\)得\(b=-1\),方程为\(y=3x-1\)。4.已知数列\(\{a_n\}\)的前\(n\)项和\(S_n=n^2+2n\),求\(a_n\)。\(n\geq2\)时,\(a_n=S_n-S_{n-1}=2n+1\)；\(n=1\)时,\(a_1=S_1=3\)也满足,所以\(a_n=2n+1\)。五、讨论题(每题5分,共4题)1.讨论函数\(y=x^3\)的单调性。\(y^\prime=3x^2\geq0\),仅当\(x=0\)时等号成立,所以在\(R\)上单调递增。2.如何判断两个向量是否平行?若存在实数\(\lambda\),使\(\vec{a}=\lambda\vec{b}\),则两向量平行；或对应坐标成比例。3.讨论等比数列的通项公