大專數(shù)學(xué)考試題庫(kù)及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)y=sinx的最小正周期是（）A.πB.2πC.3πD.4π答案：B2.若直線y=kx+b平行于直線y=2x+3，則k=（）A.1B.2C.3D.4答案：B3.一元二次方程\(ax^{2}+bx+c=0(a≠0)\)的判別式\(\Delta=\)（）A.\(b^{2}-4ac\)B.\(b^{2}+4ac\)C.\(4ac-b^{2}\)D.\(\sqrt{b^{2}-4ac}\)答案：A4.對(duì)數(shù)函數(shù)\(y=\log_{a}x(a>0,a≠1)\)，當(dāng)\(a>1\)時(shí)，函數(shù)在定義域上（）A.單調(diào)遞減B.單調(diào)遞增C.先減后增D.先增后減答案：B5.已知向量\(\vec{a}=(1,2)\)，\(\vec=(3,-1)\)，則\(\vec{a}+\vec=\)（）A.\((4,1)\)B.\((2,3)\)C.\((-2,1)\)D.\((-4,-1)\)答案：A6.定積分\(\int_{0}^{1}x^{2}dx=\)（）A.\(\frac{1}{3}\)B.\(\frac{1}{2}\)C.1D.2答案：A7.函數(shù)\(y=x^{3}\)的導(dǎo)數(shù)是（）A.\(3x^{2}\)B.\(2x^{2}\)C.\(x^{2}\)D.\(3x\)答案：A8.在等比數(shù)列\(zhòng)(\{a_{n}\}\)中，若\(a_{1}=1\)，公比\(q=2\)，則\(a_{3}=\)（）A.2B.4C.6D.8答案：B9.若\(f(x)=\frac{1}{x}\)，則\(f'(x)=\)（）A.\(-\frac{1}{x^{2}}\)B.\(\frac{1}{x^{2}}\)C.\(-\frac{1}{x}\)D.\(\frac{1}{x}\)答案：A10.對(duì)于橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)，其長(zhǎng)軸長(zhǎng)為（）A.2aB.2bC.aD.b答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)是奇函數(shù)的有（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=x+1\)D.\(y=\frac{1}{x}\)答案：ABD2.平面向量的運(yùn)算包括（）A.加法B.減法C.數(shù)乘D.除法答案：ABC3.以下關(guān)于二次函數(shù)\(y=ax^{2}+bx+c(a≠0)\)的性質(zhì)正確的是（）A.當(dāng)\(a>0\)時(shí)，開(kāi)口向上B.對(duì)稱軸為\(x=-\frac{2a}\)C.當(dāng)\(\Delta=b^{2}-4ac>0\)時(shí)，與x軸有兩個(gè)交點(diǎn)D.當(dāng)\(c=0\)時(shí)，函數(shù)過(guò)原點(diǎn)答案：ABCD4.下列等式正確的是（）A.\(\sin^{2}x+\cos^{2}x=1\)B.\(\tanx=\frac{\sinx}{\cosx}\)C.\(\sec^{2}x=1+\tan^{2}x\)D.\(\csc^{2}x=1+\cot^{2}x\)答案：ABCD5.若數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列，則（）A.\(a_{n}=a_{1}+(n-1)d\)B.\(S_{n}=\frac{n(a_{1}+a_{n})}{2}\)C.\(a_{m}+a_{n}=a_{p}+a_{q}(m+n=p+q)\)D.其通項(xiàng)公式是二次函數(shù)答案：ABC6.以下關(guān)于導(dǎo)數(shù)的應(yīng)用正確的是（）A.求函數(shù)的切線斜率B.判斷函數(shù)的單調(diào)性C.求函數(shù)的極值D.求函數(shù)的定義域答案：ABC7.在直角坐標(biāo)系中，圓的標(biāo)準(zhǔn)方程為（）A.\((x-a)^{2}+(y-b)^{2}=r^{2}\)B.\(x^{2}+y^{2}=r^{2}\)（圓心在原點(diǎn)）C.\(y=\sqrt{r^{2}-x^{2}}\)D.\(x=\sqrt{r^{2}-y^{2}}\)答案：AB8.下列關(guān)于積分的說(shuō)法正確的是（）A.\(\intkdx=kx+C(k為常數(shù))\)B.\(\intx^{n}dx=\frac{x^{n+1}}{n+1}+C(n≠-1)\)C.\(\int\sinxdx=-\cosx+C\)D.\(\int\cosxdx=\sinx+C\)答案：ABCD9.對(duì)于雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0,b>0)\)，其性質(zhì)有（）A.實(shí)軸長(zhǎng)為2aB.虛軸長(zhǎng)為2bC.漸近線方程為\(y=\pm\frac{a}x\)D.離心率\(e=\frac{c}{a}(c^{2}=a^{2}+b^{2})\)答案：ABCD10.以下屬于函數(shù)的表示方法的有（）A.解析式法B.列表法C.圖象法D.描述法答案：ABC三、判斷題（每題2分，共10題）1.函數(shù)\(y=\sqrt{x}\)的定義域是\([0,+\infty)\)。（）答案：對(duì)2.平行向量就是共線向量。（）答案：對(duì)3.二次函數(shù)\(y=ax^{2}+bx+c(a≠0)\)，當(dāng)\(a<0\)時(shí)，函數(shù)圖象開(kāi)口向下。（）答案：對(duì)4.\(\sin150^{\circ}=\frac{1}{2}\)。（）答案：對(duì)5.若數(shù)列\(zhòng)(\{a_{n}\}\)的通項(xiàng)公式\(a_{n}=2n+1\)，則該數(shù)列是等差數(shù)列。（）答案：對(duì)6.函數(shù)\(y=x^{2}\)在\((-\infty,0)\)上單調(diào)遞減。（）答案：對(duì)7.定積分的值一定是正數(shù)。（）答案：錯(cuò)8.對(duì)于橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)，離心率\(e=\frac{a}\)。（）答案：錯(cuò)9.若\(f(x)\)和\(g(x)\)都是可導(dǎo)函數(shù)，則\([f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)\)。（）答案：對(duì)10.對(duì)數(shù)函數(shù)\(y=\log_{a}x(a>0,a≠1)\)的圖象恒過(guò)點(diǎn)\((1,0)\)。（）答案：對(duì)四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=2x^{2}-3x+1\)的對(duì)稱軸。答案：對(duì)于二次函數(shù)\(y=ax^{2}+bx+c(a≠0)\)，對(duì)稱軸公式為\(x=-\frac{2a}\)，在函數(shù)\(y=2x^{2}-3x+1\)中，\(a=2\)，\(b=-3\)，所以對(duì)稱軸為\(x=\frac{3}{4}\)。2.已知向量\(\vec{a}=(2,3)\)，\(\vec=(-1,2)\)，求\(\vec{a}\cdot\vec\)。答案：若\(\vec{a}=(x_{1},y_{1})\)，\(\vec=(x_{2},y_{2})\)，則\(\vec{a}\cdot\vec=x_{1}x_{2}+y_{1}y_{2}\)，所以\(\vec{a}\cdot\vec=2\times(-1)+3\times2=4\)。3.求函數(shù)\(y=\sin2x\)的導(dǎo)數(shù)。答案：根據(jù)復(fù)合函數(shù)求導(dǎo)法則，令\(u=2x\)，則\(y=\sinu\)，\(y'_{x}=y'_{u}\cdotu'_{x}\)，\(y'_{u}=\cosu\)，\(u'_{x}=2\)，所以\(y'=\cos2x\times2=2\cos2x\)。4.計(jì)算定積分\(\int_{1}^{2}(x+1)dx\)。答案：\(\int_{1}^{2}(x+1)dx=\int_{1}^{2}xdx+\int_{1}^{2}1dx=\left[\frac{x^{2}}{2}\right]_{1}^{2}+[x]_{1}^{2}=(\frac{2^{2}}{2}-\frac{1^{2}}{2})+(2-1)=\frac{3}{2}+1=\frac{5}{2}\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x}\)在\((0,+\infty)\)和\((-\infty,0)\)上的單調(diào)性。答案：在\((0,+\infty)\)上，任取\(x_{1}<x_{2}\)，則\(y_{1}-y_{2}=\frac{1}{x_{1}}-\frac{1}{x_{2}}=\frac{x_{2}-x_{1}}{x_{1}x_{2}}>0\)，所以函數(shù)在\((0,+\infty)\)上單調(diào)遞減；在\((-\infty,0)\)上，任取\(x_{1}<x_{2}<0\)，\(y_{1}-y_{2}=\frac{1}{x_{1}}-\frac{1}{x_{2}}=\frac{x_{2}-x_{1}}{x_{1}x_{2}}>0\)，函數(shù)在\((-\infty,0)\)上單調(diào)遞減。2.闡述等差數(shù)列與等比數(shù)列的區(qū)別。答案：等差數(shù)列是后一項(xiàng)與前一項(xiàng)的差為常數(shù)，通項(xiàng)公式\(a_{n}=a_{1}+(n-1)d\)，求和公式\(S_{n}=\frac{n(a_{1}+a_{n})}{2}\)；等比數(shù)列是后一項(xiàng)與前一項(xiàng)的比為常數(shù)，通項(xiàng)公式\(a_{n}=a_{1}q^{n-1}\)，求和公式\(S_{n}=\frac{a_{1}(1-q^{n})}{1-q}(q≠1)\)。3.分析二次函數(shù)\(y=ax^{2}+bx+c(a≠0)\