安豐高中數(shù)學(xué)考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)的定義域是\([1,3]\)，則函數(shù)\(y=f(2x-1)\)的定義域是（）A.\([1,3]\)B.\([1,2]\)C.\([2,3]\)D.\([0,1]\)答案：B2.已知\(a=\log_{2}3\)，\(b=\log_{3}2\)，\(c=\log_{2}\frac{1}{3}\)，則（）A.\(a>b>c\)B.\(b>a>c\)C.\(a>c>b\)D.\(c>a>b\)答案：A3.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(\frac{2\pi}{3}\)答案：A4.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{3}=9\)，\(a_{5}=81\)，則公比\(q\)為（）A.\(3\)B.\(-3\)C.\(\pm3\)D.\(9\)答案：C5.向量\(\vec{a}=(1,2)\)，\(\vec=(x,1)\)，若\(\vec{a}\perp\vec\)，則\(x\)的值為（）A.-2B.2C.\(-\frac{1}{2}\)D.\(\frac{1}{2}\)答案：A6.在\(\triangleABC\)中，\(A=60^{\circ}\)，\(a=\sqrt{3}\)，\(b=1\)，則\(B=\)（）A.\(30^{\circ}\)B.\(45^{\circ}\)C.\(60^{\circ}\)D.\(90^{\circ}\)答案：A7.函數(shù)\(y=x^{2}+2x-3\)在\(x=1\)處的導(dǎo)數(shù)是（）A.\(4\)B.\(3\)C.\(2\)D.\(1\)答案：A8.雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0,b>0)\)的漸近線方程為\(y=\pm\frac{3}{4}x\)，則離心率\(e=\)（）A.\(\frac{5}{4}\)B.\(\frac{5}{3}\)C.\(\frac{4}{3}\)D.\(\frac{3}{4}\)答案：A9.若\(x\in(0,\frac{\pi}{2})\)，則\(y=\sinx+\cosx\)的值域是（）A.\((-1,1)\)B.\((1,\sqrt{2}]\)C.\([-\sqrt{2},\sqrt{2}]\)D.\((-\sqrt{2},-1)\)答案：B10.已知直線\(l_{1}:y=kx+1\)與直線\(l_{2}:y=3x-1\)平行，則\(k=\)（）A.\(3\)B.\(-3\)C.\(\frac{1}{3}\)D.\(-\frac{1}{3}\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中是奇函數(shù)的有（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\frac{1}{x}\)答案：ABD2.下列命題正確的是（）A.若\(a>b\)，則\(ac^{2}>bc^{2}\)B.若\(a>b\)，則\(\frac{1}{a}<\frac{1}\)C.若\(a>b\)，\(c>d\)，則\(a+c>b+d\)D.若\(a>b>0\)，\(c>d>0\)，則\(ac>bd\)答案：CD3.已知向量\(\vec{a}=(1,-1)\)，\(\vec=(2,m)\)，若\(\vec{a}\parallel\vec\)，則\(m\)的值可以是（）A.-2B.\(2\)C.\(-1\)D.\(1\)答案：A4.在等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(d=2\)，則（）A.\(a_{n}=2n-1\)B.\(S_{n}=n^{2}\)C.\(a_{5}=9\)D.\(S_{5}=25\)答案：ABCD5.函數(shù)\(y=\cos(x+\frac{\pi}{3})\)的圖象的對(duì)稱軸方程可以是（）A.\(x=\frac{\pi}{6}\)B.\(x=\frac{\pi}{3}\)C.\(x=-\frac{\pi}{6}\)D.\(x=-\frac{\pi}{3}\)答案：AC6.若圓\(C:x^{2}+y^{2}-2x=0\)，則（）A.圓心坐標(biāo)為\((1,0)\)B.半徑\(r=1\)C.過原點(diǎn)\((0,0)\)D.與\(y\)軸相切答案：ABCD7.對(duì)于二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)，下列說法正確的是（）A.當(dāng)\(a>0\)時(shí)，函數(shù)圖象開口向上B.對(duì)稱軸方程為\(x=-\frac{2a}\)C.當(dāng)\(\Delta=b^{2}-4ac=0\)時(shí)，函數(shù)圖象與\(x\)軸有一個(gè)交點(diǎn)D.當(dāng)\(c=0\)時(shí)，函數(shù)圖象過原點(diǎn)答案：ABCD8.下列關(guān)于指數(shù)函數(shù)\(y=a^{x}(a>0,a\neq1)\)的說法正確的是（）A.當(dāng)\(a>1\)時(shí)，函數(shù)在\(R\)上單調(diào)遞增B.當(dāng)\(0<a<1\)時(shí)，函數(shù)在\(R\)上單調(diào)遞減C.函數(shù)圖象恒過點(diǎn)\((0,1)\)D.函數(shù)的值域是\((0,+\infty)\)答案：ABCD9.在\(\triangleABC\)中，\(\sinA=\frac{3}{5}\)，\(\cosB=\frac{5}{13}\)，則（）A.\(\cosA=\pm\frac{4}{5}\)B.\(\sinB=\frac{12}{13}\)C.\(\sinC=\frac{63}{65}\)或\(\sinC=\frac{33}{65}\)D.\(\cosC=\frac{16}{65}\)或\(\cosC=-\frac{56}{65}\)答案：ABC10.若\(f(x)=\ln(x^{2}+1)\)，則（）A.\(f(x)\)是偶函數(shù)B.\(f(x)\)在\((0,+\infty)\)上單調(diào)遞增C.\(f(x)\)的值域是\([0,+\infty)\)D.\(f(x)\)的圖象關(guān)于\(y\)軸對(duì)稱答案：ABD三、判斷題（每題2分，共10題）1.空集是任何集合的子集。（）答案：對(duì)2.若直線\(l_{1}:y=k_{1}x+b_{1}\)與直線\(l_{2}:y=k_{2}x+b_{2}\)垂直，則\(k_{1}k_{2}=-1\)。（）答案：錯(cuò)3.函數(shù)\(y=\sqrt{x^{2}}\)與\(y=x\)是同一個(gè)函數(shù)。（）答案：錯(cuò)4.等比數(shù)列的公比不能為\(0\)。（）答案：對(duì)5.向量\(\vec{a}-\vec\)的模長等于\(\vert\vec{a}\vert-\vert\vec\vert\)。（）答案：錯(cuò)6.在\(\triangleABC\)中，\(A+B+C=\pi\)。（）答案：對(duì)7.函數(shù)\(y=\frac{1}{x}\)在\((-\infty,0)\)上單調(diào)遞減。（）答案：對(duì)8.對(duì)于函數(shù)\(y=f(x)\)，若\(f(a)=f(b)\)，則\(a=b\)。（）答案：錯(cuò)9.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e\in(0,1)\)。（）答案：對(duì)10.對(duì)數(shù)函數(shù)\(y=\log_{a}x(a>0,a\neq1)\)的定義域是\((0,+\infty)\)。（）答案：對(duì)四、簡答題（每題5分，共4題）1.求函數(shù)\(y=\frac{1}{x-1}\)的定義域。答案：要使函數(shù)\(y=\frac{1}{x-1}\)有意義，則\(x-1\neq0\)，即\(x\neq1\)，所以定義域?yàn)閈(\{x|x\neq1\}\)。2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=2\)，\(d=3\)，求\(a_{10}\)。答案：根據(jù)等差數(shù)列通項(xiàng)公式\(a_{n}=a_{1}+(n-1)d\)，\(n=10\)時(shí)，\(a_{10}=2+(10-1)\times3=2+27=29\)。3.求過點(diǎn)\((1,2)\)且斜率為\(3\)的直線方程。答案：根據(jù)直線的點(diǎn)斜式方程\(y-y_{1}=k(x-x_{1})\)，這里\(x_{1}=1\)，\(y_{1}=2\)，\(k=3\)，則直線方程為\(y-2=3(x-1)\)，即\(y=3x-1\)。4.計(jì)算\(\sin^{2}30^{\circ}+\cos^{2}30^{\circ}\)的值。答案：根據(jù)三角函數(shù)的平方關(guān)系\(\sin^{2}\alpha+\cos^{2}\alpha=1\)，當(dāng)\(\alpha=30^{\circ}\)時(shí)，\(\sin^{2}30^{\circ}+\cos^{2}30^{\circ}=1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=x^{3}-3x\)的單調(diào)性。答案：對(duì)\(y=x^{3}-3x\)求導(dǎo)得\(y'=3x^{2}-3=3(x+1)(x-1)\)。當(dāng)\(x<-1\)或\(x>1\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(-1<x<1\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減。2.若\(\triangleABC\)的三邊\(a\)，\(b\)，\(c\)滿足\(a^{2}+b^{2}=c^{2}\)，討論\(\triangleABC\)的角\(C\)的大小。答案：根據(jù)余弦定理\(\cosC=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)，因?yàn)閈(a^{2}+b^{2}=c^{2}\)，所以\(\cosC=0\)，又因?yàn)閈(0^{\circ}<C<180^{\circ}\)，所以\(C=90^{\circ}\)。3.討論指數(shù)函數(shù)\(y=2^{x}\)與對(duì)數(shù)函數(shù)\(y=\log_{2}x\)的圖象關(guān)系。答案：指數(shù)函數(shù)\(y=2^{x}\)與對(duì)數(shù)函數(shù)\(y=\log_{2}x\)的圖象關(guān)于直線\(y=x\)對(duì)稱。4.討論向量\(\vec{a}=(1,0)\)與向量\(\vec=(0