煙臺(tái)高中期末試題及答案

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.\(\pi\)B.\(2\pi\)C.\(3\pi\)D.\(4\pi\)2.直線\(y=2x+1\)的斜率是()A.1B.2C.3D.43.集合\(A=\{1,2,3\}\),集合\(B=\{2,3,4\}\),則\(A\capB\)=()A.\(\{1,2\}\)B.\(\{2,3\}\)C.\(\{3,4\}\)D.\(\{1,4\}\)4.復(fù)數(shù)\(z=3+4i\)的模是()A.5B.7C.\(\sqrt{7}\)D.\(\sqrt{5}\)5.等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(d=2\),則\(a_3\)=()A.3B.5C.7D.96.若\(\cos\alpha=\frac{1}{2}\),且\(0<\alpha<\frac{\pi}{2}\),則\(\sin\alpha\)=()A.\(\frac{1}{2}\)B.\(\frac{\sqrt{3}}{2}\)C.\(-\frac{1}{2}\)D.\(-\frac{\sqrt{3}}{2}\)7.拋物線\(y^2=4x\)的焦點(diǎn)坐標(biāo)是()A.\((1,0)\)B.\((0,1)\)C.\((2,0)\)D.\((0,2)\)8.已知向量\(\overrightarrow{a}=(1,2)\),\(\overrightarrow=(2,m)\),若\(\overrightarrow{a}\parallel\overrightarrow\),則\(m\)=()A.4B.3C.2D.19.函數(shù)\(f(x)=x^3-3x\)的極大值點(diǎn)是()A.-1B.1C.0D.210.直線\(x+y-1=0\)與圓\(x^2+y^2=1\)的位置關(guān)系是()A.相交B.相切C.相離D.不確定二、多項(xiàng)選擇題(每題2分,共10題)1.以下哪些是奇函數(shù)(）A.\(y=x\)B.\(y=x^2\)C.\(y=\sinx\)D.\(y=\cosx\)2.一個(gè)正方體的棱長為\(a\),以下說法正確的是()A.表面積為\(6a^2\)B.體積為\(a^3\)C.體對(duì)角線長為\(\sqrt{3}a\)D.面對(duì)角線長為\(\sqrt{2}a\)3.以下哪些是等比數(shù)列()A.\(1,2,4,8\cdots\)B.\(1,-1,1,-1\cdots\)C.\(1,1,1,1\cdots\)D.\(2,4,6,8\cdots\)4.已知直線\(l_1:y=k_1x+b_1\),\(l_2:y=k_2x+b_2\),若\(l_1\perpl_2\),則()A.\(k_1k_2=-1\)B.\(k_1=k_2\)C.\(k_1+k_2=0\)D.當(dāng)\(k_1\)存在時(shí),\(k_2=-\frac{1}{k_1}\)5.以下哪些點(diǎn)在橢圓\(\frac{x^2}{16}+\frac{y^2}{9}=1\)上(）A.\((4,0)\)B.\((0,3)\)C.\((2,\frac{3\sqrt{3}}{2})\)D.\((-4,0)\)6.關(guān)于函數(shù)\(y=\tanx\),說法正確的是()A.定義域是\(\{x|x\neqk\pi+\frac{\pi}{2},k\inZ\}\)B.周期是\(\pi\)C.是奇函數(shù)D.在\((-\frac{\pi}{2},\frac{\pi}{2})\)上單調(diào)遞增7.以下哪些運(yùn)算正確()A.\((a^m)^n=a^{mn}\)B.\(a^m\cdota^n=a^{m+n}\)C.\(\log_a(MN)=\log_aM+\log_aN\)(\(a>0,a\neq1,M>0,N>0\))D.\(\log_a\frac{M}{N}=\log_aM-\log_aN\)(\(a>0,a\neq1,M>0,N>0\))8.設(shè)\(a,b,c\)為實(shí)數(shù),且\(a>b\),則以下不等式一定成立的是（）A.\(a+c>b+c\)B.\(ac>bc\)C.\(a^2>b^2\)D.\(\frac{1}{a}<\frac{1}\)9.已知函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上連續(xù),以下說法正確的是()A.若\(f(a)\cdotf(b)<0\),則\(f(x)\)在\((a,b)\)內(nèi)至少有一個(gè)零點(diǎn)B.\(f(x)\)在\([a,b]\)上一定有最大值和最小值C.若\(f(x)\)在\([a,b]\)上單調(diào)遞增,則\(f(a)\)是最小值,\(f(b)\)是最大值D.\(f(x)\)的圖象在\([a,b]\)上是一條連續(xù)不斷的曲線10.以下哪些是基本不等式\(a+b\geq2\sqrt{ab}\)(\(a>0,b>0\))成立的條件()A.\(a,b\)為正實(shí)數(shù)B.當(dāng)且僅當(dāng)\(a=b\)時(shí)取等號(hào)C.\(a,b\)可以為任意實(shí)數(shù)D.\(a\geq0,b\geq0\)三、判斷題(每題2分,共10題)1.空集是任何集合的子集。()2.函數(shù)\(y=\frac{1}{x}\)在定義域內(nèi)是單調(diào)遞減函數(shù)。()3.向量\(\overrightarrow{a}\cdot\overrightarrow=|\overrightarrow{a}|\times|\overrightarrow|\times\cos\theta\),其中\(zhòng)(\theta\)為\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角。(）4.若直線\(l\)垂直于平面\(\alpha\)內(nèi)的兩條直線,則\(l\perp\alpha\)。（)5.指數(shù)函數(shù)\(y=a^x\)(\(a>0,a\neq1\))恒過點(diǎn)\((0,1)\)。()6.等差數(shù)列的前\(n\)項(xiàng)和公式為\(S_n=\frac{n(a_1+a_n)}{2}\)。()7.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)(\(a>0,b>0\))的漸近線方程是\(y=\pm\frac{a}x\)。()8.若\(z=a+bi\)（\(a,b\inR\)),則\(\overline{z}=a-bi\)。（)9.函數(shù)\(y=\sin^2x\)的最小正周期是\(\pi\)。()10.若\(a>b\),\(c>d\),則\(a-c>b-d\)。（）四、簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=2x^2-4x+3\)的對(duì)稱軸和頂點(diǎn)坐標(biāo)。答案:對(duì)于二次函數(shù)\(y=ax^2+bx+c\),對(duì)稱軸\(x=-\frac{2a}\)。此函數(shù)\(a=2\),\(b=-4\),對(duì)稱軸\(x=1\)。把\(x=1\)代入函數(shù)得\(y=1\),頂點(diǎn)坐標(biāo)為\((1,1)\)。2.已知\(\sin\alpha=\frac{3}{5}\),\(\alpha\)是第二象限角,求\(\cos\alpha\)和\(\tan\alpha\)的值。答案：因?yàn)閈(\sin^2\alpha+\cos^2\alpha=1\),\(\sin\alpha=\frac{3}{5}\),\(\alpha\)是第二象限角,所以\(\cos\alpha=-\sqrt{1-(\frac{3}{5})^2}=-\frac{4}{5}\),\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=-\frac{3}{4}\)。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\(y-y_0=k(x-x_0)\)(其中\(zhòng)((x_0,y_0)\)為直線上一點(diǎn),\(k\)為斜率)。已知點(diǎn)\((1,2)\),斜率\(k=3\),則直線方程為\(y-2=3(x-1)\),整理得\(3x-y-1=0\)。4.求數(shù)列\(zhòng)(1,3,5,7,\cdots\)的前\(n\)項(xiàng)和\(S_n\)。答案:該數(shù)列是首項(xiàng)\(a_1=1\),公差\(d=2\)的等差數(shù)列。由等差數(shù)列前\(n\)項(xiàng)和公式\(S_n=na_1+\frac{n(n-1)}{2}d\),代入得\(S_n=n\times1+\frac{n(n-1)}{2}\times2=n^2\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=x^3\)的單調(diào)性和奇偶性。答案:對(duì)\(y=x^3\)求導(dǎo)得\(y^\prime=3x^2\geq0\),\(x\neq0\)時(shí)\(y^\prime>0\),所以在\(R\)上單調(diào)遞增。又\(f(-x)=(-x)^3=-x^3=-f(x)\),所以是奇函數(shù)。2.討論直線與圓的位置關(guān)系有哪些判斷方法。答案:一是幾何法,比較圓心到直線的距離\(d\)與圓半徑\(r\)的大小,\(d>r\)相離,\(d=r\)相切,\(d<r\)相交;二是代數(shù)法,聯(lián)立直線與圓的方程,根據(jù)判別式\(\Delta\)判斷,\(\Delta>0\)相交,\(\Delta=0\)相切,\(\Delta<0\)相離。3.討論在實(shí)際生活中,哪些地方會(huì)用到等差數(shù)列或等比數(shù)列的知識(shí)。答案：如銀行存款利息計(jì)算可能用到等比數(shù)列；按一定規(guī)律增加或減少數(shù)量的問題,像每月工資固定漲幅等可用等差數(shù)列。建筑樓梯臺(tái)階高度變化符合等差數(shù)列規(guī)律,細(xì)胞分裂數(shù)量變化符合等比數(shù)列規(guī)律。4.討論如何提高數(shù)學(xué)解題能力。答案:要扎實(shí)掌握基礎(chǔ)知識(shí),理解概念定理。多做不同類型題目,總結(jié)解題方法和技巧。建立錯(cuò)題本,分析錯(cuò)誤原因。學(xué)會(huì)舉一反三,遇到難題嘗試