高中c試題和答案

單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=2^x\)的定義域是(）A.\((0,+\infty)\)B.\([0,+\infty)\)C.\((-\infty,+\infty)\)D.\((-\infty,0)\)答案：C2.已知向量\(\vec{a}=(1,2)\),\(\vec=(3,m)\),若\(\vec{a}\parallel\vec\),則\(m\)的值為()A.6B.-6C.\(\frac{3}{2}\)D.\(-\frac{3}{2}\)答案：A3.直線\(y=x+1\)的斜率為()A.-1B.0C.1D.不存在答案：C4.若\(\sin\alpha=\frac{1}{2}\),且\(\alpha\)是第一象限角,則\(\cos\alpha\)的值為()A.\(\frac{\sqrt{3}}{2}\)B.\(-\frac{\sqrt{3}}{2}\)C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案：A5.等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(a_3=5\),則公差\(d\)為()A.1B.2C.3D.4答案：B6.函數(shù)\(f(x)=x^3-3x\)的極大值點(diǎn)是()A.-1B.1C.0D.2答案：A7.拋物線\(y^2=8x\)的焦點(diǎn)坐標(biāo)是()A.\((2,0)\)B.\((0,2)\)C.\((4,0)\)D.\((0,4)\)答案：A8.已知集合\(A=\{1,2,3\}\),\(B=\{2,3,4\}\),則\(A\capB\)等于()A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1,4\}\)D.\(\varnothing\)答案：B9.若復(fù)數(shù)\(z=1+2i\),則\(\vertz\vert\)等于()A.\(\sqrt{5}\)B.5C.\(\sqrt{3}\)D.3答案：A10.一個(gè)正方體的棱長為\(2\),則其表面積為()A.8B.12C.24D.48答案：C多項(xiàng)選擇題(每題2分,共10題)1.以下哪些是奇函數(shù)(）A.\(y=x^3\)B.\(y=\sinx\)C.\(y=e^x\)D.\(y=\lnx\)答案:AB2.橢圓\(\frac{x^2}{9}+\frac{y^2}{4}=1\)的性質(zhì)正確的有()A.焦點(diǎn)在\(x\)軸B.長軸長為\(6\)C.短軸長為\(4\)D.離心率為\(\frac{\sqrt{5}}{3}\)答案:ABCD3.下列函數(shù)中,在\((0,+\infty)\)上單調(diào)遞增的有()A.\(y=x^2\)B.\(y=\log_2x\)C.\(y=3^x\)D.\(y=\frac{1}{x}\)答案:ABC4.已知\(\vec{a}=(x_1,y_1)\),\(\vec=(x_2,y_2)\),則下列向量運(yùn)算正確的是()A.\(\vec{a}+\vec=(x_1+x_2,y_1+y_2)\)B.\(\vec{a}-\vec=(x_1-x_2,y_1-y_2)\)C.\(\lambda\vec{a}=(\lambdax_1,\lambday_1)\)(\(\lambda\)為實(shí)數(shù))D.\(\vec{a}\cdot\vec=x_1x_2+y_1y_2\)答案:ABCD5.關(guān)于直線\(Ax+By+C=0\)(\(A\)、\(B\)不同時(shí)為\(0\)),說法正確的是()A.當(dāng)\(A=0\)時(shí),直線平行于\(x\)軸B.當(dāng)\(B=0\)時(shí),直線平行于\(y\)軸C.直線的斜率為\(-\frac{A}{B}\)(\(B\neq0\))D.直線在\(y\)軸上的截距為\(-\frac{C}{B}\)(\(B\neq0\))答案:ABCD6.以下屬于等比數(shù)列的是()A.\(2,4,8,16,\cdots\)B.\(1,-1,1,-1,\cdots\)C.\(1,2,3,4,\cdots\)D.\(2,2,2,2,\cdots\)答案:ABD7.函數(shù)\(y=\cosx\)的性質(zhì)有()A.周期是\(2\pi\)B.值域是\([-1,1]\)C.是偶函數(shù)D.在\([0,\pi]\)上單調(diào)遞減答案:ABCD8.已知\(a\gt0\),\(b\gt0\),且\(a+b=1\),則下列正確的是()A.\(ab\leqslant\frac{1}{4}\)B.\(\frac{1}{a}+\frac{1}\geqslant4\)C.\(a^2+b^2\geqslant\frac{1}{2}\)D.\(\sqrt{a}+\sqrt\leqslant\sqrt{2}\)答案:ABCD9.以下哪些點(diǎn)在圓\(x^2+y^2=4\)上()A.\((0,2)\)B.\((2,0)\)C.\((\sqrt{2},\sqrt{2})\)D.\((1,\sqrt{3})\)答案:ABCD10.關(guān)于導(dǎo)數(shù)的說法正確的是()A.函數(shù)在某點(diǎn)的導(dǎo)數(shù)就是該點(diǎn)切線的斜率B.導(dǎo)數(shù)大于\(0\),函數(shù)單調(diào)遞增C.導(dǎo)數(shù)小于\(0\),函數(shù)單調(diào)遞減D.函數(shù)的極值點(diǎn)處導(dǎo)數(shù)一定為\(0\)答案:ABC判斷題(每題2分,共10題)1.空集是任何集合的子集。(√)2.若\(a\gtb\),則\(a^2\gtb^2\)。（×)3.直線\(x=1\)的斜率不存在。(√)4.對(duì)數(shù)函數(shù)\(y=\log_ax\)(\(a\gt0\)且\(a\neq1\))的定義域是\((0,+\infty)\)。(√)5.向量\(\vec{a}=(1,0)\)與\(\vec=(0,1)\)垂直。(√)6.函數(shù)\(y=\sin^2x\)的最小正周期是\(\pi\)。(√)7.等差數(shù)列的通項(xiàng)公式是\(a_n=a_1+(n-1)d\)。(√)8.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)的漸近線方程是\(y=\pm\frac{a}x\)。(√)9.若\(z_1,z_2\)是復(fù)數(shù),且\(\vertz_1\vert=\vertz_2\vert\),則\(z_1=z_2\)。（×)10.正方體的體對(duì)角線長為棱長的\(\sqrt{3}\)倍。(√）簡答題(每題5分,共4題)1.求函數(shù)\(y=x^2-2x+3\)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)\(a=1\),\(b=-2\),對(duì)稱軸\(x=1\)。把\(x=1\)代入函數(shù)得\(y=2\),頂點(diǎn)坐標(biāo)為\((1,2)\)。2.已知\(\sin\alpha=\frac{3}{5}\),\(\alpha\)是第二象限角,求\(\tan\alpha\)的值。答案：因?yàn)閈(\sin\alpha=\frac{3}{5}\),\(\alpha\)是第二象限角,根據(jù)\(\sin^2\alpha+\cos^2\alpha=1\),可得\(\cos\alpha=-\sqrt{1-(\frac{3}{5})^2}=-\frac{4}{5}\),則\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=-\frac{3}{4}\)。3.求過點(diǎn)\((1,2)\)且斜率為\(3\)的直線方程。答案:由直線的點(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\(y-2=3(x-1)\),即\(y=3x-1\)。4.計(jì)算\(\int_{0}^{1}(x^2+1)dx\)。答案:根據(jù)積分公式\(\intx^ndx=\frac{1}{n+1}x^{n+1}+C\)(\(n\neq-1\)),\(\int_{0}^{1}(x^2+1)dx=(\frac{1}{3}x^3+x)\big|_{0}^{1}=(\frac{1}{3}\times1^3+1)-(\frac{1}{3}\times0^3+0)=\frac{4}{3}\)。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,并說明理由。答案：在\((0,+\infty)\)上,任取\(x_1\ltx_2\),\(f(x_1)-f(x_2)=\frac{1}{x_1}-\frac{1}{x_2}=\frac{x_2-x_1}{x_1x_2}\gt0\),所以\(y=\frac{1}{x}\)在\((0,+\infty)\)上單調(diào)遞減；同理在\((-\infty,0)\)上也單調(diào)遞減。2.探討等差數(shù)列和等比數(shù)列在實(shí)際生活中的應(yīng)用實(shí)例。答案:等差數(shù)列如每月等額還款的房貸,每月還款額構(gòu)成等差數(shù)列;等比數(shù)列如細(xì)胞分裂,每經(jīng)過一定時(shí)間細(xì)胞數(shù)量按等比數(shù)列增長。這些實(shí)例體現(xiàn)了數(shù)列在經(jīng)濟(jì)、生物等方面的應(yīng)用。3.分析直線與圓的位置關(guān)系有哪些判斷方法。答案:一是幾何法,通過比