全國(guó)二卷理數(shù)試題及答案

一、單項(xiàng)選擇題(每題2分,共10題)1.已知集合\(A=\{1,2,3\}\),\(B=\{2,3,4\}\),則\(A\cupB=(\)\)A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1,4\}\)D.\(\{1,2,3\}\)2.復(fù)數(shù)\(z=\frac{1+i}{1-i}\)(\(i\)為虛數(shù)單位)的實(shí)部是()A.\(0\)B.\(1\)C.\(-1\)D.\(2\)3.已知向量\(\vec{a}=(1,m)\),\(\vec=(3,-2)\),且\((\vec{a}+\vec)\perp\vec\),則\(m=(\)\)A.\(-8\)B.\(-6\)C.\(6\)D.\(8\)4.函數(shù)\(f(x)=\sin(2x+\frac{\pi}{3})\)的最小正周期為()A.\(4\pi\)B.\(2\pi\)C.\(\pi\)D.\(\frac{\pi}{2}\)5.若\(\log_{2}a+\log_{2}b=6\),則\(ab=(\)\)A.\(16\)B.\(32\)C.\(64\)D.\(128\)6.直線\(y=x+1\)被圓\(x^{2}+y^{2}=2\)截得的弦長(zhǎng)為()A.\(\sqrt{2}\)B.\(2\)C.\(2\sqrt{2}\)D.\(4\)7.已知\(\tan\alpha=3\),則\(\frac{2\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha}=(\)\)A.\(\frac{3}{4}\)B.\(\frac{5}{4}\)C.\(\frac{7}{4}\)D.\(2\)8.某幾何體的三視圖如圖所示,則該幾何體的體積為()(此處省略三視圖圖形）A.\(12\)B.\(18\)C.\(24\)D.\(30\)9.已知\(a=\log_{3}2\),\(b=\ln2\),\(c=5^{-\frac{1}{2}}\),則\(a\),\(b\),\(c\)的大小關(guān)系為()A.\(c<a<b\)B.\(a<c<b\)C.\(b<c<a\)D.\(c<b<a\)10.雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0,b>0)\)的離心率為\(\sqrt{3}\),則其漸近線方程為()A.\(y=\pm\sqrt{2}x\)B.\(y=\pm\sqrt{3}x\)C.\(y=\pm\frac{\sqrt{2}}{2}x\)D.\(y=\pm\frac{\sqrt{3}}{3}x\)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.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=\cosx\)D.\(y=e^{x}-e^{-x}\)2.下列說(shuō)法正確的是()A.過(guò)一點(diǎn)有且只有一條直線與已知直線垂直B.過(guò)平面外一點(diǎn)有且只有一條直線與已知平面平行C.平行于同一條直線的兩條直線平行D.垂直于同一個(gè)平面的兩條直線平行3.已知等差數(shù)列\(zhòng)(\{a_n\}\),其前\(n\)項(xiàng)和為\(S_n\),若\(a_3=5\),\(a_5=9\),則()A.\(a_1=1\)B.\(d=2\)C.\(S_7=49\)D.\(a_{10}=19\)4.對(duì)于函數(shù)\(y=\cos(2x-\frac{\pi}{3})\),下列說(shuō)法正確的是()A.最小正周期為\(\pi\)B.圖象關(guān)于點(diǎn)\((\frac{5\pi}{12},0)\)對(duì)稱C.在\([0,\frac{\pi}{2}]\)上單調(diào)遞減D.圖象關(guān)于直線\(x=\frac{\pi}{3}\)對(duì)稱5.已知\(x\),\(y\)滿足約束條件\(\begin{cases}x+y\geq1\\x-y\leq1\\y\leq1\end{cases}\),則()A.\(z=x+2y\)的最大值為\(3\)B.\(z=x+2y\)的最小值為\(\frac{3}{2}\)C.\(z=2x-y\)的最大值為\(1\)D.\(z=2x-y\)的最小值為\(-3\)6.下列函數(shù)中,在區(qū)間\((0,+\infty)\)上單調(diào)遞增的是()A.\(y=x^{\frac{1}{2}}\)B.\(y=2^{x}\)C.\(y=\log_{2}x\)D.\(y=\frac{1}{x}\)7.已知圓\(C_1\):\((x-1)^2+(y+1)^2=1\),圓\(C_2\):\((x-4)^2+(y-5)^2=9\),則()A.兩圓的圓心距為\(5\)B.兩圓外切C.兩圓相交D.兩圓內(nèi)切8.已知\(a\),\(b\),\(c\)為三角形三邊,下列能構(gòu)成直角三角形的是()A.\(a=3\),\(b=4\),\(c=5\)B.\(a=5\),\(b=12\),\(c=13\)C.\(a=7\),\(b=24\),\(c=25\)D.\(a=8\),\(b=15\),\(c=17\)9.已知\(f(x)\)是定義在\(R\)上的偶函數(shù),且在\((-\infty,0]\)上單調(diào)遞增,則（）A.\(f(-2)>f(1)\)B.\(f(3)<f(-2)\)C.\(f(x)\)在\([0,+\infty)\)上單調(diào)遞減D.若\(f(a)<f(b)\),則\(|a|>|b|\)10.設(shè)函數(shù)\(f(x)=\begin{cases}2^{x-1},x\leq1\\\log_{2}x,x>1\end{cases}\),則（）A.\(f(0)=\frac{1}{2}\)B.\(f(2)=1\)C.\(f(f(\frac{1}{2}))=0\)D.方程\(f(x)=1\)的解為\(x=2\)三、判斷題(每題2分,共10題)1.若\(a>b\),則\(a^2>b^2\)。（)2.函數(shù)\(y=\tanx\)的定義域?yàn)閈(\{x|x\neqk\pi+\frac{\pi}{2},k\inZ\}\)。()3.拋物線\(y^2=2px(p>0)\)的焦點(diǎn)坐標(biāo)為\((\frac{p}{2},0)\)。()4.若向量\(\vec{a}\cdot\vec=0\),則\(\vec{a}=\vec{0}\)或\(\vec=\vec{0}\)。()5.數(shù)列\(zhòng)(\{a_n\}\)滿足\(a_{n+1}=2a_n\),則\(\{a_n\}\)是等比數(shù)列。()6.直線\(Ax+By+C=0\)(\(A\),\(B\)不同時(shí)為\(0\)）的斜率為\(-\frac{A}{B}\)。(）7.若\(A\),\(B\)為互斥事件,則\(P(A\cupB)=P(A)+P(B)\)。（)8.函數(shù)\(y=\sinx+\cosx\)的最大值為\(2\)。()9.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e=\frac{c}{a}\)(\(c\)為半焦距)。()10.若\(f(x)\)是周期為\(T\)的函數(shù),則\(f(x+T)=f(x)\)。（）四、簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=3\sin(2x+\frac{\pi}{6})\)的單調(diào)遞增區(qū)間。答案:令\(2k\pi-\frac{\pi}{2}\leq2x+\frac{\pi}{6}\leq2k\pi+\frac{\pi}{2},k\inZ\),解不等式得\(k\pi-\frac{\pi}{3}\leqx\leqk\pi+\frac{\pi}{6},k\inZ\),所以單調(diào)遞增區(qū)間是\([k\pi-\frac{\pi}{3},k\pi+\frac{\pi}{6}],k\inZ\)。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(a_3=5\),求其通項(xiàng)公式\(a_n\)。答案：設(shè)公差為\(d\),\(a_3=a_1+2d\),即\(5=1+2d\),解得\(d=2\)。則\(a_n=a_1+(n-1)d=1+2(n-1)=2n-1\)。3.求曲線\(y=x^3\)在點(diǎn)\((1,1)\)處的切線方程。答案:對(duì)\(y=x^3\)求導(dǎo)得\(y^\prime=3x^2\),當(dāng)\(x=1\)時(shí),\(y^\prime=3\),即切線斜率為\(3\)。由點(diǎn)斜式得切線方程為\(y-1=3(x-1)\),即\(3x-y-2=0\)。4.已知\(\sin\alpha=\frac{3}{5}\),\(\alpha\in(\frac{\pi}{2},\pi)\),求\(\cos\alpha\)和\(\tan\alpha\)的值。答案：因?yàn)閈(\sin^2\alpha+\cos^2\alpha=1\),\(\alpha\in(\frac{\pi}{2},\pi)\),所以\(\cos\alpha=-\sqrt{1-\sin^2\alpha}=-\sqrt{1-(\frac{3}{5})^2}=-\frac{4}{5}\)。\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\frac{3}{5}}{-\frac{4}{5}}=-\frac{3}{4}\)。五、討論題(每題5分,共4題)1.討論直線\(y=kx+1\)與圓\(x^{2}+y^{2}=1\)的位置關(guān)系。答案:圓\(x^{2}+y^{2}=1\)圓心\((0,0)\),半徑\(r=1\)。根據(jù)點(diǎn)到直線距離公式,圓心到直線\(y=kx+1\)(即\(kx-y+1=0\))的距離\(d=\frac{1}{\sqrt{k^{2}+1}}\)。當(dāng)\(d<r\)即\(\frac{1}{\sqrt{k^{2}+1}}<1\)(\(k\neq0\))時(shí),直線與圓相交;當(dāng)\(d=r\)即\(k=0\)時(shí),直線與圓相切;當(dāng)\(d>r\)不可能成立。2.討論函數(shù)\(f(x)=x^3-3x\)的單調(diào)性與極值。答案:對(duì)\(f(x)\)求導(dǎo)得\(f^\prime(x)=3x^2-3=3(x+1)(x-1)\)。令\(f^\prime(x)>0\),得\(x<-1\)或\(x>1\),此時(shí)\(f(x)\)單調(diào)遞增;令\(f^\prime(x)<0\),得\(-1<x<1\),\(f(x)\)單調(diào)遞減。極大值\(f(-1)=2\),極小值\(f(1)=-2\)。3.討論在立體幾何中,如何證明線面垂直。答案:可以用定義,證明直線與平面內(nèi)任意一條直線垂直;也可用判定定理,若一條直線與一個(gè)平面內(nèi)的兩條相交直線都垂直,則這條直線與該平面垂直;還可利用面面垂直的性質(zhì),若兩個(gè)平面垂直,在一個(gè)平面內(nèi)垂直于交線的直線垂直于另一個(gè)平面。4.討論在概率問(wèn)題中,如何區(qū)分互斥事件和對(duì)立事件。答案：互斥事件是指兩個(gè)事件不能同時(shí)發(fā)生,即\(A\capB=\varnothing\)。對(duì)立事件是特殊的互斥事件,除了不能同時(shí)發(fā)生外,還必有一個(gè)發(fā)生,即\(A\capB=\varnothing\)且\(A\cupB=\Omega\