全國一數(shù)學(xué)試卷及答案

單項選擇題（每題2分，共10題）1.若集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|0<x<6,x\inN\}\)，則\(A\cupB=(\)\)A.\(\{1,2\}\)B.\(\{1,2,3,4,5\}\)C.\(\{0,1,2,3,4,5\}\)D.\(\varnothing\)2.已知\(i\)為虛數(shù)單位，復(fù)數(shù)\(z\)滿足\((1+i)z=2\)，則\(z\)的虛部為\((\)\)A.\(1\)B.\(-1\)C.\(i\)D.\(-i\)3.已知\(\tan\alpha=2\)，則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}=(\)\)A.\(3\)B.\(\frac{1}{3}\)C.\(2\)D.\(\frac{1}{2}\)4.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.\(8\)5.已知\(a=\log_20.2\)，\(b=2^{0.2}\)，\(c=0.2^{0.3}\)，則\((\)\)A.\(a<b<c\)B.\(a<c<b\)C.\(c<a<b\)D.\(b<c<a\)6.已知\(F_1,F_2\)是橢圓\(C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的兩個焦點，\(P\)為\(C\)上一點，且\(\angleF_1PF_2=60^{\circ}\)，\(|PF_1|=3|PF_2|\)，則\(C\)的離心率為\((\)\)A.\(\frac{\sqrt{7}}{4}\)B.\(\frac{\sqrt{13}}{4}\)C.\(\frac{\sqrt{7}}{5}\)D.\(\frac{\sqrt{13}}{5}\)7.函數(shù)\(f(x)=\frac{\sinx+x}{\cosx+x^2}\)在\([-\pi,\pi]\)的圖象大致為\((\)\)8.已知\(a\inR\)，函數(shù)\(f(x)=ax^3-x\)，若存在\(t\inR\)，使得\(|f(t+2)-f(t)|\leq\frac{2}{3}\)，則實數(shù)\(a\)的最大值是\((\)\)A.\(2\)B.\(\frac{4}{3}\)C.\(\frac{8}{3}\)D.\(3\)9.某圓柱的高為\(2\)，底面周長為\(16\)，其三視圖如右圖。圓柱表面上的點\(M\)在正視圖上的對應(yīng)點為\(A\)，圓柱表面上的點\(N\)在左視圖上的對應(yīng)點為\(B\)，則在此圓柱側(cè)面上，從\(M\)到\(N\)的路徑中，最短路徑的長度為\((\)\)A.\(2\sqrt{17}\)B.\(2\sqrt{5}\)C.\(3\)D.\(2\)10.已知\(f(x)\)是奇函數(shù)，且當(dāng)\(x<0\)時，\(f(x)=-e^{ax}\)，若\(f(\ln2)=8\)，則\(a=(\)\)A.\(3\)B.\(-3\)C.\(\frac{1}{3}\)D.\(-\frac{1}{3}\)多項選擇題（每題2分，共10題）1.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.\(y=\log_{\frac{1}{2}}x\)D.\(y=\frac{1}{x-1}\)2.已知雙曲線\(C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的離心率為\(\sqrt{3}\)，則\(C\)的漸近線方程為\((\)\)A.\(y=\pm\sqrt{2}x\)B.\(y=\pm\frac{\sqrt{2}}{2}x\)C.\(y=\pm\sqrt{3}x\)D.\(y=\pm\frac{\sqrt{3}}{3}x\)3.對于任意實數(shù)\(a,b,c,d\)，下列命題中正確的是\((\)\)A.若\(a>b\)，\(c\neq0\)，則\(ac>bc\)B.若\(a>b\)，則\(ac^2>bc^2\)C.若\(ac^2>bc^2\)，則\(a>b\)D.若\(a>b\)，\(c>d\)，則\(a-d>b-c\)4.已知\(\alpha,\beta\)是兩個不同的平面，\(m,n\)是兩條不同的直線，則下列命題正確的是\((\)\)A.若\(m\parallel\alpha\)，\(n\parallel\alpha\)，則\(m\paralleln\)B.若\(m\subset\alpha\)，\(n\subset\alpha\)，\(m\parallel\beta\)，\(n\parallel\beta\)，則\(\alpha\parallel\beta\)C.若\(m\perp\alpha\)，\(n\parallel\alpha\)，則\(m\perpn\)D.若\(\alpha\parallel\beta\)，\(m\subset\alpha\)，則\(m\parallel\beta\)5.已知\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在區(qū)間\((-\infty,0]\)上單調(diào)遞增，若實數(shù)\(a\)滿足\(f(2^{\log_3a})>f(-\sqrt{2})\)，則\(a\)的取值范圍是\((\)\)A.\((0,\sqrt{3})\)B.\((0,3)\)C.\((\sqrt{3},+\infty)\)D.\((3,+\infty)\)6.已知函數(shù)\(f(x)=2\sin(\omegax+\varphi)(\omega>0,|\varphi|<\frac{\pi}{2})\)的圖象過點\((0,1)\)，若\(f(x)\)在區(qū)間\([0,1]\)上恰好有\(zhòng)(2\)個最值點，則\(\omega\)的取值范圍為\((\)\)A.\([\frac{7\pi}{6},\frac{13\pi}{6})\)B.\([\frac{7\pi}{6},\frac{13\pi}{6}]\)C.\([\frac{7\pi}{3},\frac{13\pi}{3})\)D.\([\frac{7\pi}{3},\frac{13\pi}{3}]\)7.已知數(shù)列\(zhòng)(\{a_n\}\)是等比數(shù)列，下列結(jié)論正確的是\((\)\)A.若\(a_1a_2>0\)，則\(a_2a_3>0\)B.若\(a_1+a_3<0\)，則\(a_2+a_4<0\)C.若\(a_1a_3<0\)，則\(a_1a_2<0\)D.若\(0<a_1<a_2\)，則\(a_1+a_3>2a_2\)8.已知圓\(C:x^2+y^2-2x-4y+1=0\)上存在兩點關(guān)于直線\(l:x+my+1=0\)對稱，則下列選項正確的是\((\)\)A.\(m=-1\)B.直線\(l\)將圓\(C\)分成的兩段弧長之比為\(1:2\)C.點\((0,1)\)到直線\(l\)的距離為\(\sqrt{2}\)D.直線\(l\)與直線\(x+y+1=0\)的夾角為\(45^{\circ}\)9.已知函數(shù)\(f(x)=\left\{\begin{array}{ll}x+1,x\leq0\\\log_2x,x>0\end{array}\right.\)，則下列說法正確的是\((\)\)A.\(f(f(\frac{1}{2}))=\frac{1}{2}\)B.函數(shù)\(y=f(x)\)的圖象與\(y=-x\)的圖象有\(zhòng)(2\)個交點C.函數(shù)\(y=f(x)\)的值域為\(R\)D.函數(shù)\(y=f(x)\)在\((-\infty,0]\)上單調(diào)遞增10.已知\(a,b\)為正實數(shù)，且\(a+b=1\)，則下列結(jié)論正確的是\((\)\)A.\(ab\leq\frac{1}{4}\)B.\(\frac{1}{a}+\frac{1}\geq4\)C.\(a^2+b^2\geq\frac{1}{2}\)D.\(\sqrt{a}+\sqrt\leq\sqrt{2}\)判斷題（每題2分，共10題）1.若\(a,b\inR\)，且\(ab>0\)，則\(\frac{a}+\frac{a}\geq2\)。（）2.函數(shù)\(y=\sin^2x\)的最小正周期是\(\pi\)。（）3.若直線\(l\)與平面\(\alpha\)內(nèi)的無數(shù)條直線垂直，則\(l\perp\alpha\)。（）4.數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項和\(S_n=n^2+1\)，則\(a_n=2n-1\)。（）5.已知向量\(\vec{a}=(1,2)\)，\(\vec=(2,3)\)，則\(\vec{a}\)與\(\vec\)的夾角為銳角。（）6.函數(shù)\(y=\log_2(x^2+1)\)的值域是\([0,+\infty)\)。（）7.橢圓\(\frac{x^2}{4}+\frac{y^2}{3}=1\)的離心率\(e=\frac{1}{2}\)。（）8.若函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上的圖象是連續(xù)不斷的，且\(f(a)f(b)<0\)，則函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)至少有一個零點。（）9.若\(a>b>0\)，\(c<d<0\)，則\(ac<bd\)。（）10.函數(shù)\(y=3\sin(2x+\frac{\pi}{3})\)的圖象可由\(y=3\sin2x\)的圖象向左平移\(\frac{\pi}{3}\)個單位得到。（）簡答題（每題5分，共4題）1.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項和為\(S_n\)，\(a_3=5\)，\(S_6=36\)，求\(\{a_n\}\)的通項公式。答案：設(shè)等差數(shù)列\(zhòng)(\{a_n\}\)公差為\(d\)，由\(a_3=5\)得\(a_1+2d=5\)；\(S_6=36\)即\(6a_1+\frac{6\times5}{2}d=36\)。聯(lián)立解得\(a_1=1\)，\(d=2\)，則\(a_n=1+2(n-1)=2n-1\)。2.已知函數(shù)\(f(x)=x^3-3x^2+2\)，求\(f(x)\)的單調(diào)區(qū)間。答案：對\(f(x)\)求導(dǎo)得\(f^\prime(x)=3x^2-6x=3x(x-2)\)。令\(f^\prime(x)>0\)，解得\(x<0\)或\(x>2\)，此時\(f(x)\)單調(diào)遞增；令\(f^\prime(x)<0\)，解得\(0<x<2\)，此時\(f(x)\)單調(diào)遞減。所以增區(qū)間為\((-\infty,0)\)和\((2,+\infty)\)，減區(qū)間為\((0,2)\)。3.已知\(\triangleABC\)中，\(a=\sqrt{3}\)，\(b=1\)，\(\angleA=60^{\circ}\)，求\(\angleB\)。答案：由正弦定理\(\frac{a}{\sinA}=\frac{\sinB}\)，可得\(\sinB=\frac{b\sinA}{a}\)。將\(a=\sqrt{3}\)，\(b=1\)，\(\angleA=60^{\circ}\)代入得\(\sinB=\frac{1\times\sin60^{\circ}}{\sqrt{3}}=\frac{1}{2}\)。因為\(a>b\)，所以\(\angleA>\angleB\)，則\(\angleB=30^{\circ}\)。4.已知圓\(C\)的圓心在直線\(y=x\)上，且過點\((1,0)\)和\((0,1)\)，求圓\(C\)的方程。答案：設(shè)圓心坐標(biāo)為\((a,a)\)，圓的標(biāo)準(zhǔn)方程為\((x-a)^2+(y-a)^2=r^2\)。將\((1,0)\)和\((0,1)\)代入得\(\begin{cases}(1-a)^2+a^2=r^2\\a^2+(1-a)^2=r^2\end{cases}\)，解得\(a=\frac{1}{2}\)，\(r^2=\frac{1}{2}\)。所以圓\(C\)的方程為\((x-\frac{1}{2})^2+(y-\frac{1}{2})^2=