吉林省高考真題及答案

一、單項(xiàng)選擇題1.已知集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|0<x<6,x\inN\}\)，則滿足\(A\subseteqC\subseteqB\)的集合\(C\)的個(gè)數(shù)為（）A.4B.8C.7D.16答案：B2.已知\(i\)為虛數(shù)單位，若復(fù)數(shù)\(z=\frac{1+ai}{2-i}(a\inR)\)的實(shí)部與虛部相等，則\(a\)的值為（）A.1B.-1C.\(\frac{1}{3}\)D.\(-\frac{1}{3}\)答案：C3.函數(shù)\(f(x)=\ln(x+1)-\frac{2}{x}\)的零點(diǎn)所在的大致區(qū)間是（）A.\((0,1)\)B.\((1,2)\)C.\((2,e)\)D.\((3,4)\)答案：B4.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(3,-2)\)，且\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow\)，則\(m\)的值為（）A.-8B.-6C.6D.8答案：D5.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_3+a_5+a_7=24\)，則\(S_9\)等于（）A.36B.72C.144D.288答案：B6.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\(\sqrt{5}\)，則\(C\)的漸近線方程為（）A.\(y=\pm\frac{1}{2}x\)B.\(y=\pm2x\)C.\(y=\pm\frac{\sqrt{6}}{6}x\)D.\(y=\pm\sqrt{6}x\)答案：B7.執(zhí)行如圖所示的程序框圖，若輸入的\(x=2\)，則輸出的\(y\)值為（）A.2B.4C.8D.16答案：C8.已知函數(shù)\(f(x)=A\sin(\omegax+\varphi)(A>0,\omega>0,|\varphi|<\frac{\pi}{2})\)的部分圖象如圖所示，則\(f(x)\)的解析式為（）A.\(f(x)=2\sin(2x+\frac{\pi}{6})\)B.\(f(x)=2\sin(2x+\frac{\pi}{3})\)C.\(f(x)=2\sin(x+\frac{\pi}{6})\)D.\(f(x)=2\sin(x+\frac{\pi}{3})\)答案：A9.已知\(a=\log_{2}0.2\)，\(b=2^{0.2}\)，\(c=0.2^{0.3}\)，則（）A.\(a<b<c\)B.\(a<c<b\)C.\(c<a<b\)D.\(b<c<a\)答案：B10.已知函數(shù)\(f(x)=\begin{cases}2^{x-1}-2,x\leqslant1\\-\log_2(x+1),x>1\end{cases}\)，且\(f(a)=-3\)，則\(f(6-a)\)等于（）A.\(-\frac{7}{4}\)B.\(-\frac{5}{4}\)C.\(-\frac{3}{4}\)D.\(-\frac{1}{4}\)答案：A二、多項(xiàng)選擇題1.下列函數(shù)中，既是偶函數(shù)又在\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^3\)B.\(y=|x|+1\)C.\(y=x^2+1\)D.\(y=-\frac{1}{x}\)答案：BC2.已知直線\(l\)過點(diǎn)\((0,5)\)，且在兩坐標(biāo)軸上的截距之和為\(2\)，則直線\(l\)的方程為（）A.\(5x-3y+15=0\)B.\(5x+3y-15=0\)C.\(x-y+5=0\)D.\(x+y-5=0\)答案：AC3.已知\(a,b,c\)為三條不重合的直線，\(\alpha,\beta,\gamma\)為三個(gè)不重合的平面，下列說法正確的是（）A.若\(a\parallelc\)，\(b\parallelc\)，則\(a\parallelb\)B.若\(a\parallel\gamma\)，\(b\parallel\gamma\)，則\(a\parallelb\)C.若\(\alpha\parallelc\)，\(\beta\parallelc\)，則\(\alpha\parallel\beta\)D.若\(\alpha\parallel\gamma\)，\(\beta\parallel\gamma\)，則\(\alpha\parallel\beta\)答案：AD4.已知\(\triangleABC\)的內(nèi)角\(A,B,C\)所對(duì)的邊分別為\(a,b,c\)，下列說法正確的是（）A.若\(\sinA>\sinB\)，則\(A>B\)B.若\(\sin2A=\sin2B\)，則\(\triangleABC\)一定為等腰三角形C.若\(\cos^2A+\cos^2B-\cos^2C>1\)，則\(\triangleABC\)為鈍角三角形D.若\(b=2\sqrt{2}\)，\(B=\frac{\pi}{4}\)，若此三角形有兩解，則\(a\)的取值范圍是\((2,2\sqrt{2})\)答案：ACD5.已知函數(shù)\(f(x)=x^3-3x^2+2\)，下列說法正確的是（）A.函數(shù)\(f(x)\)在\(x=0\)處的切線方程為\(y=2\)B.函數(shù)\(f(x)\)的單調(diào)遞減區(qū)間為\((0,2)\)C.函數(shù)\(f(x)\)的極大值為\(2\)，極小值為\(-2\)D.函數(shù)\(f(x)\)與直線\(y=a\)有三個(gè)不同交點(diǎn)時(shí)，\(a\)的取值范圍是\((-2,2)\)答案：BCD6.已知橢圓\(C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的左、右焦點(diǎn)分別為\(F_1,F_2\)，離心率為\(\frac{\sqrt{3}}{3}\)，過\(F_2\)的直線\(l\)交橢圓\(C\)于\(A,B\)兩點(diǎn)，若\(\triangleAF_1B\)的周長(zhǎng)為\(4\sqrt{3}\)，則下列說法正確的是（）A.橢圓\(C\)的方程為\(\frac{x^2}{3}+\frac{y^2}{2}=1\)B.橢圓\(C\)的焦距為\(2\)C.\(|AB|\)的最小值為\(\frac{4\sqrt{3}}{3}\)D.當(dāng)\(l\)垂直于\(x\)軸時(shí)，\(\triangleAF_1B\)的面積為\(\frac{4\sqrt{3}}{3}\)答案：ABC7.已知\(a,b\)為正實(shí)數(shù)，且\(a+b=1\)，則（）A.\(\frac{1}{a}+\frac{1}\geqslant4\)B.\(a^2+b^2\geqslant\frac{1}{2}\)C.\(\sqrt{a}+\sqrt\leqslant\sqrt{2}\)D.\(\frac{1}{a}b\leqslant\frac{1}{4}\)答案：ABC8.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geqslant0\)時(shí)，\(f(x)=x^2-2x\)，則下列說法正確的是（）A.\(f(-1)=1\)B.當(dāng)\(x<0\)時(shí)，\(f(x)=-x^2-2x\)C.函數(shù)\(f(x)\)有\(zhòng)(3\)個(gè)零點(diǎn)D.函數(shù)\(f(x)\)在\((-\infty,-1)\)上單調(diào)遞減，在\((1,+\infty)\)上單調(diào)遞增答案：BCD9.已知圓\(C:x^2+y^2-2x-4y+1=0\)，直線\(l:y=kx\)，則下列說法正確的是（）A.直線\(l\)與圓\(C\)相交B.若直線\(l\)將圓\(C\)平分，則\(k=2\)C.若直線\(l\)與圓\(C\)相切，則\(k=\frac{4\pm\sqrt{7}}{3}\)D.當(dāng)\(k=1\)時(shí)，直線\(l\)被圓\(C\)截得的弦長(zhǎng)為\(\sqrt{14}\)答案：ABD10.已知函數(shù)\(f(x)=\cos(2x+\frac{\pi}{3})+\sin^2x\)，則下列說法正確的是（）A.函數(shù)\(f(x)\)的最小正周期為\(\pi\)B.函數(shù)\(f(x)\)的圖象關(guān)于直線\(x=\frac{\pi}{3}\)對(duì)稱C.函數(shù)\(f(x)\)的單調(diào)遞減區(qū)間為\([k\pi-\frac{\pi}{12},k\pi+\frac{5\pi}{12}],k\inZ\)D.函數(shù)\(f(x)\)的值域?yàn)閈([-\frac{\sqrt{3}}{2},\frac{3}{2}]\)答案：ACD三、判斷題1.若\(a,b\inR\)，則\(a^2+b^2\geqslant2ab\)，當(dāng)且僅當(dāng)\(a=b\)時(shí)等號(hào)成立。（）答案：√2.若向量\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow=\overrightarrow{0}\)。（）答案：×3.函數(shù)\(y=\log_2(x^2-4x+3)\)的單調(diào)遞增區(qū)間是\((2,+\infty)\)。（）答案：×4.若直線\(l_1:ax+y+1=0\)與直線\(l_2:x+ay+1=0\)平行，則\(a=1\)。（）答案：×5.已知\(a,b\)為異面直線，直線\(c\)與\(a\)平行，則\(c\)與\(b\)一定異面。（）答案：×6.若\(\sin\alpha=\frac{1}{2}\)，則\(\alpha=\frac{\pi}{6}\)。（）答案：×7.等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_1+a_5=10\)，則\(S_5=25\)。（）答案：√8.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\(y=\pm\frac{a}x\)。（）答案：√9.若\(f(x)\)是定義在\(R\)上的奇函數(shù)，則\(f(0)=0\)。（）答案：√10.已知\(x,y\)滿足約束條件\(\begin{cases}x+y\geqslant2\\x-y\leqslant2\\y\leqslant2\end{cases}\)，則\(z=x+2y\)的最大值為\(8\)。（）答案：√四、簡(jiǎn)答題1.已知數(shù)列\(zhòng)(\{a_n\}\)滿足\(a_1=1\)，\(a_{n+1}=2a_n+1\)。求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式。答案：由\(a_{n+1}=2a_n+1\)，可得\(a_{n+1}+1=2(a_n+1)\)。則\(\frac{a_{n+1}+1}{a_n+1}=2\)，又\(a_1+1=2\)，所以數(shù)列\(zhòng)(\{a_n+1\}\)是以\(2\)為首項(xiàng)，\(2\)為公比的等比數(shù)列。所以\(a_n+1=2\times2^{n-1}=2^n\)，則\(a_n=2^n-1\)。2.已知函數(shù)\(f(x)=x^3-3x^2-9x+1\)，求函數(shù)\(f(x)\)的單調(diào)區(qū)間和極值。答案：對(duì)\(f(x)=x^3-3x^2-9x+1\)求導(dǎo)得\(f^\prime(x)=3x^2-6x-9=3(x^2-2x-3)=3(x-3)(x+1)\)。令\(f^\prime(x)=0\)，解得\(x=-1\)或\(x=3\)。當(dāng)\(x<-1\)或\(x>3\)時(shí)，\(f^\prime(x)>0\)，函數(shù)\(f(x)\)單調(diào)遞增；當(dāng)\(-1<x<3\)時(shí)，\(f^\pr