全國(guó)1卷數(shù)學(xué)試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.已知集合\(A=\{1,2,3\}\)，\(B=\{x|x^2-3x+2=0\}\)，則\(A\capB=(\)\)A.\(\{1\}\)B.\(\{2\}\)C.\(\{1,2\}\)D.\(\{1,2,3\}\)2.已知復(fù)數(shù)\(z=\frac{2+i}{1-i}\)，則\(z\)的虛部為（）A.\(\frac{3}{2}\)B.\(\frac{3}{2}i\)C.\(\frac{1}{2}\)D.\(\frac{1}{2}i\)3.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(3,-2)\)，且\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow\)，則\(m=(\)\)A.-8B.-6C.6D.84.函數(shù)\(f(x)=\log_2(x^2+2x-3)\)的定義域是（）A.\((-3,1)\)B.\((-\infty,-3)\cup(1,+\infty)\)C.\([-3,1]\)D.\((-\infty,-3]\cup[1,+\infty)\)5.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})=(\)\)A.\(\frac{1}{7}\)B.7C.-\frac{1}{7}\)D.-76.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_3=5\)，\(S_3=12\)，則\(a_{10}=(\)\)A.18B.28C.32D.357.已知拋物線\(y^2=2px(p\gt0)\)的焦點(diǎn)為\(F\)，點(diǎn)\(M\)在拋物線上，且\(|MF|=3\)，若點(diǎn)\(M\)的橫坐標(biāo)為\(2\)，則\(p=(\)\)A.1B.2C.3D.48.已知函數(shù)\(f(x)=2\sin(\omegax+\varphi)(\omega\gt0,|\varphi|\lt\frac{\pi}{2})\)的部分圖象如圖所示，則\(\omega\)，\(\varphi\)的值分別為（）A.\(\omega=2\)，\(\varphi=\frac{\pi}{6}\)B.\(\omega=2\)，\(\varphi=-\frac{\pi}{6}\)C.\(\omega=\frac{1}{2}\)，\(\varphi=\frac{\pi}{6}\)D.\(\omega=\frac{1}{2}\)，\(\varphi=-\frac{\pi}{6}\)9.已知正方體\(ABCD-A_1B_1C_1D_1\)的棱長(zhǎng)為\(1\)，則點(diǎn)\(A\)到平面\(A_1BD\)的距離為（）A.\(\frac{\sqrt{3}}{3}\)B.\(\frac{\sqrt{6}}{3}\)C.\(\frac{\sqrt{2}}{2}\)D.\(\frac{1}{2}\)10.已知函數(shù)\(f(x)=\begin{cases}2^x-1,x\leq0\\x^{\frac{1}{2}},x\gt0\end{cases}\)，若\(f(a)=1\)，則\(a=(\)\)A.0B.1C.0或1D.1或2二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^2\)B.\(y=\sqrt{x}\)C.\(y=\log_2x\)D.\(y=\frac{1}{x}\)2.已知直線\(l_1\)：\(ax+2y+6=0\)，\(l_2\)：\(x+(a-1)y+a^2-1=0\)，若\(l_1\parallell_2\)，則\(a\)的值可能為（）A.-1B.2C.1D.03.以下關(guān)于向量的說法正確的是（）A.若\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow=\overrightarrow{0}\)B.\(|\overrightarrow{a}+\overrightarrow|\leq|\overrightarrow{a}|+|\overrightarrow|\)C.若\(\overrightarrow{a}\parallel\overrightarrow\)，則存在實(shí)數(shù)\(\lambda\)，使得\(\overrightarrow=\lambda\overrightarrow{a}\)D.已知\(\overrightarrow{a}=(x_1,y_1)\)，\(\overrightarrow=(x_2,y_2)\)，若\(\overrightarrow{a}\perp\overrightarrow\)，則\(x_1x_2+y_1y_2=0\)4.已知\(a\)，\(b\)，\(c\)滿足\(c\ltb\lta\)，且\(ac\lt0\)，則下列不等式一定成立的是（）A.\(ab\gtac\)B.\(c(b-a)\gt0\)C.\(cb^2\ltab^2\)D.\(ac(a-c)\lt0\)5.已知函數(shù)\(f(x)=\sin(2x+\varphi)\)，\((|\varphi|\lt\frac{\pi}{2})\)，則以下說法正確的是（）A.當(dāng)\(\varphi=\frac{\pi}{6}\)時(shí)，\(f(x)\)在\([0,\frac{\pi}{3}]\)上單調(diào)遞增B.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{8},0)\)對(duì)稱時(shí)，\(\varphi=-\frac{\pi}{4}\)C.\(f(x)\)的圖象的一條對(duì)稱軸方程為\(x=\frac{\pi}{2}\)時(shí)，\(\varphi=\frac{\pi}{2}\)D.\(f(x)\)在\([0,\pi]\)上的零點(diǎn)個(gè)數(shù)可能為\(2\)個(gè)6.已知橢圓\(C\)：\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a\gtb\gt0)\)的左右焦點(diǎn)分別為\(F_1,F_2\)，離心率為\(e\)，過\(F_2\)的直線與橢圓\(C\)相交于\(A\)，\(B\)兩點(diǎn)，則（）A.當(dāng)\(e=\frac{\sqrt{2}}{2}\)時(shí)，\(\triangleABF_1\)可能為直角三角形B.若\(\overrightarrow{AF_2}=3\overrightarrow{F_2B}\)，則橢圓的離心率\(e\in[\frac{\sqrt{2}}{2},1)\)C.\(\triangleABF_1\)的周長(zhǎng)為\(4a\)D.若\(AB\)的最小值為\(3b\)，則橢圓的離心率\(e=\frac{1}{2}\)7.已知函數(shù)\(f(x)=x^3-3x^2+2\)，則（）A.\(f(x)\)有兩個(gè)極值點(diǎn)B.\(f(x)\)在\(x=1\)處取得極大值C.\(f(x)\)在\(x=2\)處取得極小值D.\(f(x)\)的圖象關(guān)于點(diǎn)\((1,0)\)對(duì)稱8.已知\(a\gt0\)，\(b\gt0\)，且\(a+b=1\)，則（）A.\(a^2+b^2\geq\frac{1}{2}\)B.\(2^{a-b}\gt\frac{1}{2}\)C.\(\log_2a+\log_2b\geq-2\)D.\(\sqrt{a}+\sqrt\leq\sqrt{2}\)9.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，且滿足\(S_n=2a_n-1\)，則（）A.\(a_1=1\)B.\(a_n=2^{n-1}\)C.數(shù)列\(zhòng)(\{a_n\}\)是等比數(shù)列D.\(S_n=2^n-1\)10.已知函數(shù)\(y=f(x)\)的圖象關(guān)于直線\(x=1\)對(duì)稱，且在\((1,+\infty)\)上單調(diào)遞增，則（）A.\(f(0)\ltf(\frac{3}{2})\)B.\(f(-1)\gtf(3)\)C.\(f(-2)=f(4)\)D.\(f(2)\ltf(3)\)三、判斷題（每題2分，共10題）1.若\(a\gtb\)，則\(a^2\gtb^2\)。（）2.函數(shù)\(y=\sinx\)的最小正周期是\(2\pi\)。（）3.直線\(y=x+1\)與圓\(x^2+y^2=1\)相切。（）4.若向量\(\overrightarrow{a}\)與向量\(\overrightarrow\)的夾角為\(60^{\circ}\)，\(|\overrightarrow{a}|=2\)，\(|\overrightarrow|=1\)，則\(\overrightarrow{a}\cdot\overrightarrow=1\)。（）5.函數(shù)\(y=\log_2x\)在定義域上是單調(diào)遞減函數(shù)。（）6.等差數(shù)列\(zhòng)(\{a_n\}\)中，若\(m+n=p+q\)，則\(a_m+a_n=a_p+a_q\)。（）7.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a\gt0,b\gt0)\)的漸近線方程是\(y=\pm\frac{a}x\)。（）8.若\(x\gt0\)，\(y\gt0\)，且\(x+y=1\)，則\(xy\)的最大值是\(\frac{1}{4}\)。（）9.函數(shù)\(f(x)=x^3\)是奇函數(shù)。（）10.已知直線\(l\)的斜率\(k=\tan\alpha\)，則直線\(l\)的傾斜角為\(\alpha\)。（）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=\sin^2x+\cosx+1\)，\(x\in[-\frac{\pi}{3},\frac{2\pi}{3}]\)的值域。答案：\(y=1-\cos^2x+\cosx+1=-(\cosx-\frac{1}{2})^2+\frac{9}{4}\)。\(x\in[-\frac{\pi}{3},\frac{2\pi}{3}]\)，則\(\cosx\in[-\frac{1}{2},1]\)。當(dāng)\(\cosx=\frac{1}{2}\)時(shí)，\(y_{max}=\frac{9}{4}\)；當(dāng)\(\cosx=-\frac{1}{2}\)時(shí)，\(y_{min}=\frac{5}{4}\)，值域?yàn)閈([\frac{5}{4},\frac{9}{4}]\)。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_3=5\)，\(S_6=36\)，求\(a_n\)的通項(xiàng)公式。答案：設(shè)等差數(shù)列公差為\(d\)，由\(a_3=a_1+2d=5\)，\(S_6=6a_1+\frac{6×5}{2}d=36\)，即\(6a_1+15d=36\)。聯(lián)立解得\(a_1=1\)，\(d=2\)，則\(a_n=1+(n-1)×2=2n-1\)。3.已知圓\(C\)的圓心在直線\(x-2y=0\)上，且與\(y\)軸相切于點(diǎn)\((0,1)\)，求圓\(C\)的方程。答案：設(shè)圓心坐標(biāo)\((a,b)\)，因?yàn)閳A心在\(x-2y=0\)上，所以\(a=2b\)。又圓與\(y\)軸相切于\((0,1)\)，則圓心縱坐標(biāo)\(b=1\)，\(a=2\)，半徑\(r=2\)。圓\(C\)方程為\((x-2)^2+(y-1)^2=4\)。4.已知函數(shù)\(f(x)=x^3-3x\)，求\(f(x)\)的單調(diào)區(qū)間。答案：對(duì)\(f(x)\)求導(dǎo)得\(f^\pri