高中數(shù)學(xué)收納試卷及答案

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.函數(shù)\(y=\log_2(x^2-4x+3)\)的單調(diào)遞減區(qū)間為（）A.\((2,+\infty)\)B.\((3,+\infty)\)C.\((-\infty,2)\)D.\((-\infty,1)\)答案：D3.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(m,-1)\)，若\(\overrightarrow{a}\parallel(\overrightarrow{a}+\overrightarrow)\)，則\(m=\)（）A.\(-\frac{1}{2}\)B.\(\frac{1}{2}\)C.2D.-2答案：A4.若\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})=\)（）A.\(\frac{1}{7}\)B.7C.\(-\frac{1}{7}\)D.-7答案：C5.設(shè)\(a=\log_3\pi\)，\(b=\log_2\sqrt{3}\)，\(c=\log_3\sqrt{2}\)，則（）A.\(a>b>c\)B.\(a>c>b\)C.\(b>a>c\)D.\(b>c>a\)答案：A6.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(S_{10}=120\)，則\(a_2+a_9=\)（）A.12B.24C.36D.48答案：B7.雙曲線\(\frac{x^2}{4}-\frac{y^2}{12}=1\)的焦點(diǎn)到漸近線的距離為（）A.\(2\sqrt{3}\)B.2C.\(\sqrt{3}\)D.1答案：A8.已知函數(shù)\(f(x)=\sin(\omegax+\varphi)(\omega>0,|\varphi|<\frac{\pi}{2})\)的最小正周期為\(\pi\)，將其圖象向右平移\(\frac{\pi}{6}\)個(gè)單位后得到的圖象關(guān)于原點(diǎn)對(duì)稱(chēng)，則函數(shù)\(f(x)\)的圖象（）A.關(guān)于直線\(x=\frac{\pi}{12}\)對(duì)稱(chēng)B.關(guān)于直線\(x=\frac{5\pi}{12}\)對(duì)稱(chēng)C.關(guān)于點(diǎn)\((\frac{\pi}{12},0)\)對(duì)稱(chēng)D.關(guān)于點(diǎn)\((\frac{5\pi}{12},0)\)對(duì)稱(chēng)答案：B9.一個(gè)幾何體的三視圖如圖所示，則該幾何體的體積為（）A.\(\frac{1}{3}\)B.\(\frac{2}{3}\)C.1D.\(\frac{4}{3}\)答案：A10.已知\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\([0,+\infty)\)上單調(diào)遞增，則滿足\(f(x-1)<f(3)\)的\(x\)的取值范圍是（）A.\((-2,4)\)B.\((-3,3)\)C.\((-3,5)\)D.\((-1,3)\)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\(a,b\inR\)，且\(ab>0\)，則下列不等式中恒成立的是（）A.\(a^2+b^2\geq2ab\)B.\(a+b\geq2\sqrt{ab}\)C.\(\frac{1}{a}+\frac{1}\geq\frac{2}{\sqrt{ab}}\)D.\(\frac{a}+\frac{a}\geq2\)答案：AD3.已知\(\overrightarrow{a},\overrightarrow\)是兩個(gè)非零向量，下列說(shuō)法正確的是（）A.若\(|\overrightarrow{a}+\overrightarrow|=|\overrightarrow{a}|-|\overrightarrow|\)，則\(\overrightarrow{a}\perp\overrightarrow\)B.若\(\overrightarrow{a}\perp\overrightarrow\)，則\(|\overrightarrow{a}+\overrightarrow|=|\overrightarrow{a}|-|\overrightarrow|\)C.若\(|\overrightarrow{a}+\overrightarrow|=|\overrightarrow{a}|-|\overrightarrow|\)，則存在實(shí)數(shù)\(\lambda\)，使得\(\overrightarrow=\lambda\overrightarrow{a}\)D.若存在實(shí)數(shù)\(\lambda\)，使得\(\overrightarrow=\lambda\overrightarrow{a}\)，則\(|\overrightarrow{a}+\overrightarrow|=|\overrightarrow{a}|-|\overrightarrow|\)答案：C4.對(duì)于函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)，以下說(shuō)法正確的是（）A.函數(shù)圖象關(guān)于點(diǎn)\((\frac{\pi}{3},0)\)對(duì)稱(chēng)B.函數(shù)圖象關(guān)于直線\(x=\frac{\pi}{6}\)對(duì)稱(chēng)C.函數(shù)在\([-\frac{\pi}{6},\frac{\pi}{6}]\)上單調(diào)遞增D.函數(shù)周期為\(\pi\)答案：ACD5.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_1=1\)，\(S_{n+1}=2S_n+1\)，則（）A.數(shù)列\(zhòng)(\{S_n+1\}\)是等比數(shù)列B.\(S_n=2^n-1\)C.\(a_n=2^{n-1}\)D.\(a_n=\begin{cases}1,n=1\\2^{n-2},n\geq2\end{cases}\)答案：ABC6.已知橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的左、右焦點(diǎn)分別為\(F_1,F_2\)，點(diǎn)\(P\)在橢圓上，\(\angleF_1PF_2=60^{\circ}\)，設(shè)\(|PF_1|=m\)，\(|PF_2|=n\)，則（）A.\(m+n=2a\)B.\(m^2+n^2-mn=4c^2\)C.\(mn=\frac{4}{3}b^2\)D.\(\triangleF_1PF_2\)的面積為\(\frac{\sqrt{3}}{3}b^2\)答案：ABCD7.已知函數(shù)\(f(x)=\frac{1}{3}x^3-x^2-3x+1\)，則（）A.函數(shù)\(f(x)\)有兩個(gè)極值點(diǎn)B.函數(shù)\(f(x)\)在\((-\infty,-1)\)上單調(diào)遞增C.函數(shù)\(f(x)\)在\(x=3\)處取得極大值D.函數(shù)\(f(x)\)在\([-2,2]\)上的最小值為\(-\frac{17}{3}\)答案：ABD8.已知\(a,b,c\)分別為\(\triangleABC\)三個(gè)內(nèi)角\(A,B,C\)的對(duì)邊，\(a=2\)，且\((2+b)(\sinA-\sinB)=(c-b)\sinC\)，則（）A.\(A=60^{\circ}\)B.\(\triangleABC\)面積的最大值為\(\sqrt{3}\)C.\(\triangleABC\)周長(zhǎng)的最大值為\(6\)D.若\(b+c=4\)，則\(\triangleABC\)為等邊三角形答案：ABCD9.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時(shí)，\(f(x)=x(1+x)\)，則（）A.\(f(-2)=-6\)B.當(dāng)\(x<0\)時(shí)，\(f(x)=x(1-x)\)C.函數(shù)\(f(x)\)的圖象關(guān)于原點(diǎn)對(duì)稱(chēng)D.函數(shù)\(f(x)\)的單調(diào)遞增區(qū)間是\((-\infty,-\frac{1}{2})\)和\((\frac{1}{2},+\infty)\)答案：ABC10.已知圓\(C\)：\((x-1)^2+(y-2)^2=25\)，直線\(l\)：\((2m+1)x+(m+1)y-7m-4=0\)，則（）A.直線\(l\)恒過(guò)定點(diǎn)\((3,1)\)B.直線\(l\)與圓\(C\)相交C.直線\(l\)被圓\(C\)截得的弦長(zhǎng)的最小值為\(4\sqrt{5}\)D.當(dāng)直線\(l\)被圓\(C\)截得的弦長(zhǎng)最短時(shí)，直線\(l\)的方程為\(2x-y-5=0\)答案：ABCD三、判斷題1.若\(A\capB=A\)，則\(A\subseteqB\)。（）答案：√2.函數(shù)\(y=\frac{1}{x}\)在定義域上是減函數(shù)。（）答案：×3.若向量\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow=\overrightarrow{0}\)。（）答案：×4.若\(\sin\alpha=\sin\beta\)，則\(\alpha=\beta+2k\pi,k\inZ\)。（）答案：×5.等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_1>0\)，\(d<0\)，則\(S_n\)有最大值。（）答案：√6.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的離心率\(e=\frac{c}{a}\)，且\(0<e<1\)。（）答案：√7.函數(shù)\(y=A\sin(\omegax+\varphi)\)的最小正周期\(T=\frac{2\pi}{\omega}\)。（）答案：×（\(\omega>0\)時(shí)才成立）8.若直線\(l\)與平面\(\alpha\)內(nèi)的無(wú)數(shù)條直線垂直，則\(l\perp\alpha\)。（）答案：×9.若\(z=a+bi(a,b\inR)\)，則\(z\)為純虛數(shù)的充要條件是\(a=0\)。（）答案：×10.若\(f(x)\)是定義在\(R\)上的偶函數(shù)，則\(f(x+1)\)的圖象關(guān)于直線\(x=-1\)對(duì)稱(chēng)。（）答案：√四、簡(jiǎn)答題1.已知函數(shù)\(f(x)=\log_2(x+1)+\log_2(3-x)\)。（1）求函數(shù)\(f(x)\)的定義域；（2）求函數(shù)\(f(x)\)的值域。答案：（1）要使函數(shù)有意義，則\(\begin{cases}x+1>0\\3-x>0\end{cases}\)，解得\(-1<x<3\)，所以定義域?yàn)閈((-1,3)\)。（2）\(f(x)=\log_2[(x+1)(3-x)]=\log_2(-x^2+2x+3)\)，令\(t=-x^2+2x+3=-(x-1)^2+4\)，\(x\in(-1,3)\)，則\(0<t\leq4\)，\(y=\log_2t\)在\((0,4]\)上單調(diào)遞增，所以\(y\leq\log_24=2\)，值域?yàn)閈((-\infty,2]\)。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_3=5\)，\(S_6=36\)。（1）求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式；（2）求數(shù)列\(zhòng)(\{\frac{1}{a_na_{n+1}}\}\)的前\(n\)項(xiàng)和\(T_n\)。答案：（1）設(shè)等差數(shù)列\(zhòng)(\{a_n\}\)的公差為\(d\)，則\(\begin{cases}a_1+2d=5\\6a_1+\frac{6\times5}{2}d=36\end{cases}\)，解得\(\begin{cases}a_1=1\\d=2\end{cases}\)，所以