數(shù)學(xué)高三試卷及答案

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.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.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(m,-1)\)，若\(\overrightarrow{a}\parallel(\overrightarrow{a}+\overrightarrow)\)，則\(\overrightarrow{a}\cdot\overrightarrow\)等于（）A.\(-\frac{5}{2}\)B.\(\frac{5}{2}\)C.\(\frac{3}{2}\)D.\(-\frac{3}{2}\)答案：A4.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(S_{10}=10\)，\(S_{20}=60\)，則\(S_{30}\)等于（）A.110B.130C.150D.170答案：C5.已知函數(shù)\(f(x)=\sin(\omegax+\frac{\pi}{6})(\omega\gt0)\)的最小正周期為\(\pi\)，則\(f(x)\)圖象的一條對稱軸方程是（）A.\(x=\frac{\pi}{6}\)B.\(x=\frac{\pi}{3}\)C.\(x=\frac{\pi}{2}\)D.\(x=\frac{2\pi}{3}\)答案：B6.已知雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a\gt0,b\gt0)\)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.\(y=\pm\frac{\sqrt{3}}{2}x\)答案：A7.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時(shí)，\(f(x)=x(1+x)\)，則當(dāng)\(x\lt0\)時(shí)，\(f(x)\)的表達(dá)式為（）A.\(f(x)=x(1+x)\)B.\(f(x)=-x(1+x)\)C.\(f(x)=x(1-x)\)D.\(f(x)=-x(1-x)\)答案：C8.已知\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geq2\\x-y\leq2\\y\leq2\end{cases}\)，則\(z=x+2y\)的最大值為（）A.8B.7C.6D.5答案：B9.已知函數(shù)\(f(x)=\frac{1}{3}x^3-4x+4\)，則函數(shù)\(f(x)\)的極大值為（）A.\(\frac{28}{3}\)B.\(\frac{4}{3}\)C.\(\frac{20}{3}\)D.\(\frac{16}{3}\)答案：A10.已知\(a=\log_{0.5}3\)，\(b=2^{0.3}\)，\(c=0.3^{0.5}\)，則\(a\)，\(b\)，\(c\)的大小關(guān)系是（）A.\(a\ltb\ltc\)B.\(a\ltc\ltb\)C.\(c\lta\ltb\)D.\(c\ltb\lta\)答案：B二、多項(xiàng)選擇題1.下列說法正確的是（）A.若\(a\gtb\)，則\(ac^2\gtbc^2\)B.若\(a\gtb\)，\(c\gtd\)，則\(a-c\gtb-d\)C.若\(a\gtb\)，則\(a^3\gtb^3\)D.若\(a\gtb\)，則\(\frac{1}{a}\lt\frac{1}\)答案：C2.已知直線\(l\)過點(diǎn)\((1,0)\)且垂直于\(x\)軸，若\(l\)被拋物線\(y^2=4ax\)截得的線段長為\(4\)，則拋物線的焦點(diǎn)坐標(biāo)為（）A.\((1,0)\)B.\((2,0)\)C.\((0,1)\)D.\((0,2)\)答案：A3.已知函數(shù)\(f(x)=\cos(2x+\frac{\pi}{3})\)，則下列說法正確的是（）A.\(f(x)\)的最小正周期為\(\pi\)B.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{12},0)\)對稱C.\(f(x)\)在區(qū)間\((\frac{\pi}{3},\frac{5\pi}{6})\)上單調(diào)遞增D.\(f(x)\)的圖象可由\(y=\cos2x\)的圖象向左平移\(\frac{\pi}{3}\)個(gè)單位得到答案：ABC4.已知正方體\(ABCD-A_1B_1C_1D_1\)的棱長為\(1\)，則下列說法正確的是（）A.異面直線\(A_1C_1\)與\(B_1C\)所成的角為\(60^{\circ}\)B.直線\(A_1C_1\)與平面\(ABCD\)所成的角為\(45^{\circ}\)C.點(diǎn)\(C\)到平面\(A_1BD\)的距離為\(\frac{\sqrt{3}}{3}\)D.二面角\(A-BD-A_1\)的大小為\(60^{\circ}\)答案：AC5.已知函數(shù)\(f(x)\)的導(dǎo)函數(shù)為\(f^\prime(x)\)，且滿足\(f(x)=2xf^\prime(1)+\lnx\)，則下列說法正確的是（）A.\(f^\prime(1)=-1\)B.\(f(1)=2\)C.\(f(x)\)在\((0,+\infty)\)上單調(diào)遞減D.\(f(x)\)在\(x=1\)處取得極大值答案：ACD6.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\(F_1,F_2\)，點(diǎn)\(P\)在橢圓上，且\(\overrightarrow{PF_1}\cdot\overrightarrow{PF_2}=0\)，\(\trianglePF_1F_2\)的面積為\(9\)，則（）A.\(b=3\)B.\(a-c\gt3\)C.\(\trianglePF_1F_2\)的周長為\(6+2\sqrt{a^2-9}\)D.離心率\(e\geq\frac{\sqrt{2}}{2}\)答案：ACD7.已知函數(shù)\(y=f(x)\)是定義在\(R\)上的偶函數(shù)，且在\([0,+\infty)\)上單調(diào)遞增，則下列結(jié)論正確的是（）A.\(f(-2)\gtf(1)\gtf(0)\)B.若\(f(a)\gtf(b)\)，則\(|a|\gt|b|\)C.\(f(x)\)在\((-\infty,0]\)上單調(diào)遞減D.若\(f(x)\)在\([0,+\infty)\)上的最大值為\(f(2)\)，則\(f(x)\)在\(R\)上的最大值也為\(f(2)\)答案：ABCD8.已知數(shù)列\(zhòng)(\{a_n\}\)滿足\(a_1=1\)，\(a_{n+1}=2a_n+1\)，則下列說法正確的是（）A.\(a_3=7\)B.數(shù)列\(zhòng)(\{a_n+1\}\)是等比數(shù)列C.\(a_n=2^n-1\)D.數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=2^{n+1}-n-2\)答案：ABC9.已知函數(shù)\(f(x)=|x-a|+|x+b|(a\gt0,b\gt0)\)，則下列說法正確的是（）A.\(f(x)\)的最小值為\(a+b\)B.若\(f(x)\)的最小值為\(2\)，則\(a^2+b^2\geq2\)C.當(dāng)\(a=1\)，\(b=2\)時(shí)，\(f(x)\geq5\)的解集為\((-\infty,-3]\cup[2,+\infty)\)D.若\(f(x)\)在區(qū)間\((1,+\infty)\)上單調(diào)遞增，則\(a\leq1\)答案：ABCD10.已知\(a\)，\(b\)，\(c\)為正實(shí)數(shù)，且\(a+b+c=1\)，則下列說法正確的是（）A.\(ab+bc+ca\leq\frac{1}{3}\)B.\(a^2+b^2+c^2\geq\frac{1}{3}\)C.\(\frac{1}{a}+\frac{1}+\frac{1}{c}\geq9\)D.\(\sqrt{a}+\sqrt+\sqrt{c}\leq\sqrt{3}\)答案：ABCD三、判斷題1.若\(A\)，\(B\)為互斥事件，則\(P(A)+P(B)=1\)。（×）2.若\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow=\overrightarrow{0}\)。（×）3.函數(shù)\(y=\sinx\)的圖象的一個(gè)對稱中心是\((\pi,0)\)。（√）4.若直線\(l\)垂直于平面\(\alpha\)內(nèi)的無數(shù)條直線，則\(l\perp\alpha\)。（×）5.若\(a\gtb\gt0\)，\(c\ltd\lt0\)，則\(ac\ltbd\)。（√）6.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有零點(diǎn)，則\(f(a)\cdotf(b)\lt0\)。（×）7.雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)的漸近線方程為\(y=\pm\frac{a}x\)。（√）8.若\(a\)，\(b\)，\(c\)成等比數(shù)列，則\(b^2=ac\)。（√）9.若\(f(x)\)是奇函數(shù)，則\(f(0)=0\)。（×）10.已知\(a\)，\(b\)是實(shí)數(shù)，則“\(a\gtb\)”是“\(a^2\gtb^2\)”的充分不必要條件。（×）四、簡答題1.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_3=5\)，\(S_6=36\)。求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式。答案：設(shè)等差數(shù)列\(zhòng)(\{a_n\}\)的公差為\(d\)。由\(a_3=5\)可得\(a_1+2d=5\)；由\(S_6=36\)可得\(6a_1+\frac{6\times5}{2}d=36\)，即\(6a_1+15d=36\)。聯(lián)立方程組\(\begin{cases}a_1+2d=5\\6a_1+15d=36\end{cases}\)，解方程組得\(a_1=1\)，\(d=2\)。所以\(a_n=a_1+(n-1)d=1+2(n-1)=2n-1\)。2.已知函數(shù)\(f(x)=\sin^2x+\sqrt{3}\sinx\cosx\)。求\(f(x)\)的最小正周期及單調(diào)遞增區(qū)間。答案：先化簡\(f(x)\)，\(f(x)=\frac{1-\cos2x}{2}+\frac{\sqrt{3}}{2}\sin2x=\sin(2x-\frac{\pi}{6})+\frac{1}{2}\)。其最小正周期\(T=\frac{2\pi}{2}=\pi\)。令\(2k\pi-\frac{\pi}{2}\leq2x-\frac{\pi}{6}\leq2k\pi+\frac{\pi}{2},k\inZ\)，解不等式得\(k\pi-\frac{\pi}{6}\leqx\leqk\pi+\frac{\pi}{3},k\inZ\)，所以\(f(x)\)的單調(diào)遞增區(qū)間為\([k\pi-\frac{\pi}{6},k\pi+\frac{\pi}{3}],k\inZ\)。3.已知橢圓\(C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a\gtb\gt0)\)的離心率為\(\frac{\sqrt{2}}{2}\)，且過點(diǎn)\((\sqrt{2},1)\)。求橢圓\(C\)的方程。答案：由離心率\(e=\frac{c}{a}=\frac{\sqrt{2}}{2}\)，可得\(c^2=\frac{1}{2}a^2\)，又\(a^2=b^2+c^2\)，所以\(a^2=2b^2\)。橢圓過點(diǎn)\((\sqrt{2},1)\)，將其代入橢圓方程\(\frac{(\sqrt{2})^2}{a^2}+\frac{1^2}{b^2}=1\)，即\(\frac{2}{a^2}+\frac{1}{b^2}=1\)。把\(a^2=2b^2\)代入此方程，解