數學競賽試題及答案高中組1.已知集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|x^2-ax+a-1=0\}\)，若\(A\cupB=A\)，則實數\(a\)的值為（）A.\(2\)B.\(3\)C.\(2\)或\(3\)D.\(1\)或\(2\)答案：C2.函數\(y=\log2(x^2-4x+3)\)的單調遞增區(qū)間是（）A.\((3,+\infty)\)B.\((-\infty,1)\)C.\((2,+\infty)\)D.\((1,2)\)答案：A3.已知\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(-3,2)\)，若\(k\overrightarrow{a}+\overrightarrow\)與\(\overrightarrow{a}-3\overrightarrow\)垂直，則\(k\)的值為（）A.\(19\)B.\(-19\)C.\(20\)D.\(-20\)答案：A4.若\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan\alpha\)的值為（）A.\(\frac{3}{4}\)B.\(-\frac{3}{4}\)C.\(\frac{4}{3}\)D.\(-\frac{4}{3}\)答案：B5.已知數列\(zhòng)(\{an\}\)的前\(n\)項和\(Sn=n^2-2n\)，則\(a2+a3\)的值為（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)答案：B6.圓\(x^2+y^2-2x-4y+1=0\)的圓心到直線\(x-y=1\)的距離為（）A.\(\frac{\sqrt{2}}{2}\)B.\(\sqrt{2}\)C.\(1\)D.\(2\)答案：B7.若雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)的離心率為\(\sqrt{3}\)，則其漸近線方程為（）A.\(y=\pm\sqrt{2}x\)B.\(y=\pm\frac{\sqrt{2}}{2}x\)C.\(y=\pm2x\)D.\(y=\pm\frac{1}{2}x\)答案：A8.已知函數\(f(x)=2\sin(\omegax+\varphi)(\omega>0,|\varphi|<\frac{\pi}{2})\)的圖象經過點\((0,-1)\)，且相鄰兩條對稱軸之間的距離為\(\frac{\pi}{2}\)，則函數\(f(x)\)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\(a=\log23\)，\(b=\log34\)，\(c=\log45\)，則（）A.\(a>b>c\)B.\(a<b<c\)C.\(a=b=c\)D.\(a>c>b\)答案：A10.若\(x>0\)，\(y>0\)，且\(\frac{1}{x}+\frac{9}{y}=1\)，則\(x+y\)的最小值為（）A.\(12\)B.\(16\)C.\(18\)D.\(20\)答案：B11.已知\(f(x)\)是定義在\(R\)上的奇函數，當\(x>0\)時，\(f(x)=x^2-2x\)，則當\(x<0\)時，\(f(x)\)的解析式為（）A.\(f(x)=-x^2-2x\)B.\(f(x)=-x^2+2x\)C.\(f(x)=x^2+2x\)D.\(f(x)=x^2-2x\)答案：A12.已知拋物線\(y^2=4x\)的焦點為\(F\)，準線為\(l\)，點\(P\)在拋物線上，\(PQ\perpl\)于點\(Q\)，若\(\trianglePQF\)是等邊三角形，則\(|PF|\)的值為（）A.\(4\)B.\(6\)C.\(8\)D.\(10\)答案：A13.已知函數\(f(x)=\frac{1}{x^2+1}\)，則\(f(x)\)的值域為（）A.\((0,1)\)B.\((0,1]\)C.\([0,1)\)D.\([0,1]\)答案：B14.已知\(\triangleABC\)的內角\(A\)，\(B\)，\(C\)的對邊分別為\(a\)，\(b\)，\(c\)，若\(a=2\)，\(b=3\)，\(c=\sqrt{7}\)，則\(C\)的值為（）A.\(\frac{\pi}{6}\)B.\(\frac{\pi}{4}\)C.\(\frac{\pi}{3}\)D.\(\frac{\pi}{2}\)答案：C15.已知等差數列\(zhòng)(\{an\}\)的前\(n\)項和為\(Sn\)，若\(S3=6\)，\(S6=3\)，則\(S9\)的值為（）A.\(-3\)B.\(0\)C.\(3\)D.\(6\)答案：A16.已知函數\(f(x)=\sqrt{3}\sin2x+\cos2x\)，則\(f(x)\)的最小正周期為（）A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)答案：B17.已知\(x\in[0,\frac{\pi}{2}]\)，則函數\(y=\sinx+\cosx\)的值域為（）A.\([-1,\sqrt{2}]\)B.\([1,\sqrt{2}]\)C.\([-1,1]\)D.\([0,\sqrt{2}]\)答案：B18.已知直線\(l1\)：\(ax+2y+6=0\)，直線\(l2\)：\(x+(a-1)y+a^2-1=0\)，若\(l1\perpl2\)，則\(a\)的值為（）A.\(\frac{2}{3}\)B.\(-1\)C.\(2\)D.\(-1\)或\(2\)答案：A19.已知\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geq2\\x-y\leq2\\y\leq2\end{cases}\)，則\(z=3x+y\)的最大值為（）A.\(10\)B.\(8\)C.\(6\)D.\(4\)答案：A20.已知數列\(zhòng)(\{an\}\)滿足\(a{n+1}=\frac{1}{1-an}\)，若\(a1=\frac{1}{2}\)，則\(a{2019}\)的值為（）A.\(\frac{1}{2}\)B.\(2\)C.\(-1\)D.\(1\)答案：B1.下列函數中，既是奇函數又是增函數的是（）A.\(y=x^3\)B.\(y=\frac{1}{x}\)C.\(y=2x\)D.\(y=\sinx\)答案：AC2.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(2,-1)\)，則（）A.\(\overrightarrow{a}\perp\overrightarrow\)B.\(|\overrightarrow{a}|=\sqrt{5}\)C.\(\overrightarrow{a}\cdot\overrightarrow=0\)D.\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為\(\frac{\pi}{2}\)答案：ABCD3.已知函數\(f(x)=\sin(2x+\frac{\pi}{3})\)，則（）A.函數\(f(x)\)的最小正周期為\(\pi\)B.函數\(f(x)\)的圖象關于直線\(x=\frac{\pi}{12}\)對稱C.函數\(f(x)\)的圖象關于點\((\frac{\pi}{6},0)\)對稱D.函數\(f(x)\)在區(qū)間\([0,\frac{\pi}{2}]\)上單調遞增答案：ABC4.已知雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)的離心率為\(e\)，則（）A.\(e>1\)B.\(e=\frac{c}{a}\)C.\(e^2=1+\frac{b^2}{a^2}\)D.\(e\)越大，雙曲線的開口越小答案：ABC5.已知數列\(zhòng)(\{an\}\)是等比數列，其前\(n\)項和為\(Sn\)，則（）A.若\(a1=1\)，\(q=2\)，則\(S3=7\)B.若\(a1=1\)，\(q=2\)，則\(S6=63\)C.若\(a1=1\)，\(q=-1\)，則\(S{2019}=0\)D.若\(a1=1\)，\(q=-1\)，則\(S{2020}=0\)答案：AD6.已知函數\(f(x)=\loga(x+1)\)，\(g(x)=\loga(1-x)\)，其中\(zhòng)(a>0\)且\(a\neq1\)，則（）A.函數\(f(x)+g(x)\)的定義域為\((-1,1)\)B.函數\(f(x)-g(x)\)的定義域為\((-1,+\infty)\)C.函數\(f(x)\cdotg(x)\)的定義域為\((-1,1)\)D.函數\(\frac{f(x)}{g(x)}\)的定義域為\((-1,1)\)且\(x\neq0\)答案：ACD7.已知拋物線\(y^2=2px(p>0)\)的焦點為\(F\)，準線為\(l\)，則（）A.拋物線上的點到焦點的距離等于到準線的距離B.過焦點的直線與拋物線相交于\(A\)，\(B\)兩點，則\(|AB|\)的最小值為\(2p\)C.準線\(l\)的方程為\(x=-\frac{p}{2}\)D.拋物線的對稱軸為\(y\)軸答案：ABC8.已知函數\(f(x)=x^2-2x+3\)，則（）A.函數\(f(x)\)的圖象開口向上B.函數\(f(x)\)的對稱軸為\(x=1\)C.函數\(f(x)\)的最小值為\(2\)D.函數\(f(x)\)在區(qū)間\((-\infty,1)\)上單調遞減答案：ABCD9.已知\(\triangleABC\)的內角\(A\)，\(B\)，\(C\)的對邊分別為\(a\)，\(b\)，\(c\)，則（）A.若\(a^2+b^2>c^2\)，則\(\triangleABC\)是銳角三角形B.若\(a^2+b^2<c^2\)，則\(\triangleABC\)是鈍角三角形C.若\(a^2+b^2=c^2\)，則\(\triangleABC\)是直角三角形D.若\(a=b=c\)，則\(\triangleABC\)是等邊三角形答案：BCD10.已知等差數列\(zhòng)(\{an\}\)的公差為\(d\)，前\(n\)項和為\(Sn\)，則（）A.若\(d>0\)，則數列\(zhòng)(\{an\}\)單調遞增B.若\(d<0\)，則數列\(zhòng)(\{an\}\)單調遞減C.若\(a1>0\)，\(d<0\)，則\(Sn\)有最大值D.若\(a1<0\)，\(d>0\)，則\(Sn\)有最小值答案：ABCD1.若\(a>b\)，則\(a^2>b^2\)。（）答案：×2.函數\(y=\sinx\)的圖象關于原點對稱。（）