分析高考數(shù)學(xué)真題及答案

一、單項(xiàng)選擇題1.已知集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|0<x<5,x\inN\}\)，則滿足條件\(A\subseteqC\subseteqB\)的集合\(C\)的個(gè)數(shù)為（）A.1B.2C.3D.4答案：D2.已知\(i\)是虛數(shù)單位，則\(\frac{3+i}{1-i}\)等于（）A.\(1-2i\)B.\(2-i\)C.\(2+i\)D.\(1+2i\)答案：D3.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.\((-\infty,-3]\cup[1,+\infty)\)答案：B4.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})\)等于（）A.\(\frac{1}{7}\)B.7C.\(-\frac{1}{7}\)D.-7答案：A5.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_3+a_4+a_5=12\)，則\(S_7\)等于（）A.28B.42C.56D.14答案：A6.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(3,-2)\)，且\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow\)，則\(m\)等于（）A.-8B.-6C.6D.8答案：D7.已知拋物線\(y^2=2px(p>0)\)的焦點(diǎn)為\(F\)，點(diǎn)\(M\)在拋物線上，且\(|MF|=4\)，若以\(MF\)為直徑的圓過點(diǎn)\((0,2)\)，則\(p\)的值為（）A.2B.4C.6D.8答案：B8.已知函數(shù)\(f(x)=\begin{cases}2^x-1,x\leq0\\f(x-1)+1,x>0\end{cases}\)，把函數(shù)\(g(x)=f(x)-x\)的零點(diǎn)按從小到大的順序排成一個(gè)數(shù)列\(zhòng)(\{a_n\}\)，則該數(shù)列的通項(xiàng)公式為（）A.\(a_n=\frac{n(n-1)}{2}\)B.\(a_n=n(n-1)\)C.\(a_n=n-1\)D.\(a_n=2^n-2\)答案：C9.已知函數(shù)\(f(x)=\sin(\omegax+\varphi)(\omega>0,|\varphi|<\frac{\pi}{2})\)的最小正周期為\(4\pi\)，且對(duì)\(\forallx\inR\)，有\(zhòng)(f(x)\leqf(\frac{\pi}{3})\)成立，則\(f(x)\)圖象的一個(gè)對(duì)稱中心坐標(biāo)是（）A.\((-\frac{2\pi}{3},0)\)B.\((-\frac{\pi}{3},0)\)C.\((\frac{2\pi}{3},0)\)D.\((\frac{5\pi}{3},0)\)答案：A10.已知三棱錐\(S-ABC\)的所有頂點(diǎn)都在球\(O\)的球面上，\(\triangleABC\)是邊長(zhǎng)為\(1\)的正三角形，\(SC\)為球\(O\)的直徑，且\(SC=2\)，則此三棱錐的體積為（）A.\(\frac{\sqrt{2}}{6}\)B.\(\frac{\sqrt{3}}{6}\)C.\(\frac{\sqrt{2}}{3}\)D.\(\frac{\sqrt{2}}{2}\)答案：A二、多項(xiàng)選擇題1.下列說法正確的是（）A.若\(a>b\)，則\(ac^2>bc^2\)B.若\(a<b<0\)，則\(a^2>ab>b^2\)C.若\(a>b\)，\(c<d\)，則\(a-c>b-d\)D.若\(b<a<0\)，則\(\frac{a}<\frac{a}\)答案：BCD2.已知函數(shù)\(f(x)=\sinx+\cosx\)，則（）A.\(f(x)\)的最大值為\(\sqrt{2}\)B.\(f(x)\)的最小正周期為\(\pi\)C.\(f(x)\)的圖象關(guān)于直線\(x=\frac{5\pi}{4}\)對(duì)稱D.\(f(x)\)在區(qū)間\((\frac{\pi}{4},\frac{3\pi}{4})\)上單調(diào)遞減答案：ACD3.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\(2\)，則（）A.雙曲線\(C\)的漸近線方程為\(y=\pm\sqrt{3}x\)B.\(\frac{b^2+1}{3a}\)的最小值為\(\frac{2\sqrt{3}}{3}\)C.雙曲線\(C\)的漸近線與圓\((x-2)^2+y^2=3\)相切D.雙曲線\(C\)的漸近線被圓\((x-2)^2+y^2=4\)所截得的弦長(zhǎng)為\(2\sqrt{3}\)答案：ABCD4.已知\(a\)，\(b\)，\(c\)分別為\(\triangleABC\)三個(gè)內(nèi)角\(A\)，\(B\)，\(C\)的對(duì)邊，且\((a+b)(\sinA-\sinB)=(c-b)\sinC\)，則（）A.\(A=\frac{\pi}{6}\)B.\(B=\frac{\pi}{3}\)C.\(\cosB+\cosC\)的最大值為\(\sqrt{3}\)D.若\(\triangleABC\)的面積為\(\sqrt{3}\)，則\(a=2\)答案：CD5.已知函數(shù)\(f(x)=\begin{cases}2^x-1,x\leq0\\x^{\frac{1}{2}},x>0\end{cases}\)，則下列結(jié)論正確的是（）A.\(f(x)\)的值域?yàn)閈((-1,+\infty)\)B.\(f(x)\)在\(R\)上單調(diào)遞增C.若\(f(a)=2\)，則\(a=4\)D.方程\(f(x)-f(-x)=0\)有\(zhòng)(3\)個(gè)實(shí)數(shù)根答案：ACD6.已知\(\overrightarrow{a}\)，\(\overrightarrow\)是兩個(gè)非零向量，且\(|\overrightarrow{a}|=|\overrightarrow|=|\overrightarrow{a}-\overrightarrow|\)，則（）A.\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為\(60^{\circ}\)B.\((\overrightarrow{a}+\overrightarrow)\perp(\overrightarrow{a}-\overrightarrow)\)C.\(|\overrightarrow{a}+\overrightarrow|=\sqrt{3}|\overrightarrow{a}|\)D.若\(|\overrightarrow{a}|=1\)，則\((\overrightarrow{a}+\overrightarrow)\cdot\overrightarrow=\frac{3}{2}\)答案：ACD7.已知函數(shù)\(y=f(x)\)的圖象關(guān)于點(diǎn)\((-1,0)\)對(duì)稱，且滿足\(f(x)=-f(x+1)\)，當(dāng)\(x\in(-2,-1)\)時(shí)，\(f(x)=\frac{1}{x+2}\)，則下列結(jié)論正確的是（）A.\(f(x)\)是周期函數(shù)，周期為\(2\)B.\(f(x)\)在\((-3,-2)\)上單調(diào)遞減C.\(f(x)\)的圖象關(guān)于直線\(x=-\frac{3}{2}\)對(duì)稱D.\(f(2021)=0\)答案：ACD8.已知\(F_1\)，\(F_2\)是橢圓\(C\)：\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的兩個(gè)焦點(diǎn)，\(P\)為橢圓\(C\)上一點(diǎn)，且\(\angleF_1PF_2=60^{\circ}\)，若\(\trianglePF_1F_2\)的面積為\(9\sqrt{3}\)，則（）A.\(b=3\)B.\(a-c=\sqrt{3}\)C.\(\overrightarrow{PF_1}\cdot\overrightarrow{PF_2}=18\)D.\(\trianglePF_1F_2\)的周長(zhǎng)為\(6+6\sqrt{3}\)答案：AC9.已知函數(shù)\(f(x)=x^3-3x^2+2\)，則（）A.\(f(x)\)在\((-\infty,0)\)上單調(diào)遞增B.\(f(x)\)在\(x=2\)處取得極小值\(-2\)C.\(f(x)\)的圖象關(guān)于點(diǎn)\((1,0)\)對(duì)稱D.方程\(f(x)=k\)有三個(gè)不同實(shí)根時(shí)，\(k\)的取值范圍是\((-2,2)\)答案：ABD10.已知\(a\)，\(b\)，\(c\)滿足\(a>b>c\)，且\(ac<0\)，則下列選項(xiàng)中一定成立的是（）A.\(ab>ac\)B.\(c(b-a)>0\)C.\(cb^2<ab^2\)D.\(\frac{1}{a}>\frac{1}\)答案：AB三、判斷題1.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有零點(diǎn)，則\(f(a)\cdotf(b)<0\)。（×）2.若向量\(\overrightarrow{a}\)，\(\overrightarrow\)滿足\(\overrightarrow{a}\cdot\overrightarrow>0\)，則\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為銳角。（×）3.已知直線\(l_1\)：\(A_1x+B_1y+C_1=0\)，\(l_2\)：\(A_2x+B_2y+C_2=0\)，若\(A_1A_2+B_1B_2=0\)，則\(l_1\perpl_2\)。（√）4.若\(a\)，\(b\)，\(c\)成等比數(shù)列，則\(b^2=ac\)。（√）5.函數(shù)\(y=\sin^2x\)的最小正周期為\(\pi\)。（√）6.若\(x>0\)，\(y>0\)且\(x+y=1\)，則\(\frac{1}{x}+\frac{1}{y}\)的最小值為\(4\)。（√）7.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\(y=\pm\frac{a}x\)。（√）8.已知\(a\)，\(b\)為實(shí)數(shù)，則“\(a>b\)”是“\(a^2>b^2\)”的充分不必要條件。（×）9.若函數(shù)\(f(x)\)為奇函數(shù)，則\(f(0)=0\)。（×）10.球的體積公式為\(V=\frac{4}{3}\pir^3\)（\(r\)為球的半徑）。（√）四、簡(jiǎn)答題1.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=n^2+2n\)。求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式。當(dāng)\(n=1\)時(shí)，\(a_1=S_1=1^2+2\times1=3\)；當(dāng)\(n\geq2\)時(shí)，\(a_n=S_n-S_{n-1}=(n^2+2n)-[(n-1)^2+2(n-1)]\)，化簡(jiǎn)得\(a_n=2n+1\)。當(dāng)\(n=1\)時(shí)，\(a_1=3\)也滿足\(a_n=2n+1\)。所以數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式為\(a_n=2n+1\)。2.已知函數(shù)\(f(x)=\sin(2x+\frac{\pi}{3})\)，求\(f(x)\)的單調(diào)遞增區(qū)間。令\(2k\pi-\frac{\pi}{2}\leq2x+\frac{\pi}{3}\leq2k\pi+\frac{\pi}{2},k\inZ\)，解這個(gè)不等式：先移項(xiàng)得\(2k\pi-\frac{\pi}{2}-\frac{