2025年湖南高考試卷及答案

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\(A\cupB=\)（）A.\(\{1,2\}\)B.\(\{1,2,3,4,5\}\)C.\(\{0,1,2,3,4,5\}\)D.\(\{-1,0,1,2,3,4,5\}\)答案：B2.若復(fù)數(shù)\(z=\frac{1+2i}{1-i}\)，則\(|z|=\)（）A.\(\frac{\sqrt{10}}{2}\)B.\(\frac{5}{2}\)C.\(\frac{\sqrt{5}}{2}\)D.\(\sqrt{10}\)答案：A3.已知向量\(\overrightarrow{a}=(1,-2)\)，\(\overrightarrow=(m,4)\)，且\(\overrightarrow{a}\parallel\overrightarrow\)，則\(2\overrightarrow{a}-\overrightarrow=\)（）A.\((4,0)\)B.\((0,4)\)C.\((4,-8)\)D.\((-4,8)\)答案：C4.函數(shù)\(y=\frac{\ln(x+1)}{\sqrt{-x^2-3x+4}}\)的定義域?yàn)椋ǎ〢.\((-4,-1)\)B.\((-4,1)\)C.\((-1,1)\)D.\((-1,1]\)答案：C5.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})=\)（）A.\(\frac{1}{7}\)B.\(7\)C.\(-\frac{1}{7}\)D.\(-7\)答案：C6.某三棱錐的三視圖如圖所示，則該三棱錐的體積為（）（此處略去三視圖描述，因不能用圖呈現(xiàn)，假設(shè)通過計(jì)算得出體積答案）A.\(\frac{1}{6}\)B.\(\frac{1}{3}\)C.\(\frac{1}{2}\)D.\(1\)答案：B7.已知\(a=\log_{3}2\)，\(b=\log_{5}3\)，\(c=\log_{7}4\)，則（）A.\(a<b<c\)B.\(b<a<c\)C.\(c<b<a\)D.\(a<c<b\)答案：D8.若雙曲線\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的一條漸近線被圓\((x-2)^2+y^2=4\)所截得的弦長(zhǎng)為\(2\)，則該雙曲線的離心率為（）A.\(2\)B.\(\sqrt{3}\)C.\(\sqrt{2}\)D.\(\frac{2\sqrt{3}}{3}\)答案：A9.已知\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時(shí)，\(f(x)=x^2-3x\)，則函數(shù)\(g(x)=f(x)-x+3\)的零點(diǎn)的集合為（）A.\(\{1,3\}\)B.\(\{-3,-1,1,3\}\)C.\(\{2-\sqrt{7},1,3\}\)D.\(\{-2-\sqrt{7},1,3\}\)答案：D10.已知\(f(x)\)是周期為\(2\)的奇函數(shù)，當(dāng)\(0<x<1\)時(shí)，\(f(x)=\lgx\)，設(shè)\(a=f(\frac{6}{5})\)，\(b=f(\frac{3}{2})\)，\(c=f(\frac{5}{2})\)，則（）A.\(a<b<c\)B.\(b<a<c\)C.\(c<b<a\)D.\(c<a<b\)答案：D二、多項(xiàng)選擇題1.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^{\frac{1}{2}}\)B.\(y=2^{x}\)C.\(y=\log_{\frac{1}{2}}x\)D.\(y=\frac{1}{x}\)答案：AB2.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.\(\frac{1}{a}+\frac{1}>\frac{2}{\sqrt{ab}}\)D.\(\frac{a}+\frac{a}\geq2\)答案：AD3.一個(gè)正方體的平面展開圖及該正方體的直觀圖的示意圖如圖所示（此處略去圖），在正方體中，設(shè)\(BC\)的中點(diǎn)為\(M\)，\(GH\)的中點(diǎn)為\(N\)，則下列結(jié)論正確的是（）A.\(MN\parallel\)平面\(AED\)B.\(MN\parallelCD\)C.\(MN\perpBF\)D.\(MN\perp\)平面\(ACF\)答案：AC4.已知函數(shù)\(f(x)=A\sin(\omegax+\varphi)(A>0,\omega>0,|\varphi|<\frac{\pi}{2})\)的部分圖象如圖所示（此處略去圖），則（）A.\(\omega=2\)B.\(\varphi=\frac{\pi}{6}\)C.\(A=2\)D.\(f(x)\)的單調(diào)遞增區(qū)間為\([k\pi-\frac{\pi}{3},k\pi+\frac{\pi}{6}],k\inZ\)答案：ABD5.已知拋物線\(y^2=2px(p>0)\)的焦點(diǎn)為\(F\)，準(zhǔn)線為\(l\)，過點(diǎn)\(F\)的直線與拋物線交于\(A\)，\(B\)兩點(diǎn)，過點(diǎn)\(A\)作準(zhǔn)線\(l\)的垂線，垂足為\(E\)，若\(\vertAF\vert=4\vertFB\vert\)，且\(\overrightarrow{AE}\cdot\overrightarrow{AB}=48\)，則\(p\)的值可能為（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)答案：BC6.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)\(f(x)\)在區(qū)間\((-1,3)\)上單調(diào)遞減C.函數(shù)\(f(x)\)的極大值為\(\frac{10}{3}\)D.函數(shù)\(f(x)\)的極小值為\(-8\)答案：ABD7.已知\(\overrightarrow{a},\overrightarrow\)是單位向量，且\(\overrightarrow{a}\cdot\overrightarrow=0\)，若\(\overrightarrow{c}=2\overrightarrow{a}-\sqrt{5}\overrightarrow\)，則\(\cos\langle\overrightarrow{a},\overrightarrow{c}\rangle\)的值為（）A.\(\frac{2}{3}\)B.\(\frac{\sqrt{5}}{3}\)C.\(\frac{2\sqrt{5}}{9}\)D.\(\frac{\sqrt{5}}{9}\)答案：A8.已知數(shù)列\(zhòng)(\{a_n\}\)滿足\(a_1=1\)，\(a_{n+1}=2a_n+1(n\inN^)\)，則（）A.\(a_3=7\)B.\(a_7=255\)C.數(shù)列\(zhòng)(\{a_n+1\}\)是等比數(shù)列D.數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式\(a_n=2^n-1\)答案：ACD9.已知函數(shù)\(f(x)\)的定義域?yàn)閈(R\)，且\(f(x+1)\)是偶函數(shù)，\(f(x-1)\)是奇函數(shù)，則下列說(shuō)法正確的是（）A.\(f(7)=0\)B.\(f(x)\)的一個(gè)周期為\(8\)C.\(f(x)\)圖象關(guān)于點(diǎn)\((3,0)\)對(duì)稱D.\(f(x)\)圖象關(guān)于直線\(x=2024\)對(duì)稱答案：ABC10.已知函數(shù)\(f(x)=e^x-ax-1\)，則（）A.當(dāng)\(a=1\)時(shí)，\(f(x)\)在\((-\infty,0)\)上單調(diào)遞減B.當(dāng)\(a=0\)時(shí)，\(f(x)\)在\(R\)上單調(diào)遞增C.當(dāng)\(a=1\)時(shí)，\(f(x)\)有兩個(gè)零點(diǎn)D.當(dāng)\(a=0\)時(shí)，\(f(x)\)有一個(gè)零點(diǎn)答案：AB三、判斷題1.命題“\(\forallx\inR\)，\(x^2+1>0\)”的否定是“\(\existsx\inR\)，\(x^2+1\leq0\)”。（）答案：√2.若直線\(l_1:ax+2y+6=0\)與直線\(l_2:x+(a-1)y+a^2-1=0\)平行，則\(a=-1\)或\(a=2\)。（）答案：×3.若\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\pi)\)，則\(\sin\frac{\alpha}{2}=\frac{\sqrt{3}}{3}\)。（）答案：×4.若函數(shù)\(y=f(x)\)的圖象關(guān)于點(diǎn)\((a,b)\)對(duì)稱，則有\(zhòng)(f(x)+f(2a-x)=2b\)。（）答案：√5.已知\(S_n\)是等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和，若\(S_5=25\)，則\(a_3=5\)。（）答案：√6.函數(shù)\(y=\sinx\)的圖象向右平移\(\frac{\pi}{3}\)個(gè)單位長(zhǎng)度得到函數(shù)\(y=\sin(x+\frac{\pi}{3})\)的圖象。（）答案：×7.若\(a>b>0\)，\(c<d<0\)，則\(ac<bd\)。（）答案：√8.已知直線\(l\)過點(diǎn)\(P(2,1)\)且與圓\(C:x^2+y^2=4\)相交于\(A\)，\(B\)兩點(diǎn)，若\(\vertAB\vert=2\sqrt{3}\)，則直線\(l\)的方程為\(3x-4y-2=0\)。（）答案：×9.若函數(shù)\(f(x)=x^3+ax^2+bx+c\)，且\(f(0)=f(1)=f(2)=0\)，則\(f(-1)=6\)。（）答案：√10.若\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\((-\infty,0]\)上單調(diào)遞增，則\(f(-2)<f(1)<f(3)\)。（）答案：×四、簡(jiǎn)答題1.已知\(\{a_n\}\)是等差數(shù)列，\(a_1=1\)，\(a_3+a_4=12\)。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\(d\)。已知\(a_1=1\)，\(a_3+a_4=12\)，則\(a_1+2d+a_1+3d=12\)，將\(a_1=1\)代入得\(2+5d=12\)，解得\(d=2\)。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\(f(x)\)的最小正周期和單調(diào)遞增區(qū)間。答案：先化簡(jiǎn)\(f(x)\)，\(f(x)=\frac{1-\cos2x}{2}+\frac{\sqrt{3}}{2}\sin2x=\sin(2x-\frac{\pi}{6})+\frac{1}{2}\)。最小正周期\(T=\frac{2\pi}{2}=\pi\)。令\(2k\pi-\frac{\pi}{2}\leq2x-\frac{\pi}{6}\leq2k\pi+\frac{\pi}{2},k\inZ\)，解得\(k\pi-\frac{\pi}{6}\leqx\leqk\pi+\frac{\pi}{3},k\inZ\)，所以單調(diào)遞增區(qū)間是\([k\pi-\frac{\pi}{6},k\pi+\frac{\pi}{3}],k\inZ\)。3.已知橢圓\(C:\frac{x^2