2025年關(guān)于數(shù)學(xué)高考試題及答案

一、單項(xiàng)選擇題1.已知集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|0<x<6,x\inN\}\)，則\(A\cupB\)等于（）A.\(\{1,2\}\)B.\(\{1,2,3,4,5\}\)C.\(\{2,3,4,5\}\)D.\(\{1,3,4,5\}\)答案：B2.若復(fù)數(shù)\(z\)滿足\((1+i)z=2i\)，則\(z\)的共軛復(fù)數(shù)\(\overline{z}\)等于（）A.\(1-i\)B.\(-1-i\)C.\(1+i\)D.\(-1+i\)答案：A3.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(3,-2)\)，且\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow\)，則\(m\)的值為（）A.-8B.-6C.6D.8答案：D4.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：C5.設(shè)\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geq2\\x-y\leq2\\y\leq2\end{cases}\)，則\(z=2x-y\)的最大值為（）A.2B.4C.6D.8答案：C6.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\(\sqrt{2}\)，則\(C\)的漸近線方程為（）A.\(y=\pmx\)B.\(y=\pm\sqrt{2}x\)C.\(y=\pm\sqrt{3}x\)D.\(y=\pm2x\)答案：A7.執(zhí)行如圖所示的程序框圖，若輸入的\(x=2\)，則輸出的\(y\)值為（）A.2B.4C.8D.16答案：C8.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_3+a_5+a_7=15\)，則\(S_9\)等于（）A.45B.60C.75D.90答案：A9.已知函數(shù)\(f(x)=\sin(\omegax+\varphi)(\omega>0,|\varphi|<\frac{\pi}{2})\)的最小正周期為\(\pi\)，將其圖象向左平移\(\frac{\pi}{6}\)個(gè)單位后得到的圖象關(guān)于\(y\)軸對(duì)稱，則\(\varphi\)的值為（）A.\(\frac{\pi}{6}\)B.\(\frac{\pi}{3}\)C.-\(\frac{\pi}{6}\)D.-\(\frac{\pi}{3}\)答案：A10.已知函數(shù)\(f(x)=x^3-3x\)，過點(diǎn)\(P(2,-6)\)作曲線\(y=f(x)\)的切線，則切線方程為（）A.\(3x+y=0\)或\(24x-y-54=0\)B.\(3x-y=0\)或\(24x-y-54=0\)C.\(3x+y=0\)或\(24x+y-54=0\)D.\(3x-y=0\)或\(24x+y-54=0\)答案：A二、多項(xiàng)選擇題1.下列說法正確的是（）A.命題“\(\forallx\inR\)，\(x^2+x+1>0\)”的否定是“\(\existsx\inR\)，\(x^2+x+1\leq0\)”B.若\(a\)，\(b\inR\)，則“\(a^2+b^2\neq0\)”是“\(a\)，\(b\)不全為\(0\)”的充要條件C.若\(p\)：\(x(x-2)\leq0\)，\(q\)：\(\log_2x\leq1\)，則\(q\)是\(p\)的充分不必要條件D.若“\(p\landq\)”為假命題，“\(p\lorq\)”為真命題，則\(p\)，\(q\)一真一假答案：ABCD2.已知函數(shù)\(f(x)=\cos(2x-\frac{\pi}{3})-\cos2x\)，則（）A.\(f(x)\)的最小正周期為\(\pi\)B.\(f(x)\)的圖象關(guān)于直線\(x=\frac{2\pi}{3}\)對(duì)稱C.\(f(x)\)在\([0,\frac{\pi}{2}]\)上的值域?yàn)閈([-\frac{\sqrt{3}}{2},1]\)D.\(f(x)\)在\([0,\frac{\pi}{2}]\)上單調(diào)遞增答案：ABC3.已知\(a\)，\(b\)，\(c\)為正實(shí)數(shù)，且\(a+b+c=1\)，則（）A.\(a^2+b^2+c^2\geq\frac{1}{3}\)B.\(ab+bc+ca\leq\frac{1}{3}\)C.\(\sqrt{a}+\sqrt+\sqrt{c}\leq\sqrt{3}\)D.\(\frac{1}{a}+\frac{1}+\frac{1}{c}\geq9\)答案：ABCD4.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\(l\)恒過定點(diǎn)\((3,1)\)B.直線\(l\)與圓\(C\)相交C.當(dāng)直線\(l\)被圓\(C\)截得的弦長最短時(shí)，直線\(l\)的方程為\(2x-y-5=0\)D.直線\(l\)被圓\(C\)截得的最短弦長為\(4\sqrt{5}\)答案：ABCD5.已知函數(shù)\(f(x)=\begin{cases}x^2+2x,x\geq0\\-x^2+2x,x<0\end{cases}\)，則（）A.\(f(x)\)是奇函數(shù)B.\(f(x)\)在\(R\)上單調(diào)遞增C.不等式\(f(x)\leq3\)的解集為\([-1,1]\)D.方程\(f(x)-|x-1|=0\)有三個(gè)不同的實(shí)數(shù)根答案：ABD6.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\(\triangleF_1PF_2\)的面積為\(\sqrt{3}b^2\)，則（）A.\(a=2b\)B.橢圓\(C\)的離心率為\(\frac{1}{2}\)C.\(|PF_1|\cdot|PF_2|=4b^2\)D.\(\overrightarrow{PF_1}\cdot\overrightarrow{PF_2}=b^2\)答案：ABCD7.已知函數(shù)\(y=f(x)\)的圖象關(guān)于點(diǎn)\((1,0)\)對(duì)稱，且當(dāng)\(x>1\)時(shí)，\(f(x)=\frac{1}{x-1}\)，則（）A.\(f(-1)=-\frac{1}{2}\)B.\(f(x)\)在\((-\infty,1)\)上單調(diào)遞減C.\(f(x)\)的圖象關(guān)于直線\(x=1\)對(duì)稱D.方程\(f(x)+\frac{1}{2}=0\)的解為\(x=-1\)或\(x=3\)答案：ABD8.已知正方體\(ABCD-A_1B_1C_1D_1\)的棱長為\(2\)，\(E\)，\(F\)分別為棱\(BB_1\)，\(CC_1\)的中點(diǎn)，則（）A.\(AE\perpD_1F\)B.直線\(AE\)與平面\(A_1D_1F\)所成角的正弦值為\(\frac{\sqrt{10}}{5}\)C.平面\(AEF\)截正方體所得的截面為等腰梯形D.三棱錐\(C-AEF\)的體積為\(\frac{2}{3}\)答案：ABCD9.已知函數(shù)\(f(x)=\lnx-ax\)，若\(f(x)\)有兩個(gè)零點(diǎn)\(x_1\)，\(x_2\)（\(x_1<x_2\)），則（）A.\(0<a<\frac{1}{e}\)B.\(x_1+x_2>2e\)C.\(x_1x_2>e^2\)D.\(\frac{1}{x_1}+\frac{1}{x_2}>\frac{2}{e}\)答案：ACD10.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，且\(S_n=2a_n-2^n\)，則（）A.\(a_1=2\)B.數(shù)列\(zhòng)(\{a_n\}\)是等差數(shù)列C.\(a_n=(n+1)2^{n-1}\)D.數(shù)列\(zhòng)(\{\frac{a_n}{n+1}\}\)是等比數(shù)列答案：ACD三、判斷題1.若\(a>b\)，則\(ac^2>bc^2\)。（×）2.函數(shù)\(y=\sin^2x\)的最小正周期是\(\pi\)。（√）3.直線\(l_1\)：\(A_1x+B_1y+C_1=0\)與直線\(l_2\)：\(A_2x+B_2y+C_2=0\)垂直的充要條件是\(A_1A_2+B_1B_2=0\)。（√）4.若向量\(\overrightarrow{a}\)，\(\overrightarrow\)滿足\(\overrightarrow{a}\cdot\overrightarrow>0\)，則\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為銳角。（×）5.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有零點(diǎn)，則\(f(a)\cdotf(b)<0\)。（×）6.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)（\(a>b>0\)）的離心率\(e\)越大，橢圓越圓。（×）7.若\(a\)，\(b\)，\(c\)成等比數(shù)列，則\(b^2=ac\)。（√）8.函數(shù)\(y=\log_2(x^2+1)\)的值域是\([0,+\infty)\)。（√）9.若\(x>0\)，\(y>0\)，且\(x+y=1\)，則\(\frac{1}{x}+\frac{1}{y}\)的最小值是\(4\)。（√）10.已知\(a\)，\(b\)為異面直線，\(a\subset\alpha\)，\(b\subset\beta\)，\(\alpha\cap\beta=l\)，則\(l\)至少與\(a\)，\(b\)中的一條相交。（√）四、簡答題1.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_1=1\)，\(S_3=9\)。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\(S_3=9\)可得\(3a_1+\frac{3\times(3-1)}{2}d=9\)，將\(a_1=1\)代入得\(3+3d=9\)，解得\(d=2\)。那么\(a_n=a_1+(n-1)d=1+2(n-1)=2n-1\)。2.已知函數(shù)\(f(x)=\sin(2x+\frac{\pi}{6})+\cos(2x-\frac{\pi}{3})\)，求\(f(x)\)的最小正周期和單調(diào)遞增區(qū)間。答案：化簡\(f(x)\)，\(\cos(2x-\frac{\pi}{3})=\sin(2x-\frac{\pi}{3}+\frac{\pi}{2})=\sin(2x+\frac{\pi}{6})\)，所以\(f(x)=2\sin(2x+\frac{\pi}{6})\)。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