2025年高考數(shù)學(xué)試卷歷年及答案

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.\(\{0,1,2,3,4,5\}\)D.\(\varnothing\)答案：B2.復(fù)數(shù)\(z=\frac{1+2i}{1-i}\)（\(i\)為虛數(shù)單位），則\(z\)的共軛復(fù)數(shù)\(\overline{z}\)是（）A.\(-\frac{1}{2}+\frac{3}{2}i\)B.\(-\frac{1}{2}-\frac{3}{2}i\)C.\(\frac{1}{2}+\frac{3}{2}i\)D.\(\frac{1}{2}-\frac{3}{2}i\)答案：B3.已知\(\overrightarrow{a}=(1,-2)\)，\(\overrightarrow=(m,4)\)，且\(\overrightarrow{a}\parallel\overrightarrow\)，則\(2\overrightarrow{a}-\overrightarrow\)等于（）A.\((4,0)\)B.\((0,4)\)C.\((4,-8)\)D.\((-4,8)\)答案：C4.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.\(7\)C.\(-\frac{1}{7}\)D.\(-7\)答案：A5.函數(shù)\(y=\log_2(x^2-2x-3)\)的單調(diào)遞增區(qū)間是（）A.\((-\infty,-1)\)B.\((-\infty,1)\)C.\((1,+\infty)\)D.\((3,+\infty)\)答案：D6.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_3=5\)，\(S_9=81\)，則\(a_7\)等于（）A.\(18\)B.\(13\)C.\(9\)D.\(7\)答案：B7.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\(y=\pm\frac{3}{4}x\)，則雙曲線的離心率為（）A.\(\frac{5}{4}\)B.\(\frac{5}{3}\)C.\(\frac{\sqrt{7}}{4}\)D.\(\frac{\sqrt{7}}{3}\)答案：A8.已知一個(gè)正方體的所有頂點(diǎn)在一個(gè)球面上，若球的體積為\(\frac{9\pi}{2}\)，則正方體的棱長為（）A.\(\sqrt{3}\)B.\(\sqrt{2}\)C.\(\frac{\sqrt{3}}{2}\)D.\(1\)答案：A9.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，則\(f(x)\)的解析式為（）A.\(f(x)=2\sin(2x+\frac{\pi}{6})\)B.\(f(x)=2\sin(2x-\frac{\pi}{6})\)C.\(f(x)=2\sin(x+\frac{\pi}{6})\)D.\(f(x)=2\sin(x-\frac{\pi}{6})\)答案：A10.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時(shí)，\(f(x)=x^2-2x\)，則不等式\(f(x)>x\)的解集是（）A.\((-3,0)\cup(3,+\infty)\)B.\((-\infty,-3)\cup(0,3)\)C.\((-\infty,-3)\cup(3,+\infty)\)D.\((-3,0)\cup(0,3)\)答案：C二、多項(xiàng)選擇題1.下列函數(shù)中，既是偶函數(shù)又在\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^2+1\)B.\(y=\log_2|x|\)C.\(y=x^3\)D.\(y=2^{|x|}\)答案：ABD2.已知\(a\)，\(b\)，\(c\)是實(shí)數(shù)，則下列命題正確的是（）A.若\(a>b\)，則\(ac^2>bc^2\)B.若\(ac^2>bc^2\)，則\(a>b\)C.若\(a<b<0\)，則\(\frac{1}{a}>\frac{1}\)D.若\(a<b<0\)，則\(a^2>ab>b^2\)答案：BCD3.一個(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，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\(ABE\)B.\(MN\perp\)平面\(BDH\)C.\(MN\parallel\)平面\(BDH\)D.\(MN\perp\)平面\(CDE\)答案：AC4.已知直線\(l\)：\(y=kx+1\)與圓\(C\)：\(x^2+y^2-2x-2y+1=0\)相交于\(A\)，\(B\)兩點(diǎn)，則\(|AB|\)可能為（）A.\(1\)B.\(\sqrt{2}\)C.\(2\)D.\(2\sqrt{2}\)答案：BC5.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\(3\sqrt{3}\)，則\(b\)的值為（）A.\(\sqrt{3}\)B.\(2\)C.\(3\)D.\(4\)答案：AC6.已知函數(shù)\(f(x)=\sin(\omegax+\varphi)(\omega\gt0,|\varphi|\lt\frac{\pi}{2})\)的最小正周期為\(\pi\)，其圖象向左平移\(\frac{\pi}{6}\)個(gè)單位后得到函數(shù)\(g(x)=\cos\omegax\)的圖象，則下列說法正確的是（）A.\(\varphi=\frac{\pi}{3}\)B.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{5\pi}{12},0)\)對(duì)稱C.\(f(x)\)在\([0,\frac{\pi}{2}]\)上的值域?yàn)閈([-\frac{1}{2},1]\)D.\(f(x)\)的圖象關(guān)于直線\(x=-\frac{\pi}{12}\)對(duì)稱答案：BCD7.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，且\(S_n=2a_n-1\)，則下列說法正確的是（）A.數(shù)列\(zhòng)(\{a_n\}\)是等比數(shù)列B.\(a_n=2^{n-1}\)C.數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=2^n-1\)D.數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式\(a_n=2n-1\)答案：ABC8.已知函數(shù)\(f(x)=x^3-3x^2+2\)，\(g(x)=kx-2\)，若函數(shù)\(y=f(x)-g(x)\)有三個(gè)不同的零點(diǎn)，則實(shí)數(shù)\(k\)的取值范圍是（）A.\((-4,0)\)B.\((0,4)\)C.\((-4,4)\)D.\((-\infty,-4)\cup(4,+\infty)\)答案：BC9.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.\(a^2+b^2+c^2\geq\frac{1}{3}\)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：ABCD10.已知函數(shù)\(f(x)\)滿足\(f(x+2)=f(x)\)，且\(f(x)\)是偶函數(shù)，當(dāng)\(x\in[0,1]\)時(shí)，\(f(x)=x^2\)，若在區(qū)間\([-1,3]\)內(nèi)，函數(shù)\(g(x)=f(x)-kx-k\)有\(zhòng)(4\)個(gè)零點(diǎn)，則實(shí)數(shù)\(k\)的取值范圍是（）A.\((0,\frac{1}{4})\)B.\((0,\frac{1}{3})\)C.\((0,\frac{1}{2})\)D.\((0,1)\)答案：AB三、判斷題1.若\(a\)，\(b\)為實(shí)數(shù)，則“\(a>b\)”是“\(a^2>b^2\)”的充分不必要條件。（×）2.若向量\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow=\overrightarrow{0}\)。（×）3.函數(shù)\(y=\sinx\)在區(qū)間\([0,\pi]\)上的圖象與\(x\)軸圍成的面積為\(0\)。（×）4.若直線\(l\)與平面\(\alpha\)內(nèi)的無數(shù)條直線垂直，則\(l\perp\alpha\)。（×）5.等比數(shù)列\(zhòng)(\{a_n\}\)的公比\(q>1\)時(shí)，數(shù)列\(zhòng)(\{a_n\}\)為遞增數(shù)列。（×）6.若\(f(x)\)是奇函數(shù)，則\(f(0)=0\)。（×）7.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。（×）8.函數(shù)\(y=\log_2(x^2+1)\)的值域是\([0,+\infty)\)。（√）9.若\(a\)，\(b\)，\(c\)成等差數(shù)列，則\(a+c=2b\)。（√）10.已知函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)可導(dǎo)，若\(f^\prime(x)>0\)，則\(y=f(x)\)在區(qū)間\((a,b)\)上單調(diào)遞增。（√）四、簡答題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\(a_1=1\)代入可得\(2+5d=12\)，解得\(d=2\)。根據(jù)等差數(shù)列通項(xiàng)公式\(a_n=a_1+(n-1)d\)，可得\(a_n=1+2(n-1)=2n-1\)。2.已知函數(shù)\(f(x)=x^3-3x^2+2\)，求函數(shù)\(f(x)\)的單調(diào)區(qū)間。答案：對(duì)\(f(x)=x^3-3x^2+2\)求導(dǎo)得\(f^\prime(x)=3x^2-6x\)。令\(f^\prime(x)=0\)，即\(3x^2-6x=0\)，\(3x(x-2)=0\)，解得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)或\(x>2\)時(shí)，\(f^\prime(x)>0\)，\(f(x)\)單調(diào)遞增；當(dāng)\(0<x<2\)時(shí)，\(f^\prime(x)<0\)，\(f(x)\)單調(diào)遞減。所以\(f(x)\)的單調(diào)遞增區(qū)間為\((-\infty,0)\)和\((2,+\infty)\)，單調(diào)遞減區(qū)間為\((0,2)\)。3.已知圓\(C\)的圓心在直線\(y=x\)上，且過點(diǎn)\((1,0)\)和\((0,\sqrt{2})\)，求圓\(C\)的方程。答案：設(shè)圓\(C\)的圓心坐標(biāo)為\((a,a)\)，半徑為\(r\)。圓的標(biāo)準(zhǔn)方程為\((x-a)^2+(y-a)^2=r^2\)。因?yàn)閳A過點(diǎn)\((1,0)\)和\((0,\sqrt{2})\)，則\((1-a)^2+a^2=r^2\)，\((0-a)^2+(\sqrt{2}-a)^2=r^2\)。由\((1-a)^2+a^2=a^2+(\sqrt{2}-a)^2\)，展開得\(1-2a+a^2+a^2=a^2+2-2\sqrt{2}a+a^2\)