2025年高考試卷及參考答案數(shù)學(xué)

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(mǎn)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\(i\)為虛數(shù)單位，若復(fù)數(shù)\(z=\frac{1+ai}{2-i}(a\inR)\)的實(shí)部與虛部相等，則\(a=（）\)A.1B.-1C.\(\frac{1}{3}\)D.-\(\frac{1}{3}\)答案：C3.已知\(\vec{a}=(1,2)\)，\(\vec=(-2,m)\)，若\(\vec{a}\parallel\vec\)，則\(2\vec{a}+3\vec=（）\)A.\((-5,-10)\)B.\((-4,-8)\)C.\((-3,-6)\)D.\((-2,-4)\)答案：B4.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\(S_7=（）\)A.28B.42C.56D.14答案：A5.函數(shù)\(f(x)=\frac{\lnx}{x}\)的單調(diào)遞增區(qū)間是（）A.\((0,1)\)B.\((0,e)\)C.\((1,+\infty)\)D.\((e,+\infty)\)答案：B6.已知直線(xiàn)\(l\)過(guò)點(diǎn)\((1,0)\)且垂直于\(x\)軸，若\(l\)被拋物線(xiàn)\(y^2=4ax\)截得的線(xiàn)段長(zhǎng)為\(4\)，則拋物線(xiàn)的焦點(diǎn)坐標(biāo)為（）A.\((1,0)\)B.\((2,0)\)C.\((0,1)\)D.\((0,2)\)答案：A7.一個(gè)四面體的三視圖如圖所示，則該四面體的表面積是（）A.\(1+\sqrt{3}\)B.\(2+\sqrt{3}\)C.\(1+2\sqrt{3}\)D.\(2\sqrt{3}\)答案：B8.若\(\tan\alpha=3\)，則\(\frac{\sin2\alpha}{\cos^2\alpha}\)的值等于（）A.2B.3C.4D.6答案：D9.已知雙曲線(xiàn)\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的離心率為\(\sqrt{5}\)，則其漸近線(xiàn)方程為（）A.\(y=\pm2x\)B.\(y=\pm\frac{1}{2}x\)C.\(y=\pm\frac{1}{3}x\)D.\(y=\pm\frac{1}{4}x\)答案：A10.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列成一個(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：C二、多項(xiàng)選擇題1.下列說(shuō)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\(\frac{1}{a}<\frac{1}\)答案：BD2.已知函數(shù)\(f(x)=\sin(2x+\varphi)(0<\varphi<\frac{\pi}{2})\)的圖象經(jīng)過(guò)點(diǎn)\((0,\frac{3}{5})\)，則（）A.\(f(\frac{\pi}{8})=\frac{\sqrt{2}}{10}\)B.函數(shù)\(f(x)\)在\([-\frac{\pi}{6},\frac{\pi}{3}]\)上單調(diào)遞增C.函數(shù)\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{5\pi}{12},0)\)對(duì)稱(chēng)D.函數(shù)\(f(x)\)的圖象可以由\(g(x)=\sin2x\)的圖象向左平移\(\frac{\pi}{12}\)個(gè)單位長(zhǎng)度得到答案：ACD3.已知圓\(C_1:x^2+y^2-2x-4y+4=0\)與圓\(C_2:x^2+y^2+2x-6y+9=0\)，則（）A.兩圓的圓心距為\(\sqrt{10}\)B.兩圓的外公切線(xiàn)有\(zhòng)(3\)條C.兩圓相交，且公共弦所在的直線(xiàn)方程為\(4x-2y+5=0\)D.兩圓相交，且公共弦的長(zhǎng)度為\(\frac{\sqrt{26}}{2}\)答案：ACD4.已知\(a,b\)為正實(shí)數(shù)，且\(a+b=1\)，則（）A.\(a^2+b^2\geq\frac{1}{2}\)B.\(2^{a-b}>\frac{1}{2}\)C.\(\log_2a+\log_2b\geq-2\)D.\(\sqrt{a}+\sqrt\leq\sqrt{2}\)答案：ABD5.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\(\angleF_1PF_2=60^{\circ}\)，則（）A.當(dāng)\(a=\sqrt{2}b\)時(shí)，滿(mǎn)足條件的點(diǎn)\(P\)有兩個(gè)B.當(dāng)\(a>\sqrt{2}b\)時(shí)，滿(mǎn)足條件的點(diǎn)\(P\)有四個(gè)C.\(\triangleF_1PF_2\)的面積為\(\frac{\sqrt{3}}{3}b^2\)D.\(\overrightarrow{PF_1}\cdot\overrightarrow{PF_2}\)的取值范圍是\([b^2,2b^2)\)答案：ABCD6.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時(shí)，\(f(x)=x(1+x)\)，則（）A.當(dāng)\(x<0\)時(shí)，\(f(x)=-x(1-x)\)B.\(f(-2)=-6\)C.函數(shù)\(f(x)\)的單調(diào)遞增區(qū)間是\((-\infty,-\frac{1}{2})\)和\((\frac{1}{2},+\infty)\)D.函數(shù)\(y=f(x)-\frac{1}{4}\)的零點(diǎn)個(gè)數(shù)是\(3\)答案：ABD7.已知\(a_n\)是等比數(shù)列，公比\(q>1\)，前\(n\)項(xiàng)和為\(S_n\)，若\(a_2+a_3=12\)，\(a_1\cdota_4=27\)，則（）A.\(q=3\)B.\(a_n=3^{n-1}\)C.\(S_n=\frac{3^n-1}{2}\)D.\(\{a_n\}\)的前\(n\)項(xiàng)積\(T_n=3^{\frac{n(n-1)}{2}}\)答案：ABCD8.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\(x=0\)處取得極大值\(2\)C.\(f(x)\)在\((1,+\infty)\)上單調(diào)遞減D.\(f(x)\)在\(x=2\)處取得極小值\(-2\)答案：ABD9.已知拋物線(xiàn)\(y^2=2px(p>0)\)的焦點(diǎn)為\(F\)，過(guò)點(diǎn)\(F\)的直線(xiàn)\(l\)交拋物線(xiàn)于\(A(x_1,y_1)\)，\(B(x_2,y_2)\)兩點(diǎn)，則（）A.\(y_1y_2=-p^2\)B.若弦\(AB\)的中點(diǎn)為\(M(x_0,y_0)\)，則直線(xiàn)\(AB\)的斜率為\(\frac{p}{y_0}\)C.以\(AF\)為直徑的圓與\(y\)軸相切D.若\(\overrightarrow{AF}=2\overrightarrow{FB}\)，則直線(xiàn)\(AB\)的斜率為\(\pm2\sqrt{2}\)答案：ABCD10.已知函數(shù)\(f(x)=\begin{cases}e^x,x\leq0\\\lnx,x>0\end{cases}\)，若\(F(x)=f(x)+\frac{1}{2}x-m\)有兩個(gè)零點(diǎn)，則實(shí)數(shù)\(m\)的取值范圍可以是（）A.\((-\infty,1]\)B.\((-\infty,1)\)C.\([1,+\infty)\)D.\((-\infty,\frac{1}{2})\)答案：BD三、判斷題1.若\(A\)，\(B\)是互斥事件，則\(P(A\cupB)=P(A)+P(B)\)。（）答案：正確2.若向量\(\vec{a}\)，\(\vec\)滿(mǎn)足\(\vec{a}\cdot\vec=0\)，則\(\vec{a}=\vec{0}\)或\(\vec=\vec{0}\)。（）答案：錯(cuò)誤3.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{3},0)\)對(duì)稱(chēng)。（）答案：錯(cuò)誤4.若直線(xiàn)\(l\)垂直于平面\(\alpha\)內(nèi)的無(wú)數(shù)條直線(xiàn)，則\(l\perp\alpha\)。（）答案：錯(cuò)誤5.若\(a>b\)，則\(a^3>b^3\)。（）答案：正確6.函數(shù)\(y=\log_2(x^2+1)\)的值域是\([0,+\infty)\)。（）答案：正確7.等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，若\(a_1>0\)，\(S_5=S_9\)，則\(S_n\)在\(n=7\)時(shí)取得最大值。（）答案：正確8.若\(z=a+bi(a,b\inR)\)，則\(z\)為純虛數(shù)的充要條件是\(a=0\)。（）答案：錯(cuò)誤9.雙曲線(xiàn)\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的漸近線(xiàn)方程為\(y=\pm\frac{a}x\)。（）答案：正確10.若函數(shù)\(f(x)\)在區(qū)間\([a,b]\)上連續(xù)，且\(f(a)f(b)<0\)，則函數(shù)\(f(x)\)在區(qū)間\((a,b)\)內(nèi)至少有一個(gè)零點(diǎn)。（）答案：正確四、簡(jiǎn)答題1.已知數(shù)列\(zhòng)(\{a_n\}\)滿(mǎn)足\(a_1=1\)，\(a_{n+1}=2a_n+1\)。（1）求證數(shù)列\(zhòng)(\{a_n+1\}\)是等比數(shù)列；（2）求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式。答案：（1）由\(a_{n+1}=2a_n+1\)可得\(a_{n+1}+1=2(a_n+1)\)。又\(a_1+1=2\neq0\)，所以\(\frac{a_{n+1}+1}{a_n+1}=2\)，所以數(shù)列\(zhòng)(\{a_n+1\}\)是以\(2\)為首項(xiàng)，\(2\)為公比的等比數(shù)列。（2）由（1）知\(a_n+1=2\times2^{n-1}=2^n\)，所以\(a_n=2^n-1\)。2.已知函數(shù)\(f(x)=x^3-3x^2+2\)。（1）求函數(shù)\(f(x)\)的單調(diào)區(qū)間；（2）求函數(shù)\(f(x)\)在區(qū)間\([-1,3]\)上的最值。答案：（1）對(duì)\(f(x)=x^3-3x^2+2\)求導(dǎo)得\(f^\prime(x)=3x^2-6x=3x(x-2)\)。令\(f^\prime(x)>0\)，解得\(x<0\)或\(x>2\)；令\(f^\prime(x)<0\)，解得\(0<x<2\)。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\((0,2)\)。（2）由（1）知\(f(x)\)在\([-1,0)\)遞增，\((0,2)\)遞減，\((2,3]\)遞增。\(f(-1)=-2\)，\(f(0)=2\)，\(f(2)=-2\)，\(f(3)=2\)。所以\(f(x)