2025年新課標(biāo)理試題及答案

一、單項(xiàng)選擇題1.已知集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|0<x<5,x\inN\}\)，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.設(shè)復(fù)數(shù)\(z\)滿足\((1+i)z=2i\)，則\(\vertz\vert=（）A.\(\frac{1}{2}\)B.\(\frac{\sqrt{2}}{2}\)C.\(\sqrt{2}\)D.2答案：C3.執(zhí)行如圖所示的程序框圖，若輸入\(n=10\)，則輸出的\(S=（）A.\(\frac{5}{11}\)B.\(\frac{10}{11}\)C.\(\frac{36}{55}\)D.\(\frac{72}{55}\)答案：A4.已知\(\sin\alpha+\cos\alpha=\frac{1}{5}\)，\(\alpha\in(0,\pi)\)，則\(\tan\alpha=（）A.\(-\frac{4}{3}\)B.\(-\frac{3}{4}\)C.\(\frac{3}{4}\)D.\(\frac{4}{3}\)答案：A5.設(shè)\(F_1,F_2\)是橢圓\(E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的左、右焦點(diǎn)，\(P\)為直線\(x=\frac{3a}{2}\)上一點(diǎn)，\(\triangleF_2PF_1\)是底角為\(30^{\circ}\)的等腰三角形，則\(E\)的離心率為（）A.\(\frac{1}{2}\)B.\(\frac{2}{3}\)C.\(\frac{3}{4}\)D.\(\frac{4}{5}\)答案：C6.已知函數(shù)\(f(x)=2\sin(\omegax+\varphi)(\omega>0)\)的圖象關(guān)于直線\(x=\frac{\pi}{12}\)對(duì)稱，且\(f(\frac{\pi}{3})=0\)，則\(\omega\)的最小值為（）A.2B.4C.6D.8答案：A7.某幾何體的三視圖如圖所示，則該幾何體的體積為（）A.\(16+8\pi\)B.\(8+8\pi\)C.\(16+16\pi\)D.\(8+16\pi\)答案：A8.已知\(a=\log_36\)，\(b=\log_510\)，\(c=\log_714\)，則\(a,b,c\)的大小關(guān)系為（）A.\(c>b>a\)B.\(b>c>a\)C.\(a>c>b\)D.\(a>b>c\)答案：D9.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時(shí)，\(f(x)=x(1+x)\)，則當(dāng)\(x<0\)時(shí)，\(f(x)=（）A.\(x(1+x)\)B.\(-x(1+x)\)C.\(x(1-x)\)D.\(-x(1-x)\)答案：C10.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\(\sqrt{3}\)，則\(C\)的漸近線方程為（）A.\(y=\pm\sqrt{2}x\)B.\(y=\pm\frac{\sqrt{2}}{2}x\)C.\(y=\pm2x\)D.\(y=\pm\frac{1}{2}x\)答案：A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.已知向量\(\vec{a}=(1,2)\)，\(\vec=(m,1)\)，若\(\vec{a}\parallel\vec\)，則實(shí)數(shù)\(m\)的值為（）A.\(\frac{1}{2}\)B.\(-\frac{1}{2}\)C.2D.-2答案：A3.已知函數(shù)\(f(x)=\sin(2x+\varphi)\)，其中\(zhòng)(\varphi\in(-\pi,\pi)\)，若\(f(x)\leq\vertf(\frac{\pi}{6})\vert\)對(duì)\(x\inR\)恒成立，且\(f(\frac{\pi}{2})>f(\pi)\)，則\(\varphi\)的值可以是（）A.\(-\frac{\pi}{6}\)B.\(\frac{\pi}{6}\)C.\(\frac{\pi}{3}\)D.\(\frac{7\pi}{6}\)答案：BD4.已知數(shù)列\(zhòng)(\{a_n\}\)是等差數(shù)列，其前\(n\)項(xiàng)和為\(S_n\)，若\(a_3=6\)，\(S_3=12\)，則（）A.\(a_n=2n\)B.\(a_n=3n-3\)C.\(S_n=n^2+n\)D.\(S_n=\frac{3n^2-3n}{2}\)答案：AC5.已知直線\(l:y=kx+1\)與圓\(C:x^2+y^2-2x-3=0\)相交于\(A,B\)兩點(diǎn)，則弦長(zhǎng)\(\vertAB\vert\)的值可能為（）A.2B.3C.4D.5答案：ABC6.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（）A.\(f(x)\)是周期函數(shù)B.\(f(3)=0\)C.\(f(x)=f(x+4)\)D.\(f(x+7)\)是奇函數(shù)答案：ACD7.已知\(a,b\inR\)，且\(a+b=1\)，則下列不等式一定成立的是（）A.\(a^2+b^2\geq\frac{1}{2}\)B.\(ab\leq\frac{1}{4}\)C.\(\sqrt{a}+\sqrt\leq\sqrt{2}\)D.\(\frac{1}{a}+\frac{1}\geq4\)答案：ABC8.已知函數(shù)\(f(x)=x^3-3x\)，則（）A.\(f(x)\)是奇函數(shù)B.\(f(x)\)在\(x=1\)處取得極大值C.\(f(x)\)在\((-1,1)\)上單調(diào)遞增D.\(y=f(x)\)的圖象關(guān)于點(diǎn)\((0,0)\)對(duì)稱答案：ABCD9.已知橢圓\(C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的左、右焦點(diǎn)分別為\(F_1,F_2\)，點(diǎn)\(P\)在橢圓\(C\)上，\(\angleF_1PF_2=60^{\circ}\)，且\(\triangleF_1PF_2\)的面積為\(\sqrt{3}\)，則下列說(shuō)法正確的是（）A.\(b=1\)B.\(a=2\)C.\(\triangleF_1PF_2\)的周長(zhǎng)為\(4+2\sqrt{3}\)D.橢圓\(C\)的離心率為\(\frac{1}{2}\)答案：ACD10.已知函數(shù)\(f(x)=e^x-ax\)有兩個(gè)零點(diǎn)\(x_1,x_2(x_1<x_2)\)，則（）A.\(a>e\)B.\(x_1+x_2>2\)C.\(x_1x_2>1\)D.\(f(x)\)在\((-\infty,\lna)\)上單調(diào)遞減答案：ABD三、判斷題1.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\(2\)。（）答案：錯(cuò)誤2.若\(\sin\alpha=\frac{1}{2}\)，則\(\alpha=\frac{\pi}{6}\)。（）答案：錯(cuò)誤3.已知\(a,b\inR\)，則“\(a>b\)”是“\(a^2>b^2\)”的充分不必要條件。（）答案：錯(cuò)誤4.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)有零點(diǎn)，則\(f(a)f(b)<0\)。（）答案：錯(cuò)誤5.已知數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n=n^2+1\)，則\(a_n=2n-1\)。（）答案：錯(cuò)誤6.若向量\(\vec{a}\cdot\vec=0\)，則\(\vec{a}\perp\vec\)。（）答案：錯(cuò)誤（當(dāng)\(\vec{a}=\vec{0}\)或\(\vec=\vec{0}\)時(shí)不成立）7.橢圓\(\frac{x^2}{4}+\frac{y^2}{3}=1\)的離心率\(e=\frac{1}{2}\)。（）答案：正確8.函數(shù)\(y=\cos^2x-\sin^2x\)的最小正周期為\(\pi\)。（）答案：正確9.若直線\(l\)垂直于平面\(\alpha\)內(nèi)的無(wú)數(shù)條直線，則\(l\perp\alpha\)。（）答案：錯(cuò)誤10.已知函數(shù)\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\((0,+\infty)\)上單調(diào)遞減，則\(f(-2)>f(1)\)。（）答案：正確四、簡(jiǎn)答題1.求函數(shù)\(y=\sin^2x+\sqrt{3}\sinx\cosx\)的最小正周期和單調(diào)遞增區(qū)間。答案：先化簡(jiǎn)\(y=\frac{1-\cos2x}{2}+\frac{\sqrt{3}}{2}\sin2x=\sin(2x-\frac{\pi}{6})+\frac{1}{2}\)。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\([k\pi-\frac{\pi}{6},k\pi+\frac{\pi}{3}],k\inZ\)。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_3=5\)，\(S_5=25\)，求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式。答案：設(shè)等差數(shù)列\(zhòng)(\{a_n\}\)的首項(xiàng)為\(a_1\)，公差為\(d\)。由\(a_3=5\)得\(a_1+2d=5\)，由\(S_5=25\)得\(5a_1+\frac{5\times4}{2}d=25\)，即\(a_1+2d=5\)。聯(lián)立解得\(a_1=1\)，\(d=2\)。所以\(a_n=a_1+(n-1)d=1+2(n-1)=2n-1\)。3.已知圓\(C\)的圓心在直線\(y=x\)上，且與\(x\)軸相切于點(diǎn)\((1,0)\)，求圓\(C\)的方程。答案：因?yàn)閳A\(C\)與\(x\)軸相切于點(diǎn)\((1,0)\)，且圓心在直線\(y=x\)上，所以圓心坐標(biāo)為\((1,1)\)，半徑\(r=1\)。根據(jù)圓的標(biāo)準(zhǔn)方程\((x-a)^2+(y-b)^2=r^2\)（\((a,b)\)為圓心坐標(biāo)），可得圓\(C\)的方程為\((x-1)^2+(y-1)^2=1\)。4.已知函數(shù)\(f(x)=x^3-3x^2+2\)，求函數(shù)\(f(x)\)的極值。答案：對(duì)\(f(x)\)求導(dǎo)得\(f^\prime(x)=3x^2-6x=3x(x-2)\)。令\(f^\prime(x)=0\)，解得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)時(shí)，\(f^\prime(x)>0\)，\(f(x)\)遞增；當(dāng)\(0<x<2\)時(shí)，\(f^\prime(x)<0\)，\(f(x)\)遞減；當(dāng)\(x>2\)時(shí)，\(f^\prime(x)>0\)，\(f(x)\)遞增。所以極大值\(f(0)=2\)，極小值\(f(2)=-2\)。五、討論題1.討論直線\(y=kx+1\)與橢圓\(\frac{x^2}{4}+\frac{y^2}{3}=1\)的位置關(guān)系。答案：將\(y=kx+1\)代入橢圓方程\(\frac{x^2}{4}+\frac{y^2}{3}=1\)，得到\(\frac{x^2}{4}+\frac{(kx+1)^2}{