2025年高一下學(xué)期數(shù)學(xué)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.\(\{2,3,4,5\}\)D.\(\{1,3,4,5\}\)答案：B2.函數(shù)\(y=\sqrt{\log_{0.5}(4x-3)}\)的定義域?yàn)椋ǎ〢.\((\frac{3}{4},1]\)B.\((\frac{3}{4},+\infty)\)C.\((1,+\infty)\)D.\((\frac{3}{4},1)\)答案：A3.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan\alpha=\)（）A.\(\frac{3}{4}\)B.\(-\frac{3}{4}\)C.\(\frac{4}{3}\)D.\(-\frac{4}{3}\)答案：B4.向量\(\overrightarrow{a}=(1,-2)\)，\(\overrightarrow=(3,4)\)，則\(\overrightarrow{a}\)在\(\overrightarrow\)方向上的投影為（）A.\(-1\)B.\(1\)C.\(-\frac{1}{5}\)D.\(\frac{1}{5}\)答案：A5.在等差數(shù)列\(zhòng)(\{a_n\}\)中，若\(a_3+a_5+a_7=15\)，則\(a_5=\)（）A.\(5\)B.\(10\)C.\(15\)D.\(20\)答案：A6.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，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：A7.直線\(l\)過(guò)點(diǎn)\((1,2)\)且與直線\(2x-y+1=0\)垂直，則直線\(l\)的方程為（）A.\(x+2y-5=0\)B.\(x-2y+3=0\)C.\(2x+y-4=0\)D.\(2x-y=0\)答案：A8.已知圓\(C\)：\((x-1)^2+(y-2)^2=4\)，過(guò)點(diǎn)\(P(0,1)\)的直線\(l\)與圓\(C\)相交于\(A\)，\(B\)兩點(diǎn)，當(dāng)\(|AB|\)最小時(shí)，直線\(l\)的方程為（）A.\(x-y+1=0\)B.\(x+y-1=0\)C.\(x-2y+2=0\)D.\(2x-y+1=0\)答案：B9.若\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geqslant1\\x-y\leqslant1\\y\leqslant1\end{cases}\)，則\(z=3x-y\)的最大值為（）A.\(1\)B.\(3\)C.\(5\)D.\(7\)答案：C10.已知\(a=\log_{3}2\)，\(b=\log_{5}3\)，\(c=\log_{7}4\)，則\(a\)，\(b\)，\(c\)的大小關(guān)系為（）A.\(a<b<c\)B.\(a<c<b\)C.\(c<a<b\)D.\(c<b<a\)答案：B二、多項(xiàng)選擇題1.下列函數(shù)中，在\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^2\)B.\(y=\sqrt{x}\)C.\(y=2^x\)D.\(y=\log_{0.5}x\)答案：ABC2.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(2,1)\)，則下列說(shuō)法正確的是（）A.若\(\overrightarrow{a}\parallel\overrightarrow\)，則\(m=\frac{1}{2}\)B.若\(\overrightarrow{a}\perp\overrightarrow\)，則\(m=-2\)C.若\(|\overrightarrow{a}|=|\overrightarrow|\)，則\(m=\pm\sqrt{2}\)D.若\(m=1\)，則\(\overrightarrow{a}\)與\(\overrightarrow\)夾角的余弦值為\(\frac{3\sqrt{10}}{10}\)答案：ABD3.下列關(guān)于等差數(shù)列的說(shuō)法正確的是（）A.若\(\{a_n\}\)是等差數(shù)列，\(a_m=n\)，\(a_n=m\)，則\(a_{m+n}=0\)B.若\(\{a_n\}\)是等差數(shù)列，\(S_n\)是其前\(n\)項(xiàng)和，則\(S_m\)，\(S_{2m}-S_m\)，\(S_{3m}-S_{2m}\)仍成等差數(shù)列C.若\(\{a_n\}\)是等差數(shù)列，\(a_1<0\)，\(d>0\)，則\(S_n\)有最小值D.若\(\{a_n\}\)是等差數(shù)列，\(S_n\)是其前\(n\)項(xiàng)和，\(S_{12}<0\)，\(S_{13}>0\)，則\(n=6\)時(shí)\(S_n\)取得最小值答案：ABCD4.已知函數(shù)\(f(x)=\sin(2x+\varphi)(-\frac{\pi}{2}<\varphi<\frac{\pi}{2})\)的圖象關(guān)于直線\(x=\frac{\pi}{6}\)對(duì)稱(chēng)，則（）A.\(\varphi=\frac{\pi}{6}\)B.\(f(x)\)在\([0,\frac{\pi}{6}]\)上單調(diào)遞增C.\(f(x)\)的圖象向左平移\(\frac{\pi}{6}\)個(gè)單位長(zhǎng)度得到\(g(x)=\cos2x\)的圖象D.若\(f(x)\)在\([0,a]\)上的值域?yàn)閈([-\frac{1}{2},1]\)，則\(\frac{\pi}{6}\leqslanta\leqslant\frac{\pi}{2}\)答案：ABCD5.已知圓\(C_1\)：\(x^2+y^2-2x-4y+4=0\)，圓\(C_2\)：\(x^2+y^2-4x-6y+9=0\)，則（）A.兩圓的圓心距\(d=\sqrt{5}\)B.兩圓相交C.兩圓的公切線有\(zhòng)(2\)條D.兩圓公共弦所在直線方程為\(2x+2y-5=0\)答案：ABCD6.已知\(a\)，\(b\)，\(c\)為正實(shí)數(shù)，且\(a+b+c=1\)，則（）A.\(a^2+b^2+c^2\geqslant\frac{1}{3}\)B.\(ab+bc+ca\leqslant\frac{1}{3}\)C.\(\sqrt{a}+\sqrt+\sqrt{c}\leqslant\sqrt{3}\)D.\(\frac{1}{a}+\frac{1}+\frac{1}{c}\geqslant9\)答案：ABCD7.已知函數(shù)\(f(x)=\begin{cases}2^x-1,x\leqslant0\\x^{\frac{1}{2}},x>0\end{cases}\)，則下列說(shuō)法正確的是（）A.\(f(x)\)的值域?yàn)閈((-1,+\infty)\)B.\(f(x)\)在\(R\)上單調(diào)遞增C.若\(f(a)=1\)，則\(a=1\)D.\(f(x)\)的圖象關(guān)于點(diǎn)\((0,0)\)對(duì)稱(chēng)答案：ABC8.已知直線\(l\)：\(y=kx+m\)與圓\(C\)：\((x-1)^2+(y-2)^2=4\)相交于\(A\)，\(B\)兩點(diǎn)，則（）A.若\(|AB|=2\sqrt{3}\)，則圓心到直線\(l\)的距離為\(1\)B.若\(m=0\)，則\(k\)的取值范圍是\((-\infty,-\frac{2\sqrt{5}}{5})\cup(\frac{2\sqrt{5}}{5},+\infty)\)C.若\(k=1\)，則\(m\)的取值范圍是\((-2-\sqrt{2},-2+\sqrt{2})\)D.若直線\(l\)被圓\(C\)所截得的弦長(zhǎng)最短，則\(k=-\frac{1}{2}\)答案：ABD9.已知\(a\)，\(b\)，\(c\)分別為\(\triangleABC\)三個(gè)內(nèi)角\(A\)，\(B\)，\(C\)的對(duì)邊，且\((a+b)(\sinA-\sinB)=(c-b)\sinC\)，則（）A.\(A=\frac{\pi}{3}\)B.\(a^2=b^2+c^2-bc\)C.若\(b+c=4\)，則\(a\)的最小值為\(2\)D.若\(a=\sqrt{3}\)，則\(b+c\)的最大值為\(2\sqrt{3}\)答案：ABCD10.已知函數(shù)\(f(x)=x^3-3x^2+2\)，則（）A.\(f(x)\)的極大值為\(2\)B.\(f(x)\)的極小值為\(-2\)C.\(f(x)\)在\((-\infty,0)\)上單調(diào)遞增D.\(f(x)\)在\((2,+\infty)\)上單調(diào)遞增答案：ABCD三、判斷題1.空集是任何集合的真子集。（×）2.函數(shù)\(y=\frac{1}{x}\)在其定義域上是減函數(shù)。（×）3.若\(\overrightarrow{a}\cdot\overrightarrow=0\)，則\(\overrightarrow{a}=\overrightarrow{0}\)或\(\overrightarrow=\overrightarrow{0}\)。（×）4.等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和公式\(S_n=\frac{n(a_1+a_n)}{2}\)。（√）5.函數(shù)\(y=A\sin(\omegax+\varphi)\)的最小正周期\(T=\frac{2\pi}{\omega}\)。（√）6.若直線\(l_1\)：\(A_1x+B_1y+C_1=0\)與直線\(l_2\)：\(A_2x+B_2y+C_2=0\)平行，則\(\frac{A_1}{A_2}=\frac{B_1}{B_2}\neq\frac{C_1}{C_2}\)。（√）7.圓\(x^2+y^2=r^2\)的圓心為\((0,0)\)，半徑為\(r\)。（√）8.若\(a>b>0\)，\(c>d>0\)，則\(ac>bd\)。（√）9.函數(shù)\(y=\log_{a}x\)（\(a>0\)且\(a\neq1\)）的圖象恒過(guò)點(diǎn)\((1,0)\)。（√）10.若\(a\)，\(b\)，\(c\)成等比數(shù)列，則\(b^2=ac\)。（√）四、簡(jiǎn)答題1.已知函數(shù)\(f(x)=\log_{2}(x^2-2x-3)\)，求其定義域。答案：要使函數(shù)有意義，則\(x^2-2x-3>0\)，即\((x-3)(x+1)>0\)。解得\(x<-1\)或\(x>3\)。所以函數(shù)\(f(x)\)的定義域?yàn)閈((-\infty,-1)\cup(3,+\infty)\)。2.已知向量\(\overrightarrow{a}=(2,-1)\)，\(\overrightarrow=(m,3)\)，若\(\overrightarrow{a}\perp\overrightarrow\)，求\(m\)的值。答案：因?yàn)閈(\overrightarrow{a}\perp\overrightarrow\)，根據(jù)向量垂直的性質(zhì)，\(\overrightarrow{a}\cdot\overrightarrow=0\)。又\(\overrightarrow{a}\cdot\overrightarrow=2m+(-1)\times3=