高一數(shù)學(xué)期中考試題及答案

一、單項(xiàng)選擇題1.設(shè)集合\(A=\{x|1<x<3\}\)，\(B=\{x|2<x<4\}\)，則\(A\cupB=\)（）A.\(\{x|1<x<4\}\)B.\(\{x|2<x<3\}\)C.\(\{x|1<x<2\}\)D.\(\{x|3<x<4\}\)答案：A2.函數(shù)\(y=\sqrt{x-1}\)的定義域是（）A.\((-\infty,1]\)B.\([1,+\infty)\)C.\((-\infty,1)\)D.\((1,+\infty)\)答案：B3.已知\(f(x)=x^2-2x\)，則\(f(2)\)的值為（）A.0B.1C.2D.3答案：A4.函數(shù)\(y=2^x\)的圖象經(jīng)過的點(diǎn)是（）A.\((0,0)\)B.\((0,1)\)C.\((1,0)\)D.\((1,\frac{1}{2})\)答案：B5.若\(a=0.3^2\)，\(b=2^{0.3}\)，\(c=\log_2{0.3}\)，則\(a\)，\(b\)，\(c\)的大小關(guān)系是（）A.\(a<b<c\)B.\(c<a<b\)C.\(c<b<a\)D.\(a<c<b\)答案：B6.函數(shù)\(y=\log_{\frac{1}{2}}(x^2-4)\)的單調(diào)遞增區(qū)間是（）A.\((2,+\infty)\)B.\((-\infty,-2)\)C.\((0,+\infty)\)D.\((-\infty,0)\)答案：B7.已知\(\sin\alpha=\frac{3}{5}\)，且\(\alpha\)是第二象限角，則\(\cos\alpha\)的值為（）A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)答案：B8.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(\frac{\pi}{4}\)答案：A9.若\(\tan\alpha=2\)，則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值為（）A.3B.-3C.\(\frac{1}{3}\)D.\(-\frac{1}{3}\)答案：A10.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（）（本題雖無(wú)圖，但按正常考點(diǎn)設(shè)置）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：A二、多項(xiàng)選擇題1.下列集合運(yùn)算正確的是（）A.\(A\capA=A\)B.\(A\cup\varnothing=A\)C.\(A\cap\varnothing=\varnothing\)D.\(A\cupA=A\)答案：ABCD2.下列函數(shù)中，是偶函數(shù)的有（）A.\(y=x^2\)B.\(y=|x|\)C.\(y=x^3\)D.\(y=\frac{1}{x}\)答案：AB3.以下關(guān)于指數(shù)函數(shù)\(y=a^x\)（\(a>0\)且\(a\neq1\)）的說法正確的是（）A.當(dāng)\(0<a<1\)時(shí)，函數(shù)在\(R\)上單調(diào)遞減B.當(dāng)\(a>1\)時(shí)，函數(shù)在\(R\)上單調(diào)遞增C.函數(shù)的圖象恒過點(diǎn)\((0,1)\)D.函數(shù)的值域是\((0,+\infty)\)答案：ABCD4.已知\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x>0\)時(shí)，\(f(x)=x^2-4x\)，則下列說法正確的是（）A.\(f(0)=0\)B.當(dāng)\(x<0\)時(shí)，\(f(x)=-x^2-4x\)C.\(f(x)\)的圖象關(guān)于原點(diǎn)對(duì)稱D.方程\(f(x)=0\)有三個(gè)實(shí)數(shù)根答案：ACD5.對(duì)于函數(shù)\(y=\cosx\)，以下說法正確的是（）A.函數(shù)的最小正周期是\(2\pi\)B.函數(shù)在\([0,\pi]\)上單調(diào)遞減C.函數(shù)的值域是\([-1,1]\)D.函數(shù)圖象關(guān)于\(y\)軸對(duì)稱答案：ABCD6.下列三角函數(shù)值為正的是（）A.\(\sin120^{\circ}\)B.\(\cos(-60^{\circ})\)C.\(\tan135^{\circ}\)D.\(\sin(-210^{\circ})\)答案：ABD7.已知\(\alpha\)是第三象限角，\(\sin\alpha=-\frac{3}{5}\)，則以下正確的是（）A.\(\cos\alpha=-\frac{4}{5}\)B.\(\tan\alpha=\frac{3}{4}\)C.\(\sin(\alpha+\frac{\pi}{2})=-\frac{4}{5}\)D.\(\cos(\alpha-\frac{\pi}{2})=-\frac{3}{5}\)答案：ABC8.函數(shù)\(y=\sin(2x-\frac{\pi}{3})\)的圖象可以由\(y=\sin2x\)的圖象經(jīng)過怎樣的變換得到（）A.向右平移\(\frac{\pi}{6}\)個(gè)單位長(zhǎng)度B.向左平移\(\frac{\pi}{6}\)個(gè)單位長(zhǎng)度C.向右平移\(\frac{\pi}{3}\)個(gè)單位長(zhǎng)度D.先將\(y=\sin2x\)圖象上所有點(diǎn)的橫坐標(biāo)伸長(zhǎng)到原來的\(2\)倍（縱坐標(biāo)不變），再向右平移\(\frac{\pi}{3}\)個(gè)單位長(zhǎng)度答案：A9.已知函數(shù)\(f(x)=\log_a(x+1)\)（\(a>0\)且\(a\neq1\)），若\(f(1)=1\)，則（）A.\(a=2\)B.函數(shù)\(f(x)\)在\((-1,+\infty)\)上單調(diào)遞增C.\(f(3)=2\)D.函數(shù)\(f(x)\)的圖象過點(diǎn)\((0,0)\)答案：ABD10.以下關(guān)于函數(shù)\(y=\tanx\)的說法正確的是（）A.函數(shù)的定義域是\(\{x|x\neqk\pi+\frac{\pi}{2},k\inZ\}\)B.函數(shù)的最小正周期是\(\pi\)C.函數(shù)在區(qū)間\((-\frac{\pi}{2},\frac{\pi}{2})\)上單調(diào)遞增D.函數(shù)圖象關(guān)于點(diǎn)\((\frac{k\pi}{2},0)\)（\(k\inZ\)）對(duì)稱答案：ABCD三、判斷題1.空集是任何集合的子集。（√）2.函數(shù)\(y=\frac{1}{x}\)在其定義域上是單調(diào)遞減函數(shù)。（×）3.若\(a^m=a^n\)（\(a>0\)且\(a\neq1\)），則\(m=n\)。（√）4.函數(shù)\(y=\sinx\)與\(y=\cosx\)的圖象形狀相同，只是位置不同。（√）5.對(duì)數(shù)函數(shù)\(y=\log_ax\)（\(a>0\)且\(a\neq1\)）的定義域是\((0,+\infty)\)。（√）6.若\(f(x)\)是奇函數(shù)，且\(f(0)\)有意義，則\(f(0)=0\)。（√）7.函數(shù)\(y=A\sin(\omegax+\varphi)\)的最大值是\(A\)。（×）（當(dāng)\(A<0\)時(shí)，最大值是\(|A|\)）8.對(duì)于任意角\(\alpha\)，都有\(zhòng)(\sin^2\alpha+\cos^2\alpha=1\)。（√）9.函數(shù)\(y=\cos(x+\frac{\pi}{2})\)與\(y=-\sinx\)是同一個(gè)函數(shù)。（√）10.若\(\tan\alpha=1\)，則\(\alpha=\frac{\pi}{4}\)。（×）（\(\alpha=k\pi+\frac{\pi}{4},k\inZ\)）四、簡(jiǎn)答題1.已知集合\(A=\{x|x^2-3x+2=0\}\)，\(B=\{x|x^2-ax+a-1=0\}\)，若\(B\subseteqA\)，求實(shí)數(shù)\(a\)的值。答案：先解方程\(x^2-3x+2=0\)，即\((x-1)(x-2)=0\)，得\(A=\{1,2\}\)。再解方程\(x^2-ax+a-1=0\)，可得\((x-1)[x-(a-1)]=0\)，則\(x=1\)或\(x=a-1\)，所以\(B=\{1,a-1\}\)。因?yàn)閈(B\subseteqA\)，當(dāng)\(a-1=1\)時(shí)，\(a=2\)；當(dāng)\(a-1=2\)時(shí)，\(a=3\)。所以\(a=2\)或\(a=3\)。2.求函數(shù)\(y=\log_2(x^2-2x-3)\)的定義域。答案：要使函數(shù)有意義，則\(x^2-2x-3>0\)。因式分解得\((x-3)(x+1)>0\)。則有\(zhòng)(\begin{cases}x-3>0\\x+1>0\end{cases}\)或\(\begin{cases}x-3<0\\x+1<0\end{cases}\)。解第一個(gè)不等式組得\(x>3\)，解第二個(gè)不等式組得\(x<-1\)。所以函數(shù)的定義域?yàn)閈((-\infty,-1)\cup(3,+\infty)\)。3.已知\(\sin\alpha=\frac{1}{3}\)，\(\alpha\)是第二象限角，求\(\cos\alpha\)和\(\tan\alpha\)的值。答案：因?yàn)閈(\sin^2\alpha+\cos^2\alpha=1\)，\(\sin\alpha=\frac{1}{3}\)，所以\(\cos^2\alpha=1-\sin^2\alpha=1-(\frac{1}{3})^2=\frac{8}{9}\)。又因?yàn)閈(\alpha\)是第二象限角，\(\cos\alpha<0\)，所以\(\cos\alpha=-\frac{2\sqrt{2}}{3}\)。則\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\frac{1}{3}}{-\frac{2\sqrt{2}}{3}}=-\frac{\sqrt{2}}{4}\)。4.簡(jiǎn)述函數(shù)\(y=\sinx\)到\(y=2\sin(2x+\frac{\pi}{6})\)的圖象變換過程。答案：先將\(y=\sinx\)圖象上所有點(diǎn)的橫坐標(biāo)縮短到原來的\(\frac{1}{2}\)（縱坐標(biāo)不變），得到\(y=\sin2x\)的圖象；再將\(y=\sin2x\)的圖象向左平移\(\frac{\pi}{12}\)個(gè)單位長(zhǎng)度，得到\(y=\sin(2x+\frac{\pi}{6})\)的圖象；最后將\(y=\sin(2x+\frac{\pi}{6})\)圖象上所有點(diǎn)的縱坐標(biāo)伸長(zhǎng)到原來的\(2\)倍（橫坐標(biāo)不變），就得到\(y=2\sin(2x+\frac{\pi}{6})\)的圖象。五、討論題1.已知函數(shù)\(f(x)=x^2+bx+c\)，且\(f(1)=0\)，\(f(3)=0\)。討論函數(shù)\(f(x)\)的單調(diào)性，并求其最值。答案：因?yàn)閈(f(1)=0\)，\(f(3)=0\)，將\(x=1\)，\(x=3\)代入\(f(x)=x^2+bx+c\)可得\(\begin{cases}1+b+c=0\\9+3b+c=0\end{cases}\)，解方程組得\(b=-4\)，\(c=3\)，所以\(f(x)=x^2-4x+3=(x-2)^2-1\)。函數(shù)對(duì)稱軸為\(x=2\)，開口向上。在\((-\infty,2)\)上單調(diào)遞減，在\((2,+\infty)\)上單調(diào)遞增。當(dāng)\(x=2\)時(shí)，函數(shù)