揚(yáng)州高一數(shù)學(xué)期中考試及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.已知集合\(A=\{1,2,3\}\)，\(B=\{2,3,4\}\)，則\(A\capB\)等于（）A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1,2,3\}\)D.\(\{2,3,4\}\)2.函數(shù)\(y=\sqrt{x-1}\)的定義域是（）A.\((1,+\infty)\)B.\([1,+\infty)\)C.\((-\infty,1)\)D.\((-\infty,1]\)3.已知\(\alpha\)終邊過點(diǎn)\((-3,4)\)，則\(\sin\alpha\)的值為（）A.\(\frac{3}{5}\)B.\(\frac{4}{5}\)C.\(-\frac{3}{5}\)D.\(-\frac{4}{5}\)4.若\(a=2^{0.3}\)，\(b=0.3^{2}\)，\(c=\log_{2}0.3\)，則（）A.\(a>b>c\)B.\(b>a>c\)C.\(c>a>b\)D.\(b>c>a\)5.函數(shù)\(y=\log_{2}(x+1)\)的圖象經(jīng)過（）A.\((0,0)\)B.\((0,1)\)C.\((1,0)\)D.\((1,1)\)6.已知\(\cos\alpha=\frac{1}{3}\)，且\(\alpha\)是第四象限角，則\(\sin\alpha\)的值為（）A.\(\frac{2\sqrt{2}}{3}\)B.\(-\frac{2\sqrt{2}}{3}\)C.\(\frac{2}{3}\)D.\(-\frac{2}{3}\)7.函數(shù)\(y=2\sin(2x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(4\pi\)8.若\(f(x)\)是奇函數(shù)，且\(f(1)=2\)，則\(f(-1)\)等于（）A.\(2\)B.\(-2\)C.\(1\)D.\(-1\)9.已知\(\tan\alpha=2\)，則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值為（）A.\(3\)B.\(\frac{1}{3}\)C.\(2\)D.\(\frac{1}{2}\)10.函數(shù)\(y=\frac{1}{x-1}\)在區(qū)間\([2,3]\)上的最大值是（）A.\(1\)B.\(\frac{1}{2}\)C.\(\frac{1}{3}\)D.\(-\frac{1}{2}\)二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，是偶函數(shù)的有（）A.\(y=x^{2}\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=|x|\)2.以下哪些區(qū)間是函數(shù)\(y=\sinx\)的單調(diào)遞增區(qū)間（）A.\([-\frac{\pi}{2},\frac{\pi}{2}]\)B.\([\frac{\pi}{2},\frac{3\pi}{2}]\)C.\([-\frac{3\pi}{2},-\frac{\pi}{2}]\)D.\([0,\pi]\)3.已知集合\(M=\{x|x^{2}-3x+2=0\}\)，\(N=\{1,2\}\)，則（）A.\(M=N\)B.\(M\subseteqN\)C.\(N\subseteqM\)D.\(M\capN=\varnothing\)4.下列函數(shù)中，值域是\((0,+\infty)\)的有（）A.\(y=2^{x}\)B.\(y=\sqrt{x}\)C.\(y=\log_{2}x\)D.\(y=\frac{1}{x^{2}}\)5.若\(a>b>0\)，則下列不等式成立的是（）A.\(a^{2}>b^{2}\)B.\(\frac{1}{a}<\frac{1}\)C.\(\log_{2}a>\log_{2}b\)D.\(a^{3}>b^{3}\)6.函數(shù)\(y=\log_{a}x\)（\(a>0\)且\(a\neq1\)）的圖象經(jīng)過的定點(diǎn)是（）A.\((1,0)\)B.\((0,1)\)C.當(dāng)\(a>1\)時(shí)過\((1,0)\)D.當(dāng)\(0<a<1\)時(shí)過\((1,0)\)7.已知\(\alpha\)是第二象限角，\(\sin\alpha=\frac{3}{5}\)，則（）A.\(\cos\alpha=-\frac{4}{5}\)B.\(\tan\alpha=-\frac{3}{4}\)C.\(\cos\alpha=\frac{4}{5}\)D.\(\tan\alpha=\frac{3}{4}\)8.下列函數(shù)中，與函數(shù)\(y=x\)是同一函數(shù)的有（）A.\(y=\sqrt{x^{2}}\)B.\(y=\frac{x^{2}}{x}\)C.\(y=\sqrt[3]{x^{3}}\)D.\(y=(\sqrt{x})^{2}\)9.對(duì)于函數(shù)\(y=3\sin(2x-\frac{\pi}{6})\)，下列說法正確的是（）A.圖象關(guān)于直線\(x=\frac{\pi}{3}\)對(duì)稱B.圖象關(guān)于點(diǎn)\((\frac{\pi}{6},0)\)對(duì)稱C.最小正周期是\(\pi\)D.在區(qū)間\([0,\frac{\pi}{2}]\)上單調(diào)遞增10.若函數(shù)\(f(x)\)滿足\(f(x+2)=f(x)\)，則\(f(x)\)的周期可能是（）A.\(2\)B.\(4\)C.\(6\)D.\(8\)三、判斷題（每題2分，共10題）1.空集是任何集合的子集。（）2.函數(shù)\(y=\tanx\)的定義域是\(x\neqk\pi+\frac{\pi}{2},k\inZ\)。（）3.若\(a>b\)，則\(a^{2}>b^{2}\)。（）4.函數(shù)\(y=\log_{a}x\)（\(a>0\)且\(a\neq1\)）在\((0,+\infty)\)上一定是增函數(shù)。（）5.若\(f(x)\)是奇函數(shù)，\(f(0)\)有意義，則\(f(0)=0\)。（）6.函數(shù)\(y=\sinx\)與\(y=\cosx\)的圖象形狀相同，只是位置不同。（）7.集合\(A=\{x|x^{2}-1=0\}\)的子集個(gè)數(shù)是\(4\)個(gè)。（）8.函數(shù)\(y=2x+1\)是奇函數(shù)。（）9.若\(\sin\alpha=\sin\beta\)，則\(\alpha=\beta\)。（）10.函數(shù)\(y=x^{3}\)在\(R\)上是單調(diào)遞增函數(shù)。（）四、簡答題（每題5分，共4題）1.求函數(shù)\(y=\sqrt{3-2x-x^{2}}\)的定義域。答案：要使根式有意義，則\(3-2x-x^{2}\geq0\)，即\(x^{2}+2x-3\leq0\)，因式分解得\((x+3)(x-1)\leq0\)，解得\(-3\leqx\leq1\)，定義域?yàn)閈([-3,1]\)。2.已知\(\sin\alpha=\frac{1}{2}\)，且\(\alpha\)是第一象限角，求\(\cos\alpha\)和\(\tan\alpha\)的值。答案：因?yàn)閈(\sin^{2}\alpha+\cos^{2}\alpha=1\)，\(\sin\alpha=\frac{1}{2}\)，\(\alpha\)是第一象限角，所以\(\cos\alpha=\sqrt{1-\sin^{2}\alpha}=\frac{\sqrt{3}}{2}\)，\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\sqrt{3}}{3}\)。3.求函數(shù)\(y=\log_{2}(x^{2}-3x+2)\)的單調(diào)遞增區(qū)間。答案：先求定義域\(x^{2}-3x+2>0\)，得\(x>2\)或\(x<1\)。令\(t=x^{2}-3x+2\)，其對(duì)稱軸為\(x=\frac{3}{2}\)，在\((2,+\infty)\)上\(t\)遞增，又\(y=\log_{2}t\)遞增，所以單調(diào)遞增區(qū)間是\((2,+\infty)\)。4.已知函數(shù)\(f(x)=x^{2}+2ax+1\)在區(qū)間\([-1,2]\)上的最小值為\(f(2)\)，求\(a\)的取值范圍。答案：函數(shù)\(f(x)\)圖象開口向上，對(duì)稱軸為\(x=-a\)。因?yàn)樵趨^(qū)間\([-1,2]\)上最小值為\(f(2)\)，說明對(duì)稱軸\(-a\geq2\)，即\(a\leq-2\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=a^{x}\)（\(a>0\)且\(a\neq1\)）與\(y=\log_{a}x\)（\(a>0\)且\(a\neq1\)）的關(guān)系。答案：二者互為反函數(shù)。圖象關(guān)于直線\(y=x\)對(duì)稱。\(y=a^{x}\)定義域?yàn)閈(R\)，值域?yàn)閈((0,+\infty)\)；\(y=\log_{a}x\)定義域?yàn)閈((0,+\infty)\)，值域?yàn)閈(R\)。單調(diào)性一致，\(a>1\)時(shí)都遞增，\(0<a<1\)時(shí)都遞減。2.結(jié)合函數(shù)圖象，討論\(y=\sinx\)與\(y=\cosx\)在\([0,2\pi]\)上的交點(diǎn)情況。答案：令\(\sinx=\cosx\)，即\(\tanx=1\)，在\([0,2\pi]\)上，\(x=\frac{\pi}{4}\)或\(x=\frac{5\pi}{4}\)。從圖象看，二者在\([0,2\pi]\)上有兩個(gè)交點(diǎn)\((\frac{\pi}{4},\frac{\sqrt{2}}{2})\)和\((\frac{5\pi}{4},-\frac{\sqrt{2}}{2})\)。3.討論如何根據(jù)函數(shù)單調(diào)性定義證明函數(shù)\(f(x)=x^{3}\)在\(R\)上是增函數(shù)。答案：設(shè)\(x_{1},x_{2}\inR\)且\(x_{1}<x_{2}\)，則\(f(x_{1})-f(x_{2})=x_{1}^{3}-x_{2}^{3}=(x_{1}-x_{2})(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2})\)。\(x_{1}-x_{2}<0\)，\(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}>0\)，所以\(f(x_{1})<f(x_{2})\)，得證\(f(x)\)在\(R\)上遞增。4.討論函數(shù)\(y=A\sin(\omegax+\varphi)\)（\(A>0\)，\(\omega>0\)）中\(zhòng)(A\)、\(\omega\)、\(\varphi\)對(duì)函數(shù)圖象的影響。答案：\(A\)影響函數(shù)的振幅，\(A\)越大，圖象縱向拉伸程度越大；\(\omega\)影響周期，\(T=\frac{2\pi}{\omega}\)，\(\omega\)越大周期越小，圖象橫向壓縮；\(\varphi\)影響相位，決定圖象左右平移，\(\varphi>0\)時(shí)向左平移\(\frac{\varphi}{\omega}\)個(gè)