高一半期考試題及答案

一、單項選擇題（每題2分，共10題）1.已知集合\(A=\{1,2,3\}\)，\(B=\{2,3,4\}\)，則\(A\capB=\)（）A.\(\{1\}\)B.\(\{2,3\}\)C.\(\{1,2,3,4\}\)D.\(\varnothing\)答案：B2.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.若\(a=\log_{2}3\)，\(b=\log_{3}2\)，\(c=\log_{2}\frac{1}{3}\)，則（）A.\(a>b>c\)B.\(b>a>c\)C.\(a>c>b\)D.\(c>a>b\)答案：A4.函數(shù)\(y=3\sin(2x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(\frac{2\pi}{3}\)答案：A5.已知向量\(\vec{a}=(1,2)\)，\(\vec=(x,1)\)，若\(\vec{a}\perp\vec\)，則\(x=\)（）A.\(-2\)B.\(-\frac{1}{2}\)C.\(\frac{1}{2}\)D.\(2\)答案：A6.在等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(d=2\)，則\(a_{5}=\)（）A.\(9\)B.\(11\)C.\(13\)D.\(15\)答案：A7.直線\(l:y=kx+1\)與圓\(C:x^{2}+y^{2}=1\)的位置關(guān)系是（）A.相交B.相切C.相離D.與\(k\)的值有關(guān)答案：A8.若\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\cos\alpha=\)（）A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)答案：B9.已知函數(shù)\(f(x)=x^{2}+2x-3\)，則\(f(1)=\)（）A.\(0\)B.\(1\)C.\(2\)D.\(3\)答案：A10.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=2\)，\(q=3\)，則\(a_{3}=\)（）A.\(18\)B.\(12\)C.\(6\)D.\(2\)答案：A二、多項選擇題（每題2分，共10題）1.下列函數(shù)是奇函數(shù)的是（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\frac{1}{x}\)答案：ABD2.下列向量中與向量\(\vec{a}=(1,2)\)平行的向量是（）A.\((2,4)\)B.\((-1,-2)\)C.\((\frac{1}{2},1)\)D.\((-2,-3)\)答案：AB3.以下關(guān)于對數(shù)函數(shù)\(y=\log_{a}x(a>0,a\neq1)\)的說法正確的是（）A.當(dāng)\(a>1\)時，函數(shù)在\((0,+\infty)\)上單調(diào)遞增B.當(dāng)\(0<a<1\)時，函數(shù)在\((0,+\infty)\)上單調(diào)遞減C.函數(shù)圖象恒過點(diǎn)\((1,0)\)D.函數(shù)定義域為\(R\)答案：ABC4.在\(\triangleABC\)中，\(a,b,c\)分別為角\(A,B,C\)所對的邊，若\(A=60^{\circ}\)，\(a=\sqrt{3}\)，根據(jù)余弦定理可以得到（）A.\(b^{2}+c^{2}-bc=3\)B.\(b^{2}+c^{2}=3+bc\)C.\(b^{2}+c^{2}-2bc=3\)D.\(b^{2}+c^{2}=3+2bc\)答案：AB5.下列等式正確的是（）A.\(\sin^{2}\alpha+\cos^{2}\alpha=1\)B.\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\)C.\(\sin(A+B)=\sinA+\sinB\)D.\(\cos(A-B)=\cosA\cosB+\sinA\sinB\)答案：ABD6.對于二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)，其對稱軸為（）A.\(x=-\frac{2a}\)B.\(x=\frac{2a}\)C.當(dāng)\(a>0\)時，函數(shù)在\(x=-\frac{2a}\)處取得最小值D.當(dāng)\(a<0\)時，函數(shù)在\(x=-\frac{2a}\)處取得最大值答案：ACD7.若直線\(l_{1}:y=k_{1}x+b_{1}\)與直線\(l_{2}:y=k_{2}x+b_{2}\)平行，則（）A.\(k_{1}=k_{2}\)B.\(b_{1}=b_{2}\)C.\(k_{1}k_{2}=-1\)D.\(k_{1}=k_{2}\)且\(b_{1}\neqb_{2}\)答案：AD8.已知數(shù)列\(zhòng)(\{a_{n}\}\)是等比數(shù)列，公比為\(q\)，則（）A.若\(q>1\)，數(shù)列\(zhòng)(\{a_{n}\}\)單調(diào)遞增B.若\(0<q<1\)，數(shù)列\(zhòng)(\{a_{n}\}\)單調(diào)遞減C.若\(a_{1}<0,q>1\)，則數(shù)列\(zhòng)(\{a_{n}\}\)單調(diào)遞減D.若\(a_{1}>0,0<q<1\)，則數(shù)列\(zhòng)(\{a_{n}\}\)單調(diào)遞減答案：CD9.下列函數(shù)在\((0,+\infty)\)上是增函數(shù)的是（）A.\(y=x^{2}\)B.\(y=\log_{2}x\)C.\(y=\frac{1}{x}\)D.\(y=2^{x}\)答案：ABD10.對于函數(shù)\(y=A\sin(\omegax+\varphi)(A>0,\omega>0)\)，以下說法正確的是（）A.\(A\)決定函數(shù)的振幅B.\(\omega\)決定函數(shù)的周期\(T=\frac{2\pi}{\omega}\)C.\(\varphi\)決定函數(shù)的初相D.函數(shù)的圖象可由\(y=\sinx\)平移伸縮得到答案：ABCD三、判斷題（每題2分，共10題）1.空集是任何集合的子集。（）答案：對2.函數(shù)\(y=2^{x}\)與\(y=\log_{2}x\)的圖象關(guān)于直線\(y=x\)對稱。（）答案：對3.若\(\vec{a}\cdot\vec=0\)，則\(\vec{a}=\vec{0}\)或者\(yùn)(\vec=\vec{0}\)。（）答案：錯4.在等差數(shù)列\(zhòng)(\{a_{n}\}\)中，若\(m+n=p+q\)，則\(a_{m}+a_{n}=a_{p}+a_{q}\)。（）答案：對5.直線的傾斜角\(\theta\)的范圍是\([0,\pi)\)。（）答案：對6.函數(shù)\(y=\cosx\)在\((0,\pi)\)上是減函數(shù)。（）答案：對7.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{n}=a_{1}q^{n-1}\)。（）答案：對8.對于函數(shù)\(y=f(x)\)，若\(f(-x)=f(x)\)，則函數(shù)為偶函數(shù)。（）答案：對9.向量\(\vec{a}-\vec\)的相反向量是\(\vec-\vec{a}\)。（）答案：對10.圓\(x^{2}+y^{2}=r^{2}\)的圓心為\((0,0)\)，半徑為\(r\)。（）答案：對四、簡答題（每題5分，共4題）1.求函數(shù)\(y=\frac{1}{x-1}\)的漸近線。答案：對于函數(shù)\(y=\frac{1}{x-1}\)，當(dāng)\(x\rightarrow1\)時，\(y\rightarrow\infty\)，所以\(x=1\)是其垂直漸近線；當(dāng)\(x\rightarrow\pm\infty\)時，\(y\rightarrow0\)，所以\(y=0\)是其水平漸近線。2.已知向量\(\vec{a}=(2,3)\)，\(\vec=(-1,2)\)，求\(\vec{a}+\vec\)與\(\vec{a}-\vec\)。答案：\(\vec{a}+\vec=(2-1,3+2)=(1,5)\)，\(\vec{a}-\vec=(2+1,3-2)=(3,1)\)。3.若\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\frac{\pi}{2})\)，求\(\sin\alpha\)的值。答案：因為\(\sin^{2}\alpha+\cos^{2}\alpha=1\)，\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\frac{\pi}{2})\)，所以\(\sin\alpha=\sqrt{1-\cos^{2}\alpha}=\sqrt{1-(\frac{1}{3})^{2}}=\frac{2\sqrt{2}}{3}\)。4.求等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和公式\(S_{n}\)。答案：對于等差數(shù)列\(zhòng)(\{a_{n}\}\)，首項\(a_{1}\)，公差\(d\)，\(S_{n}=na_{1}+\frac{n(n-1)}{2}d\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=x^{3}-3x^{2}\)的單調(diào)性。答案：對\(y=x^{3}-3x^{2}\)求導(dǎo)得\(y'=3x^{2}-6x=3x(x-2)\)。令\(y'=0\)，得\(x=0\)或\(x=2\)。當(dāng)\(x<0\)或\(x>2\)時，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(0<x<2\)時，\(y'<0\)，函數(shù)單調(diào)遞減。2.討論等比數(shù)列\(zhòng)(\{a_{n}\}\)中，當(dāng)公比\(q<0\)時，數(shù)列的各項符號規(guī)律。答案：當(dāng)公比\(q<0\)時，等比數(shù)列\(zhòng)(\{a_{n}\}\)的各項符號正負(fù)交替。因為\(a_{n}=a_{1}q^{n-1}\)，當(dāng)\(n\)為奇數(shù)時，若\(a_{1}>0\)，則\(a_{n}\)與\(q\)異號；若\(a_{1}<0\)，則\(a_{n}\)與\(q\)同號。當(dāng)\(n\)為偶數(shù)時，情況相反。3.討論直線\(y=kx+b\)中\(zhòng)(k\)和\(b\)的幾何意義。答案：在直線\(y=kx+b\)中，\(k\)表示直線的斜率，它決定直線的傾斜程度，\(k=\tan\theta\)（\(\theta\)為直線傾斜角）；\(b\)表示