職高數(shù)學(xué)考試題庫及答案

一、單項選擇題（每題2分，共10題）1.若函數(shù)\(y=f(x)\)的定義域是\([1,3]\)，則\(y=f(x+1)\)的定義域是（）A.[0,2]B.[1,3]C.[2,4]D.[-1,1]2.已知向量\(\vec{a}=(1,2)\)，\(\vec=(2,k)\)，若\(\vec{a}\parallel\vec\)，則\(k=(\)\)A.1B.2C.3D.43.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(d=2\)，則\(a_{5}=(\)\)A.9B.10C.11D.124.在\(\triangleABC\)中，\(a=3\)，\(b=4\)，\(\angleC=60^{\circ}\)，則\(c=(\)\)A.\(\sqrt{13}\)B.\(\sqrt{19}\)C.\(\sqrt{37}\)D.\(\sqrt{21}\)5.函數(shù)\(y=\sin(x+\frac{\pi}{3})\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(3\pi\)D.\(4\pi\)6.若\(log_{a}2=m\)，\(log_{a}3=n\)，則\(a^{m+n}=(\)\)A.5B.6C.8D.97.過點\((1,2)\)且斜率為\(3\)的直線方程是（）A.\(y-2=3(x-1)\)B.\(y+2=3(x+1)\)C.\(y-2=-3(x-1)\)D.\(y+2=-3(x+1)\)8.二次函數(shù)\(y=x^{2}-2x-3\)的頂點坐標(biāo)是（）A.\((1,-4)\)B.\((-1,-4)\)C.\((1,4)\)D.\((-1,4)\)9.若\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\frac{\pi}{2})\)，則\(\sin\alpha=(\)\)A.\(\frac{2\sqrt{2}}{3}\)B.\(-\frac{2\sqrt{2}}{3}\)C.\(\frac{\sqrt{2}}{3}\)D.\(-\frac{\sqrt{2}}{3}\)10.集合\(A=\{x|x^{2}-x-2=0\}\)，則\(A=(\)\)A.\(\{-1,2\}\)B.\(\{1,2\}\)C.\(\{-1,-2\}\)D.\(\{1,-2\}\)答案：1.A2.D3.A4.A5.B6.B7.A8.A9.A10.A二、多項選擇題（每題2分，共10題）1.下列函數(shù)中，是奇函數(shù)的有（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\frac{1}{x}\)2.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.\(b_{1}\neqb_{2}\)3.以下數(shù)列是等比數(shù)列的有（）A.\(1,2,4,8,\cdots\)B.\(2,2,2,2,\cdots\)C.\(-1,1,-1,1,\cdots\)D.\(1,3,9,27,\cdots\)4.在\(\triangleABC\)中，根據(jù)正弦定理\(\frac{a}{\sinA}=\frac{\sinB}=\frac{c}{\sinC}\)，下列等式成立的有（）A.\(a\sinB=b\sinA\)B.\(b\sinC=c\sinB\)C.\(a\sinC=c\sinA\)D.\(c\sinA=a\sinC\)5.下列函數(shù)的圖象關(guān)于\(y\)軸對稱的有（）A.\(y=x^{2}\)B.\(y=\cosx\)C.\(y=|x|\)D.\(y=\log_{2}x\)6.向量\(\vec{a}=(x_{1},y_{1})\)，\(\vec=(x_{2},y_{2})\)，則\(\vec{a}+\vec=(\)\)A.\((x_{1}+x_{2},y_{1}+y_{2})\)B.\((x_{1}-x_{2},y_{1}-y_{2})\)C.\((x_{2}+x_{1},y_{2}+y_{1})\)D.\((x_{2}-x_{1},y_{2}-y_{1})\)7.二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)的對稱軸為\(x=-\frac{2a}\)，當(dāng)\(a>0\)時（）A.函數(shù)在\((-\infty,-\frac{2a})\)上單調(diào)遞減B.函數(shù)在\((-\infty,-\frac{2a})\)上單調(diào)遞增C.函數(shù)在\((-\frac{2a},+\infty)\)上單調(diào)遞減D.函數(shù)在\((-\frac{2a},+\infty)\)上單調(diào)遞增8.對于指數(shù)函數(shù)\(y=a^{x}(a>0,a\neq1)\)，當(dāng)\(a>1\)時（）A.函數(shù)在\(R\)上單調(diào)遞增B.函數(shù)在\(R\)上單調(diào)遞減C.當(dāng)\(x>0\)時，\(y>1\)D.當(dāng)\(x<0\)時，\(y>1\)9.已知圓的方程為\((x-a)^{2}+(y-b)^{2}=r^{2}\)，則（）A.圓心坐標(biāo)為\((a,b)\)B.半徑為\(r\)C.當(dāng)\(r=0\)時表示一個點D.當(dāng)\(a=b=0\)時，圓心在原點10.在平行四邊形\(ABCD\)中，設(shè)\(\overrightarrow{AB}=\vec{a}\)，\(\overrightarrow{AD}=\vec\)，則\(\overrightarrow{AC}=(\)\)A.\(\vec{a}+\vec\)B.\(\vec+\vec{a}\)C.\(\overrightarrow{BC}+\overrightarrow{AB}\)D.\(\overrightarrow{CD}+\overrightarrow{DA}\)答案：1.ABD2.AD3.ABCD4.ABCD5.ABC6.AC7.AD8.AC9.ABCD10.AB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)\(y=\frac{1}{x}\)在\((0,+\infty)\)上單調(diào)遞增。（）3.若\(\vec{a}\cdot\vec=0\)，則\(\vec{a}=\vec{0}\)或\(\vec=\vec{0}\)。（）4.等比數(shù)列中，公比\(q\)不能為\(0\)。（）5.直線\(y=kx+b\)的斜率\(k\)不存在時，直線垂直于\(x\)軸。（）6.\(\sin^{2}\alpha+\cos^{2}\alpha=1\)對任意\(\alpha\)都成立。（）7.對數(shù)函數(shù)\(y=\log_{a}x(a>0,a\neq1)\)的定義域為\((0,+\infty)\)。（）8.兩個向量相等當(dāng)且僅當(dāng)它們的模相等。（）9.二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)的圖象開口方向由\(a\)的正負(fù)決定。（）10.等差數(shù)列的通項公式\(a_{n}=a_{1}+(n-1)d\)。（）答案：1.對2.錯3.錯4.對5.對6.對7.對8.錯9.對10.對四、簡答題（每題5分，共4題）1.求函數(shù)\(y=2x^{2}-4x+1\)的最小值。-答案：-對于二次函數(shù)\(y=ax^{2}+bx+c(a\neq0)\)，這里\(a=2\)，\(b=-4\)，\(c=1\)。-根據(jù)頂點坐標(biāo)公式\(x=-\frac{2a}=-\frac{-4}{2\times2}=1\)。-把\(x=1\)代入函數(shù)得\(y=2\times1^{2}-4\times1+1=-1\)，所以函數(shù)最小值為\(-1\)。2.已知向量\(\vec{a}=(3,4)\)，\(\vec=(-2,1)\)，求\(\vec{a}-2\vec\)。-答案：-因為\(\vec=(-2,1)\)，所以\(2\vec=(-4,2)\)。-則\(\vec{a}-2\vec=(3,4)-(-4,2)=(3+4,4-2)=(7,2)\)。3.求等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和公式，其中\(zhòng)(a_{1}\)為首項，\(d\)為公差。-答案：-等差數(shù)列前\(n\)項和\(S_{n}=na_{1}+\frac{n(n-1)}{2}d\)。4.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，求\(\cos\alpha\)。-答案：-因為\(\sin^{2}\alpha+\cos^{2}\alpha=1\)，所以\(\cos\alpha=-\sqrt{1-\sin^{2}\alpha}\)。-把\(\sin\alpha=\frac{3}{5}\)代入得\(\cos\alpha=-\sqrt{1-(\frac{3}{5})^{2}}=-\frac{4}{5}\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=kx+b\)的圖象在\(k>0\)和\(k<0\)時的單調(diào)性。-答案：-當(dāng)\(k>0\)時，函數(shù)\(y=kx+b\)在\(R\)上單調(diào)遞增。-當(dāng)\(k<0\)時，函數(shù)\(y=kx+b\)在\(R\)上單調(diào)遞減。2.討論等比數(shù)列\(zhòng)(\{a_{n}\}\)中，當(dāng)公比\(q=1\)和\(q\neq1\)時，前\(n\)項和公式的區(qū)別。-答案：-當(dāng)\(q=1\)時，\(S_{n}=na_{1}\)。-當(dāng)\(q\neq1\)時，\(S_{n}=\frac{a_{1}(1-q^{n})}{1-q}\)。3.討論在\(\triangleABC\)中，銳角三角形、直角三角形和鈍角三角形的余弦定理的不同表現(xiàn)形式。-答案：-對于\(\triangleABC\)，\(a^{2}=b^{2}+c^{2}-2bc\cosA\)，\(b^{2}=a^{2}+c^{2}-2ac\cosB\)，\(c^{2}=a^{2}+b^{2}-2ab\cosC\)。-在銳角三角形中，\(\cosA>0\)，\(\cosB>0\)，\(\cosC>0\)。-在直角三角形中，例如\(C=90^{\circ}\)時，\(c^{2}=a^{2}+b^{2}\