高二數(shù)學考試題庫及答案

高二數(shù)學考試題庫及答案

一、單項選擇題（每題2分，共10題）1.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：A2.若向量\(\vec{a}=(1,2)\)，\(\vec=(x,1)\)，且\(\vec{a}\perp\vec\)，則\(x=(\))A.-2B.2C.\(-\frac{1}{2}\)D.\(\frac{1}{2}\)答案：A3.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=2\)，公比\(q=3\)，則\(a_{3}=(\))A.18B.6C.2D.\(\frac{2}{3}\)答案：A4.過點\((1,2)\)且斜率為\(3\)的直線方程為（）A.\(y-2=3(x-1)\)B.\(y+2=3(x+1)\)C.\(y-1=3(x-2)\)D.\(y+1=3(x+2)\)答案：A5.在\(\triangleABC\)中，\(a=3\)，\(b=4\)，\(C=60^{\circ}\)，則\(c=(\))A.\(\sqrt{13}\)B.\(\sqrt{37}\)C.\(\sqrt{21}\)D.\(\sqrt{19}\)答案：A6.橢圓\(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\)的離心率\(e=(\))A.\(\frac{\sqrt{7}}{4}\)B.\(\frac{\sqrt{7}}{3}\)C.\(\frac{3}{4}\)D.\(\frac{4}{3}\)答案：A7.函數(shù)\(y=\lnx\)在\(x=e\)處的切線方程為（）A.\(y=\frac{1}{e}x\)B.\(y=x\)C.\(y=\frac{1}{e}x+1\)D.\(y=\frac{1}{e}x-1\)答案：A8.已知\(f(x)=x^{3}+ax^{2}+bx+c\)，\(f(1)=1\)，\(f(2)=2\)，則\(f(3)=(\))A.3B.4C.5D.6答案：A9.若雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)的漸近線方程為\(y=\pm\frac{3}{4}x\)，則離心率\(e=(\))A.\(\frac{5}{4}\)B.\(\frac{5}{3}\)C.\(\frac{\sqrt{7}}{4}\)D.\(\frac{\sqrt{7}}{3}\)答案：A10.若\(\log_{a}2\lt\log_{a}3\)，則\(a\)的取值范圍是（）A.\((0,1)\)B.\((1,+\infty)\)C.\((0,+\infty)\)D.\((-\infty,0)\)答案：B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)的是（）A.\(y=x^{3}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\frac{1}{x}\)答案：ABD2.已知向量\(\vec{a}=(1,1)\)，\(\vec=(2,-1)\)，則（）A.\(\vec{a}+\vec=(3,0)\)B.\(\vec{a}-\vec=(-1,2)\)C.\(2\vec{a}-\vec=(0,3)\)D.\(\vec{a}\cdot\vec=1\)答案：ABD3.等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和為\(S_{n}\)，若\(a_{1}=1\)，\(d=2\)，則（）A.\(a_{n}=2n-1\)B.\(S_{n}=n^{2}\)C.\(a_{3}=5\)D.\(S_{3}=9\)答案：ABCD4.直線\(l_{1}:y=k_{1}x+b_{1}\)與直線\(l_{2}:y=k_{2}x+b_{2}\)平行的條件是（）A.\(k_{1}=k_{2}\)B.\(b_{1}\neqb_{2}\)C.\(k_{1}\neqk_{2}\)D.\(b_{1}=b_{2}\)答案：AB5.在\(\triangleABC\)中，根據(jù)正弦定理，下列等式成立的是（）A.\(\frac{a}{\sinA}=\frac{\sinB}\)B.\(\frac{a}{\sinB}=\frac{\sinA}\)C.\(\frac{a}{\cosA}=\frac{\cosB}\)D.\(\frac{a}{\cosB}=\frac{\cosA}\)答案：A6.橢圓的標準方程可以是（）A.\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)B.\(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1(a>b>0)\)C.\(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1(a>b>0)\)D.\(\frac{(x-k)^{2}}{b^{2}}+\frac{(y-h)^{2}}{a^{2}}=1(a>b>0)\)答案：ABCD7.函數(shù)\(y=x^{2}-2x-3\)的零點是（）A.\(x=3\)B.\(x=-1\)C.\(x=1\)D.\(x=0\)答案：AB8.對于函數(shù)\(y=f(x)\)，其導數(shù)\(f'(x)\)的幾何意義是（）A.函數(shù)\(y=f(x)\)在某點處的切線斜率B.函數(shù)\(y=f(x)\)的變化率C.函數(shù)\(y=f(x)\)的平均變化率D.函數(shù)\(y=f(x)\)圖象的彎曲程度答案：AB9.若\(a,b\inR\)，則\(a^{2}+b^{2}\geqslant2ab\)成立的條件是（）A.\(a=b\)B.\(a\neqb\)C.\(a,b\inR\)D.\(a,b>0\)答案：C10.下列關(guān)于數(shù)列極限的說法正確的是（）A.\(\lim_{n\rightarrow\infty}\frac{1}{n}=0\)B.\(\lim_{n\rightarrow\infty}n=+\infty\)C.\(\lim_{n\rightarrow\infty}(-1)^{n}\)不存在D.\(\lim_{n\rightarrow\infty}2^{n}=+\infty\)答案：ABCD三、判斷題（每題2分，共10題）1.函數(shù)\(y=\cosx\)是偶函數(shù)。（）答案：正確2.向量\(\vec{a}\)與\(\vec\)平行，則\(\vec{a}=\lambda\vec(\lambda\inR)\)。（）答案：錯誤3.等差數(shù)列的通項公式\(a_{n}=a_{1}+(n-1)d\)。（）答案：正確4.直線\(x=1\)的斜率不存在。（）答案：正確5.在\(\triangleABC\)中，\(a^{2}=b^{2}+c^{2}-2bc\cosA\)。（）答案：正確6.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的長軸長為\(2a\)。（）答案：正確7.函數(shù)\(y=x^{3}\)的單調(diào)遞增區(qū)間是\((-\infty,+\infty)\)。（）答案：正確8.若\(a>b\)，則\(ac^{2}>bc^{2}\)。（）答案：錯誤9.等比數(shù)列\(zhòng)(\{a_{n}\}\)的公比\(q\neq0\)。（）答案：正確10.函數(shù)\(y=\log_{a}x(a>0,a\neq1)\)的圖象恒過定點\((1,0)\)。（）答案：正確四、簡答題（每題5分，共4題）1.求函數(shù)\(y=x^{2}-4x+3\)的頂點坐標。答案：對于二次函數(shù)\(y=ax^{2}+bx+c\)，其頂點橫坐標\(x=-\frac{2a}\)，這里\(a=1\)，\(b=-4\)，則\(x=2\)，把\(x=2\)代入函數(shù)得\(y=4-8+3=-1\)，所以頂點坐標為\((2,-1)\)。2.已知向量\(\vec{a}=(3,4)\)，\(\vec=(-2,1)\)，求\(\vec{a}+\vec\)與\(\vec{a}-\vec\)的坐標。答案：\(\vec{a}+\vec=(3-2,4+1)=(1,5)\)，\(\vec{a}-\vec=(3+2,4-1)=(5,3)\)。3.求等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和\(S_{n}\)的公式（\(a_{1}\)為首項，\(d\)為公差）。答案：\(S_{n}=na_{1}+\frac{n(n-1)}{2}d\)。4.求過點\((2,3)\)且與直線\(y=2x-1\)平行的直線方程。答案：因為平行直線斜率相等，已知直線斜率為\(2\)，設(shè)所求直線方程為\(y=2x+b\)，把點\((2,3)\)代入得\(3=4+b\)，解得\(b=-1\)，所以直線方程為\(y=2x-1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\sinx\)在\([0,2\pi]\)上的單調(diào)性。答案：在\([0,\frac{\pi}{2}]\)上單調(diào)遞增，在\([\frac{\pi}{2},\frac{3\pi}{2}]\)上單調(diào)遞減，在\([\frac{3\pi}{2},2\pi]\)上單調(diào)遞增。2.討論向量\(\vec{a}\)與\(\vec\)垂直的條件。答案：若\(\vec{a}=(x_{1},y_{1})\)，\(\vec=(x_{2},y_{2})\)，則\(\vec{a}\perp\vec\)的條件是\(x_{1}