2o20綿陽(yáng)二診數(shù)學(xué)試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.函數(shù)\(y=\sinx\)的最小正周期是（）A.\(\pi\)B.\(2\pi\)C.\(4\pi\)D.\(\frac{\pi}{2}\)2.若集合\(A=\{1,2,3\}\)，\(B=\{2,3,4\}\)，則\(A\capB=（）\)A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1,2,3\}\)D.\(\{2,3,4\}\)3.直線\(y=2x+1\)的斜率是（）A.\(1\)B.\(2\)C.\(\frac{1}{2}\)D.\(-2\)4.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(3,4)\)，則\(\overrightarrow{a}+\overrightarrow=（）\)A.\((4,6)\)B.\((-2,-2)\)C.\((2,2)\)D.\((3,5)\)5.拋物線\(y^{2}=4x\)的焦點(diǎn)坐標(biāo)是（）A.\((1,0)\)B.\((0,1)\)C.\((2,0)\)D.\((0,2)\)6.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(a_{3}=5\)，則\(a_{2}=（）\)A.\(3\)B.\(4\)C.\(5\)D.\(6\)7.若\(\sin\alpha=\frac{3}{5}\)，且\(\alpha\)是第二象限角，則\(\cos\alpha=（）\)A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)8.函數(shù)\(f(x)=x^{3}\)在\(x=1\)處的導(dǎo)數(shù)是（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)9.不等式\(x^{2}-3x+2\lt0\)的解集是（）A.\(\{x|1\ltx\lt2\}\)B.\(\{x|x\lt1\)或\(x\gt2\}\)C.\(\{x|-1\ltx\lt-2\}\)D.\(\{x|x\lt-1\)或\(x\gt-2\}\)10.已知\(\tan\alpha=2\)，則\(\frac{\sin\alpha}{\cos\alpha}=（）\)A.\(2\)B.\(\frac{1}{2}\)C.\(-2\)D.\(-\frac{1}{2}\)二、多項(xiàng)選擇題（每題2分，共10題）1.以下哪些是基本初等函數(shù)（）A.冪函數(shù)B.指數(shù)函數(shù)C.對(duì)數(shù)函數(shù)D.三角函數(shù)2.下列向量中，與向量\((1,1)\)平行的有（）A.\((2,2)\)B.\((-1,-1)\)C.\((1,-1)\)D.\((-2,-2)\)3.橢圓\(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\)的性質(zhì)正確的是（）A.長(zhǎng)軸長(zhǎng)為\(6\)B.短軸長(zhǎng)為\(4\)C.焦點(diǎn)在\(x\)軸上D.離心率為\(\frac{\sqrt{5}}{3}\)4.下列函數(shù)中，是偶函數(shù)的有（）A.\(y=x^{2}\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=x^{3}\)5.等差數(shù)列\(zhòng)(\{a_{n}\}\)的通項(xiàng)公式\(a_{n}=a_{1}+(n-1)d\)，其中哪些量確定后數(shù)列就確定了（）A.\(a_{1}\)B.\(d\)C.\(n\)D.\(a_{n}\)6.以下哪些點(diǎn)在直線\(y=x+1\)上（）A.\((0,1)\)B.\((1,2)\)C.\((-1,0)\)D.\((2,3)\)7.對(duì)于\(y=\log_{a}x\)（\(a\gt0\)且\(a\neq1\)），下列說(shuō)法正確的是（）A.當(dāng)\(a\gt1\)時(shí)，函數(shù)在\((0,+\infty)\)上單調(diào)遞增B.當(dāng)\(0\lta\lt1\)時(shí)，函數(shù)在\((0,+\infty)\)上單調(diào)遞減C.函數(shù)過(guò)定點(diǎn)\((1,0)\)D.函數(shù)的值域是\(R\)8.已知\(\sin\alpha=\frac{1}{2}\)，則\(\alpha\)可能的值為（）A.\(\frac{\pi}{6}\)B.\(\frac{5\pi}{6}\)C.\(\frac{7\pi}{6}\)D.\(\frac{11\pi}{6}\)9.函數(shù)\(y=2\sinx\)的性質(zhì)正確的是（）A.最大值為\(2\)B.最小值為\(-2\)C.周期為\(2\pi\)D.對(duì)稱軸方程為\(x=k\pi+\frac{\pi}{2}(k\inZ)\)10.直線\(l_{1}:A_{1}x+B_{1}y+C_{1}=0\)與\(l_{2}:A_{2}x+B_{2}y+C_{2}=0\)平行的條件是（）A.\(A_{1}B_{2}-A_{2}B_{1}=0\)B.\(A_{1}C_{2}-A_{2}C_{1}\neq0\)C.\(B_{1}C_{2}-B_{2}C_{1}\neq0\)D.\(\frac{A_{1}}{A_{2}}=\frac{B_{1}}{B_{2}}=\frac{C_{1}}{C_{2}}\)三、判斷題（每題2分，共10題）1.空集是任何集合的子集。（）2.若\(a\gtb\)，則\(a^{2}\gtb^{2}\)。（）3.函數(shù)\(y=\frac{1}{x}\)在定義域內(nèi)是單調(diào)遞減函數(shù)。（）4.向量\(\overrightarrow{a}\)與\(-\overrightarrow{a}\)長(zhǎng)度相等，方向相反。（）5.圓\(x^{2}+y^{2}=r^{2}\)的圓心是\((0,0)\)，半徑是\(r\)。（）6.若\(\sin\alpha=0\)，則\(\alpha=0\)。（）7.等差數(shù)列的前\(n\)項(xiàng)和公式\(S_{n}=na_{1}+\frac{n(n-1)}{2}d\)。（）8.函數(shù)\(y=e^{x}\)的導(dǎo)數(shù)是\(y^\prime=e^{x}\)。（）9.不等式\(x^{2}+1\gt0\)的解集是\(R\)。（）10.直線\(y=kx+b\)（\(k\neq0\)）的斜率為\(k\)，在\(y\)軸上的截距為\(b\)。（）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=3x^{2}+2x+1\)的對(duì)稱軸和頂點(diǎn)坐標(biāo)。-對(duì)稱軸公式為\(x=-\frac{2a}\)，此函數(shù)\(a=3\)，\(b=2\)，則對(duì)稱軸\(x=-\frac{2}{2\times3}=-\frac{1}{3}\)。-把\(x=-\frac{1}{3}\)代入函數(shù)得\(y=3\times(-\frac{1}{3})^{2}+2\times(-\frac{1}{3})+1=\frac{2}{3}\)，頂點(diǎn)坐標(biāo)為\((-\frac{1}{3},\frac{2}{3})\)。2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(d=2\)，求\(a_{5}\)的值。-根據(jù)等差數(shù)列通項(xiàng)公式\(a_{n}=a_{1}+(n-1)d\)。-當(dāng)\(n=5\)，\(a_{1}=1\)，\(d=2\)時(shí)，\(a_{5}=1+(5-1)\times2=1+8=9\)。3.求\(\sin15^{\circ}\)的值。-\(\sin15^{\circ}=\sin(45^{\circ}-30^{\circ})\)。-由兩角差正弦公式\(\sin(A-B)=\sinA\cosB-\cosA\sinB\)，得\(\sin15^{\circ}=\sin45^{\circ}\cos30^{\circ}-\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}\)。4.已知直線\(l\)過(guò)點(diǎn)\((1,2)\)且斜率為\(3\)，求直線\(l\)的方程。-由直線點(diǎn)斜式方程\(y-y_{0}=k(x-x_{0})\)（\((x_{0},y_{0})\)為直線上一點(diǎn)，\(k\)為斜率）。-這里\(x_{0}=1\)，\(y_{0}=2\)，\(k=3\)，則直線\(l\)方程為\(y-2=3(x-1)\)，即\(y=3x-1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\sinx\)與\(y=\cosx\)在\([0,2\pi]\)上的單調(diào)性差異。-\(y=\sinx\)在\([0,\frac{\pi}{2}]\)遞增，\([\frac{\pi}{2},\frac{3\pi}{2}]\)遞減，\([\frac{3\pi}{2},2\pi]\)遞增；\(y=\cosx\)在\([0,\pi]\)遞減，\([\pi,2\pi]\)遞增。二者單調(diào)性區(qū)間不同。2.探討直線與圓的位置關(guān)系有哪些判斷方法。-代數(shù)法：聯(lián)立直線與圓方程，根據(jù)判別式判斷。幾何法：比較圓心到直線距離\(d\)與半徑\(r\)大小，\(d\gtr\)相離，\(d=r\)相切，\(d\ltr\)相交。3.說(shuō)一說(shuō)在等比數(shù)列中，公比\(q\)對(duì)數(shù)列性質(zhì)的影響。-當(dāng)\(|q|\gt1\)，數(shù)列遞增（\(a_{1}\gt0\)）或遞減（\(a_{1}\lt0\)）；當(dāng)\(|q|=1\)，數(shù)列為常數(shù)列；當(dāng)\(0\lt|q|\lt1\)，數(shù)列遞減（\(a_{1}\gt0\)）或遞增（\(a_{1}\lt0\)）；當(dāng)\(q\lt0\)，數(shù)列正負(fù)交替。4.如何根據(jù)函數(shù)圖象判斷函數(shù)的奇偶性？-若函數(shù)圖象關(guān)于原點(diǎn)對(duì)稱，則函數(shù)是奇函數(shù)；若函數(shù)圖象關(guān)于\(y\)軸對(duì)稱，則函數(shù)是偶函數(shù)。奇函數(shù)滿足\(f(-x)=-f(x)\)，偶函數(shù)滿足\(f(-x)=f(x)\)