平遠(yuǎn)高考數(shù)學(xué)試題及答案

一、單項選擇題(每題2分,共10題)1.函數(shù)\(y=x^2\)的導(dǎo)數(shù)是(）A.\(2x\)B.\(x^2\)C.\(2\)D.\(0\)2.集合\(A=\{1,2,3\}\),\(B=\{2,4\}\),則\(A\capB\)等于()A.\(\{1,2\}\)B.\(\{2\}\)C.\(\{2,3\}\)D.\(\{1,2,3,4\}\)3.直線\(y=2x+1\)的斜率是()A.\(1\)B.\(2\)C.\(\frac{1}{2}\)D.\(-2\)4.已知\(\sin\alpha=\frac{1}{2}\),且\(0<\alpha<\frac{\pi}{2}\),則\(\cos\alpha\)的值為()A.\(\frac{\sqrt{3}}{2}\)B.\(-\frac{\sqrt{3}}{2}\)C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)5.拋物線\(y^2=4x\)的焦點坐標(biāo)是()A.\((0,1)\)B.\((1,0)\)C.\((0,-1)\)D.\((-1,0)\)6.等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(a_3=5\),則\(a_2\)等于()A.\(3\)B.\(4\)C.\(5\)D.\(6\)7.若\(a>b\),則下列不等式成立的是()A.\(a^2>b^2\)B.\(\frac{1}{a}<\frac{1}\)C.\(a+c>b+c\)D.\(ac>bc\)8.復(fù)數(shù)\(z=3+4i\)的模\(\vertz\vert\)是()A.\(3\)B.\(4\)C.\(5\)D.\(7\)9.函數(shù)\(y=\log_2x\)的定義域是()A.\((0,+\infty)\)B.\([0,+\infty)\)C.\((-\infty,0)\)D.\((-\infty,+\infty)\)10.從\(5\)個不同元素中取出\(2\)個元素的組合數(shù)\(C_{5}^2\)是()A.\(10\)B.\(20\)C.\(15\)D.\(25\)二、多項選擇題(每題2分,共10題)1.以下哪些是奇函數(shù)(）A.\(y=x^3\)B.\(y=\sinx\)C.\(y=e^x\)D.\(y=\lnx\)2.以下哪些是橢圓的標(biāo)準(zhǔn)方程形式()A.\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)(\(a>b>0\)）B.\(\frac{y^2}{a^2}+\frac{x^2}{b^2}=1\)(\(a>b>0\)）C.\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)D.\(\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\)3.下列命題正確的是()A.垂直于同一條直線的兩條直線平行B.平行于同一個平面的兩條直線平行C.過直線外一點有且只有一條直線與已知直線平行D.過平面外一點有且只有一個平面與已知平面平行4.已知向量\(\vec{a}=(1,2)\),\(\vec=(-1,x)\),則以下正確的是()A.若\(\vec{a}\parallel\vec\),則\(x=-2\)B.若\(\vec{a}\perp\vec\),則\(x=\frac{1}{2}\)C.\(\vec{a}+\vec=(0,2+x)\)D.\(\vert\vec{a}\vert=\sqrt{5}\)5.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.對數(shù)函數(shù)D.三角函數(shù)6.對于數(shù)列\(zhòng)(\{a_n\}\),以下說法正確的是()A.若\(a_{n+1}-a_n=d\)(\(d\)為常數(shù)),則\(\{a_n\}\)是等差數(shù)列B.若\(\frac{a_{n+1}}{a_n}=q\)（\(q\)為常數(shù)）,則\(\{a_n\}\)是等比數(shù)列C.等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項和\(S_n=\frac{n(a_1+a_n)}{2}\)D.等比數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項和\(S_n=\frac{a_1(1-q^n)}{1-q}\)(\(q\neq1\))7.直線\(Ax+By+C=0\)(\(A\)、\(B\)不同時為\(0\))的斜率\(k\)和在\(y\)軸上的截距\(b\)可能是()A.\(k=-\frac{A}{B}\)(\(B\neq0\))B.\(b=-\frac{C}{B}\)(\(B\neq0\))C.當(dāng)\(B=0\)時,直線垂直于\(x\)軸D.當(dāng)\(A=0\)時,直線垂直于\(y\)軸8.以下哪些是三角函數(shù)的性質(zhì)(）A.\(y=\sinx\)的周期是\(2\pi\)B.\(y=\cosx\)是偶函數(shù)C.\(y=\tanx\)的定義域是\(x\neqk\pi+\frac{\pi}{2},k\inZ\)D.\(y=\sinx\)的值域是\([-1,1]\)9.已知函數(shù)\(y=f(x)\),以下說法正確的是()A.若\(f(x)\)在\(x=x_0\)處可導(dǎo),則\(f(x)\)在\(x=x_0\)處連續(xù)B.若\(f(x)\)在\(x=x_0\)處連續(xù),則\(f(x)\)在\(x=x_0\)處可導(dǎo)C.函數(shù)的極值點可能是導(dǎo)數(shù)為\(0\)的點D.函數(shù)的最值一定在端點處取得10.關(guān)于二項式\((a+b)^n\)展開式,正確的是()A.展開式共有\(zhòng)(n+1\)項B.二項式系數(shù)之和為\(2^n\)C.通項公式\(T_{r+1}=C_{n}^ra^{n-r}b^r\)D.當(dāng)\(n\)為偶數(shù)時,中間一項的二項式系數(shù)最大三、判斷題(每題2分,共10題)1.空集是任何集合的真子集。()2.函數(shù)\(y=2^x\)是增函數(shù)。()3.圓\(x^2+y^2=r^2\)的圓心是\((0,0)\),半徑是\(r\)。(）4.若\(a\cdotb=0\),則\(\vec{a}\perp\vec\)（\(\vec{a}\)、\(\vec\)為非零向量)。（)5.直線\(x+y+1=0\)與直線\(x-y+1=0\)垂直。()6.等差數(shù)列的通項公式是\(a_n=a_1+(n-1)d\)。()7.函數(shù)\(y=\cos^2x-\sin^2x\)的最小正周期是\(\pi\)。()8.若\(a>b>0\),\(c>d>0\),則\(ac>bd\)。(）9.復(fù)數(shù)\(z=a+bi\)（\(a,b\inR\)),當(dāng)\(b=0\)時,\(z\)是實數(shù)。（)10.組合數(shù)\(C_{n}^m=C_{n}^{n-m}\)。(）四、簡答題(每題5分,共4題)1.求函數(shù)\(y=3x^2-2x+1\)的對稱軸和頂點坐標(biāo)。-答案:對于二次函數(shù)\(y=ax^2+bx+c\),對稱軸為\(x=-\frac{2a}\),此函數(shù)\(a=3\),\(b=-2\),則對稱軸\(x=\frac{1}{3}\)。把\(x=\frac{1}{3}\)代入函數(shù)得\(y=3\times(\frac{1}{3})^2-2\times\frac{1}{3}+1=\frac{2}{3}\),頂點坐標(biāo)為\((\frac{1}{3},\frac{2}{3})\)。2.已知\(\tan\alpha=2\),求\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值。-答案：分子分母同時除以\(\cos\alpha\),則\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}=\frac{\tan\alpha+1}{\tan\alpha-1}\),將\(\tan\alpha=2\)代入得\(\frac{2+1}{2-1}=3\)。3.求過點\((1,2)\)且斜率為\(3\)的直線方程。-答案:由直線點斜式方程\(y-y_0=k(x-x_0)\)(\((x_0,y_0)\)為直線上一點,\(k\)為斜率),已知點\((1,2)\),斜率\(k=3\),則直線方程為\(y-2=3(x-1)\),整理得\(3x-y-1=0\)。4.已知等差數(shù)列\(zhòng)(\{a_n\}\)中,\(a_1=1\),\(d=2\),求\(a_5\)和\(S_5\)。-答案：\(a_n=a_1+(n-1)d\),所以\(a_5=1+(5-1)\times2=9\)。\(S_n=\frac{n(a_1+a_n)}{2}\),\(S_5=\frac{5\times(1+9)}{2}=25\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=x^3-3x\)的單調(diào)性與極值。-答案:對函數(shù)求導(dǎo)得\(y^\prime=3x^2-3=3(x+1)(x-1)\)。令\(y^\prime>0\),得\(x<-1\)或\(x>1\),函數(shù)在\((-\infty,-1)\)和\((1,+\infty)\)上遞增;令\(y^\prime<0\),得\(-1<x<1\),函數(shù)在\((-1,1)\)上遞減。當(dāng)\(x=-1\)時取極大值\(2\),\(x=1\)時取極小值\(-2\)。2.討論直線與圓的位置關(guān)系有哪些判定方法。-答案:一是幾何法,比較圓心到直線的距離\(d\)與圓半徑\(r\)的大小,\(d>r\)時相離,\(d=r\)時相切,\(d<r\)時相交;二是代數(shù)法,聯(lián)立直線與圓的方程得方程組,消元后看所得一元二次方程的判別式\(\Delta\),\(\Delta>0\)相交,\(\Delta=0\)相切,\(\Delta<0\)相離。3.討論如何根據(jù)數(shù)列的前\(n\)項和\(S_n\)求數(shù)列的通項公式\(a_n\)。-答案:當(dāng)\(n=1\)時,\(a_1=S_1\);當(dāng)\(n\geqslant2\)時,\(a_n=S_n-S_{n-1}\)。最后要檢驗\(n=1\)時\(a_1\)是否滿足\(n\geqslant2\)時的\(a_n\)表達(dá)式,若滿足則統(tǒng)一寫,不滿足則分段寫。4.討論在實際問題中建立函數(shù)模型的一般步驟。-答案:首先要審題,明確問題的背景和要求;接著設(shè)出變量,分析各變量間的關(guān)系;然后根據(jù)已知條件建立函數(shù)關(guān)系式;再確定函數(shù)的定