三基理論試題和答案高中

一、單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=\log_{2}(x+1)\)的定義域是(）A.\((-1,+\infty)\)B.\((-\infty,-1)\)C.\((0,+\infty)\)D.\((1,+\infty)\)2.已知向量\(\overrightarrow{a}=(1,2)\),\(\overrightarrow=(-2,m)\),若\(\overrightarrow{a}\parallel\overrightarrow\),則\(m\)的值為()A.\(1\)B.\(-1\)C.\(4\)D.\(-4\)3.等差數(shù)列\(zhòng)(\{a_{n}\}\)中,\(a_{3}=5\),\(a_{5}=9\),則\(a_{7}\)等于()A.\(11\)B.\(13\)C.\(15\)D.\(17\)4.已知\(\sin\alpha=\frac{3}{5}\),\(\alpha\)是第二象限角,則\(\cos\alpha\)的值為()A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)5.直線(xiàn)\(3x+4y-5=0\)的斜率是()A.\(\frac{3}{4}\)B.\(-\frac{3}{4}\)C.\(\frac{4}{3}\)D.\(-\frac{4}{3}\)6.拋物線(xiàn)\(y^{2}=8x\)的焦點(diǎn)坐標(biāo)是()A.\((2,0)\)B.\((0,2)\)C.\((-2,0)\)D.\((0,-2)\)7.若\(x\gt0\),則\(x+\frac{4}{x}\)的最小值是()A.\(2\)B.\(4\)C.\(6\)D.\(8\)8.已知\(\alpha\),\(\beta\)是兩個(gè)不同平面,\(m\),\(n\)是兩條不同直線(xiàn),若\(m\subset\alpha\),\(n\subset\beta\),且\(m\paralleln\),則\(\alpha\)與\(\beta\)的位置關(guān)系是()A.平行B.相交C.平行或相交D.垂直9.函數(shù)\(f(x)=x^{3}-3x\)的單調(diào)遞增區(qū)間是()A.\((-\infty,-1)\)和\((1,+\infty)\)B.\((-1,1)\)C.\((-\infty,0)\)D.\((0,+\infty)\)10.已知\(a=\log_{3}2\),\(b=\log_{2}3\),\(c=\log_{2}5\),則\(a\),\(b\),\(c\)的大小關(guān)系是()A.\(a\ltb\ltc\)B.\(a\ltc\ltb\)C.\(b\lta\ltc\)D.\(b\ltc\lta\)二、多項(xiàng)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.\(y=x+1\)D.\(y=\frac{1}{x}\)2.以下屬于基本不等式應(yīng)用的有()A.求\(y=x+\frac{1}{x}(x\gt0)\)最小值B.求\(y=2x+\frac{8}{x}(x\gt0)\)最小值C.求\(y=x^{2}+\frac{1}{x^{2}}\)最小值D.求\(y=x+\frac{1}{x}(x\lt0)\)最大值3.關(guān)于橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a\gtb\gt0)\),正確的有()A.長(zhǎng)軸長(zhǎng)為\(2a\)B.短軸長(zhǎng)為\(2b\)C.焦距為\(2c\)(\(c^{2}=a^{2}-b^{2}\))D.離心率\(e=\frac{c}{a}\)4.下列直線(xiàn)中,與直線(xiàn)\(2x-y+1=0\)平行的有（）A.\(4x-2y-3=0\)B.\(2x-y-2=0\)C.\(x+2y-1=0\)D.\(-4x+2y+5=0\)5.以下哪些是等比數(shù)列的性質(zhì)()A.\(a_{n}^{2}=a_{n-1}a_{n+1}(n\gt1)\)B.若\(m+n=p+q\),則\(a_{m}a_{n}=a_{p}a_{q}\)C.\(S_{n}\)為前\(n\)項(xiàng)和,則\(S_{n}\),\(S_{2n}-S_{n}\),\(S_{3n}-S_{2n}\)成等比數(shù)列D.公比\(q\neq0\)6.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.\(\frac{11\pi}{6}\)7.對(duì)于函數(shù)\(y=\cosx\),正確的是()A.周期是\(2\pi\)B.對(duì)稱(chēng)軸是\(x=k\pi(k\inZ)\)C.對(duì)稱(chēng)中心是\((k\pi+\frac{\pi}{2},0)(k\inZ)\)D.在\([0,\pi]\)上單調(diào)遞減8.以下哪些是直線(xiàn)的方程形式()A.點(diǎn)斜式\(y-y_{0}=k(x-x_{0})\)B.斜截式\(y=kx+b\)C.兩點(diǎn)式\(\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\)D.一般式\(Ax+By+C=0\)9.已知\(a\),\(b\),\(c\)為實(shí)數(shù),下列命題正確的是()A.若\(a\gtb\),則\(ac^{2}\gtbc^{2}\)B.若\(a\gtb\),\(c\gtd\),則\(a-c\gtb-d\)C.若\(a\gtb\gt0\),則\(\frac{1}{a}\lt\frac{1}\)D.若\(a\gtb\),\(c\lt0\),則\(ac\ltbc\)10.以下哪些函數(shù)的定義域是\(R\)(）A.\(y=x^{2}\)B.\(y=\sinx\)C.\(y=e^{x}\)D.\(y=\lgx\)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)\(y=\tanx\)的周期是\(\pi\)。()4.圓\(x^{2}+y^{2}=r^{2}\)的圓心是\((0,0)\),半徑是\(r\)。()5.若向量\(\overrightarrow{a}\cdot\overrightarrow=0\),則\(\overrightarrow{a}\perp\overrightarrow\)。(）6.數(shù)列\(zhòng)(1\),\(2\),\(3\),\(4\),\(5\)是等差數(shù)列也是等比數(shù)列。（)7.直線(xiàn)\(x=1\)的斜率不存在。()8.對(duì)數(shù)函數(shù)\(y=\log_{a}x(a\gt0,a\neq1)\)的定義域是\((0,+\infty)\)。()9.若\(\alpha\),\(\beta\)是銳角,\(\sin\alpha=\sin\beta\),則\(\alpha=\beta\)。（)10.函數(shù)\(y=2x^{2}-4x+3\)的最小值是\(1\)。(）四、簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=\frac{1}{\sqrt{x-2}}\)的定義域。答案:要使函數(shù)有意義,則\(x-2\gt0\),解得\(x\gt2\),所以定義域?yàn)閈((2,+\infty)\)。2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)中,\(a_{1}=1\),\(d=2\),求\(a_{5}\)。答案：根據(jù)等差數(shù)列通項(xiàng)公式\(a_{n}=a_{1}+(n-1)d\),則\(a_{5}=a_{1}+4d=1+4\times2=9\)。3.求直線(xiàn)\(3x-4y+5=0\)的斜率和在\(y\)軸上的截距。答案:將直線(xiàn)方程化為斜截式\(y=\frac{3}{4}x+\frac{5}{4}\),斜率\(k=\frac{3}{4}\),在\(y\)軸上截距為\(\frac{5}{4}\)。4.求\(\sin15^{\circ}\)的值。答案:\(\sin15^{\circ}=\sin(45^{\circ}-30^{\circ})=\sin45^{\circ}\cos30^{\circ}-\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=x^{2}-2x+3\)的單調(diào)性。答案:對(duì)函數(shù)\(y=x^{2}-2x+3\)配方得\(y=(x-1)^{2}+2\)。其對(duì)稱(chēng)軸為\(x=1\),開(kāi)口向上。所以在\((-\infty,1)\)上單調(diào)遞減,在\((1,+\infty)\)上單調(diào)遞增。2.討論直線(xiàn)\(y=kx+1\)與圓\(x^{2}+y^{2}=1\)的位置關(guān)系。答案:圓心\((0,0)\)到直線(xiàn)\(y=kx+1\)即\(kx-y+1=0\)的距離\(d=\frac{1}{\sqrt{k^{2}+1}}\)。當(dāng)\(d\lt1\)即\(k\neq0\)時(shí),直線(xiàn)與圓相交;當(dāng)\(d=1\)即\(k=0\)時(shí),直線(xiàn)與圓相切;不存在\(d\gt1\)的情況。3.討論等比數(shù)列\(zhòng)(\{a_{n}\}\)中,\(a_{1}\gt0\),公比\(q\)的取值對(duì)數(shù)列單調(diào)性的影響。答案：當(dāng)\(q\gt1\)時(shí),數(shù)列單調(diào)遞增；當(dāng)\(0\ltq\lt1\)時(shí),數(shù)列單調(diào)遞減；當(dāng)\(q=1\)時(shí),數(shù)列為常數(shù)列；當(dāng)\(q\lt0\)時(shí),數(shù)列不具有單調(diào)性。4.討論\(y=\sinx\)與\(y=\cosx\)在\([0,2\pi]\)上的交點(diǎn)情況。答案:令\(\sinx=\cosx\),即\(\tanx=1\),在\([0,2\pi]\)上,\(x=\frac{\pi}{4}\)或\(x=\frac{5\pi}{4}\),所以交點(diǎn)坐標(biāo)為\((\frac{\pi}{4},\frac{\sqrt{2}}{2})\)和\((\frac{5\pi