高三考試題目及答案講解

一、單項(xiàng)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.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)答案：B2.已知向量\(\vec{a}=(1,2)\)，\(\vec=(-1,m)\)，若\(\vec{a}\parallel\vec\)，則\(m\)的值為（）A.\(2\)B.\(-2\)C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案：B3.雙曲線\(\frac{x^{2}}{4}-\frac{y^{2}}{5}=1\)的漸近線方程為（）A.\(y=\pm\frac{\sqrt{5}}{2}x\)B.\(y=\pm\frac{2\sqrt{5}}{5}x\)C.\(y=\pm\frac{3}{2}x\)D.\(y=\pm\frac{2}{3}x\)答案：A4.已知\(\cos\alpha=-\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\sin\alpha\)的值為（）A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{5}\)D.\(-\frac{3}{5}\)答案：A5.若復(fù)數(shù)\(z=1+i\)，則\(\vertz^{2}\vert\)的值為（）A.\(2\)B.\(2\sqrt{2}\)C.\(4\)D.\(4\sqrt{2}\)答案：C6.已知\(a=\log_{3}2\)，\(b=\log_{5}2\)，\(c=\log_{2}3\)，則（）A.\(a>c>b\)B.\(b>c>a\)C.\(c>a>b\)D.\(c>b>a\)答案：C7.函數(shù)\(f(x)=x^{3}-3x+1\)的單調(diào)遞減區(qū)間為（）A.\((-1,1)\)B.\((1,+\infty)\)C.\((-\infty,-1)\)D.\((-\infty,-1)\cup(1,+\infty)\)答案：A8.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{3}+a_{5}=10\)，則\(a_{4}\)的值為（）A.\(5\)B.\(6\)C.\(8\)D.\(10\)答案：A9.若\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geqslant1\\x-y\leqslant1\\y\leqslant1\end{cases}\)，則\(z=3x-y\)的最大值為（）A.\(1\)B.\(2\)C.\(3\)D.\(4\)答案：C10.已知直線\(l\)過(guò)點(diǎn)\((0,-1)\)，且與圓\(x^{2}+y^{2}-2x-4y=0\)相切，則直線\(l\)的方程為（）A.\(x+y+1=0\)B.\(x-y-1=0\)C.\(x+2y+2=0\)D.\(2x-y-1=0\)答案：B二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，是偶函數(shù)的有（）A.\(y=x^{2}\)B.\(y=\cosx\)C.\(y=\ln(x^{2}+1)\)D.\(y=2^{x}\)答案：ABC2.一個(gè)正方體的棱長(zhǎng)為\(2\)，則以下說(shuō)法正確的是（）A.正方體的表面積為\(24\)B.正方體的體積為\(8\)C.正方體的外接球半徑為\(\sqrt{3}\)D.正方體的內(nèi)切球半徑為\(1\)答案：ABCD3.已知\(a\)，\(b\)，\(c\)為實(shí)數(shù)，則下列命題正確的是（）A.若\(a>b\)，則\(ac^{2}>bc^{2}\)B.若\(a>b\)，\(c>d\)，則\(a-c>b-d\)C.若\(a>b\)，\(c>0\)，則\(ac>bc\)D.若\(a>b\)，\(c<0\)，則\(ac<bc\)答案：CD4.以下哪些是橢圓\(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)的性質(zhì)（）A.長(zhǎng)軸長(zhǎng)為\(10\)B.短軸長(zhǎng)為\(6\)C.離心率為\(\frac{4}{5}\)D.焦點(diǎn)坐標(biāo)為\((\pm4,0)\)答案：ABCD5.已知函數(shù)\(f(x)=2\sin(2x-\frac{\pi}{6})\)，則下列說(shuō)法正確的是（）A.\(f(x)\)的最小正周期為\(\pi\)B.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{12},0)\)對(duì)稱C.\(f(x)\)在\([0,\frac{\pi}{2}]\)上的值域?yàn)閈([-1,2]\)D.\(f(x)\)的圖象關(guān)于直線\(x=\frac{\pi}{3}\)對(duì)稱答案：ACD6.下列導(dǎo)數(shù)運(yùn)算正確的是（）A.\((x^{n})^\prime=nx^{n-1}\)B.\((\sinx)^\prime=\cosx\)C.\((e^{x})^\prime=e^{x}\)D.\((\lnx)^\prime=\frac{1}{x}\)答案：ABCD7.已知\(\{a_{n}\}\)是等比數(shù)列，公比為\(q\)，則下列說(shuō)法正確的是（）A.若\(q>1\)，則\(\{a_{n}\}\)是遞增數(shù)列B.若\(a_{1}>0\)，\(0<q<1\)，則\(\{a_{n}\}\)是遞減數(shù)列C.若\(a_{1}<0\)，\(0<q<1\)，則\(\{a_{n}\}\)是遞增數(shù)列D.若\(q<0\)，則\(\{a_{n}\}\)是擺動(dòng)數(shù)列答案：BCD8.從\(4\)名男生和\(3\)名女生中選\(3\)人參加活動(dòng)，下列說(shuō)法正確的是（）A.若全選男生，有\(zhòng)(C_{4}^{3}\)種選法B.若選\(2\)男\(zhòng)(1\)女，有\(zhòng)(C_{4}^{2}C_{3}^{1}\)種選法C.若女生甲必須參加，有\(zhòng)(C_{6}^{2}\)種選法D.若至少有\(zhòng)(1\)名女生，有\(zhòng)(C_{7}^{3}-C_{4}^{3}\)種選法答案：ABCD9.已知直線\(l_{1}\)：\(ax+y+1=0\)，\(l_{2}\)：\(x+ay+1=0\)，則下列說(shuō)法正確的是（）A.當(dāng)\(a=1\)時(shí)，\(l_{1}\)與\(l_{2}\)重合B.當(dāng)\(a=-1\)時(shí)，\(l_{1}\)與\(l_{2}\)平行C.當(dāng)\(l_{1}\perpl_{2}\)時(shí)，\(a=0\)D.當(dāng)\(l_{1}\)與\(l_{2}\)相交時(shí)，\(a\neq\pm1\)答案：ABCD10.已知\(A\)，\(B\)，\(C\)是球\(O\)表面上的三點(diǎn)，\(AB=BC=CA=2\)，球心\(O\)到平面\(ABC\)的距離為\(\sqrt{3}\)，則下列說(shuō)法正確的是（）A.\(\triangleABC\)的外接圓半徑為\(\frac{2\sqrt{3}}{3}\)B.球\(O\)的半徑為\(\frac{\sqrt{21}}{3}\)C.球\(O\)的表面積為\(\frac{28\pi}{3}\)D.球\(O\)的體積為\(\frac{28\sqrt{21}\pi}{27}\)答案：ACD三、判斷題（每題2分，共10題）1.若\(a>b\)，則\(a^{2}>b^{2}\)。（×）2.函數(shù)\(y=\frac{1}{x}\)在定義域內(nèi)是減函數(shù)。（×）3.若向量\(\vec{a}\cdot\vec=0\)，則\(\vec{a}\perp\vec\)。（×）4.拋物線\(y^{2}=4x\)的焦點(diǎn)坐標(biāo)為\((1,0)\)。（√）5.若\(\sin\alpha=\sin\beta\)，則\(\alpha=\beta\)。（×）6.等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=na_{1}+\frac{n(n-1)d}{2}\)。（√）7.函數(shù)\(y=\cosx\)的圖象關(guān)于\(y\)軸對(duì)稱。（√）8.若\(z=a+bi\)（\(a\)，\(b\inR\)），則\(\vertz\vert=\sqrt{a^{2}+b^{2}}\)。（√）9.圓\((x-a)^{2}+(y-b)^{2}=r^{2}\)的圓心坐標(biāo)為\((a,b)\)。（√）10.若\(f(x)\)是奇函數(shù)，則\(f(0)=0\)。（×）四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(f(x)=x^{2}-2x+3\)在區(qū)間\([0,3]\)上的最值。答案：對(duì)\(f(x)=x^{2}-2x+3\)配方得\(f(x)=(x-1)^{2}+2\)。對(duì)稱軸為\(x=1\)，\(f(1)=2\)，\(f(0)=3\)，\(f(3)=6\)，所以最小值為\(2\)，最大值為\(6\)。2.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，求\(\cos\alpha\)與\(\tan\alpha\)的值。答案：因?yàn)閈(\sin^{2}\alpha+\cos^{2}\alpha=1\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，所以\(\cos\alpha=-\sqrt{1-\sin^{2}\alpha}=-\frac{4}{5}\)，\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=-\frac{3}{4}\)。3.求雙曲線\(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\)的實(shí)軸長(zhǎng)、虛軸長(zhǎng)、焦距和離心率。答案：\(a=3\)，\(b=4\)，\(c=\sqrt{a^{2}+b^{2}}=5\)。實(shí)軸長(zhǎng)\(2a=6\)，虛軸長(zhǎng)\(2b=8\)，焦距\(2c=10\)，離心率\(e=\frac{c}{a}=\frac{5}{3}\)。4.已知數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=n^{2}\)，求\(a_{n}\)。答案：當(dāng)\(n=1\)時(shí)，\(a_{1}=S_{1}=1\)；當(dāng)\(n\geqslant2\)時(shí)，\(a_{n}=S_{n}-S_{n-1}=n^{2}-(n-1)^{2}=2n-1\)。\(n=1\)時(shí)也滿足\(a_{n}=2n-1\)，所以\(a_{n}=2n-1\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(f(x)=ax^{2}+bx+c\)（\(a\neq0\)）的單調(diào)性與\(a\)，\(b\)，\(c\)的關(guān)系。答案：\(a>0\)時(shí)，對(duì)稱軸\(x=-\frac{2a}\)，\((-\infty,-\frac{2a})\)遞減，\((-\frac{2a},+\infty)\)遞增；\(a<0\)時(shí)，\((-\infty,-\frac{2a})\)遞增，\((-\frac{2a},+\infty)\)遞減，\(c\)不影響單調(diào)性。2.討論直線與圓的位置關(guān)系有哪些判斷方法。答案：一是代數(shù)法，聯(lián)立直線與圓方程，消元得一元二次方程，根據(jù)判別式判斷；二是幾何法，