高三考試題及答案

一、單項(xiàng)選擇題（每題2分，共10題）1.已知集合\(A=\{1,2,3\}\)，\(B=\{x|x^{2}-3x+2=0\}\)，則\(A\capB=(\)\)A.\(\{1\}\)B.\(\{2\}\)C.\(\{1,2\}\)D.\(\{1,2,3\}\)答案：C2.若\(a=\log_{2}3\)，\(b=\log_{3}2\)，\(c=\log_{2}\frac{1}{3}\)，則\(a,b,c\)的大小關(guān)系為（）A.\(a>b>c\)B.\(b>a>c\)C.\(a>c>b\)D.\(c>a>b\)答案：A3.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.\(\frac{2\pi}{3}\)答案：A4.若向量\(\vec{a}=(1,2)\)，\(\vec=(x,-1)\)，且\(\vec{a}\perp\vec\)，則\(x=(\)\)A.\(2\)B.\(-2\)C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)答案：A5.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，公比\(q=2\)，則\(a_{5}=(\)\)A.\(16\)B.\(32\)C.\(8\)D.\(64\)答案：A6.在\(\triangleABC\)中，\(A=60^{\circ}\)，\(a=\sqrt{3}\)，\(b=1\)，則\(B=(\)\)A.\(30^{\circ}\)B.\(45^{\circ}\)C.\(60^{\circ}\)D.\(90^{\circ}\)答案：A7.過點(diǎn)\((1,0)\)且與直線\(x-2y-2=0\)平行的直線方程是（）A.\(x-2y-1=0\)B.\(x-2y+1=0\)C.\(2x+y-2=0\)D.\(x+2y-1=0\)答案：A8.若函數(shù)\(y=f(x)\)的圖象關(guān)于直線\(x=1\)對(duì)稱，且當(dāng)\(x\leqslant1\)時(shí)，\(f(x)=x^{2}+1\)，則當(dāng)\(x>1\)時(shí)，\(f(x)=(\)\)A.\((x-2)^{2}+1\)B.\((x-1)^{2}+1\)C.\((x+2)^{2}+1\)D.\((x+1)^{2}+1\)答案：A9.從\(5\)名男生和\(3\)名女生中選\(3\)人參加某項(xiàng)活動(dòng)，要求既有男生又有女生，則不同的選法有（）種。A.\(45\)B.\(30\)C.\(40\)D.\(35\)答案：A10.已知橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\)的離心率\(e=\frac{\sqrt{3}}{2}\)，且過點(diǎn)\((\sqrt{3},\frac{1}{2})\)，則橢圓方程為（）A.\(\frac{x^{2}}{4}+y^{2}=1\)B.\(\frac{x^{2}}{3}+y^{2}=1\)C.\(\frac{x^{2}}{2}+y^{2}=1\)D.\(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\)答案：A二、多項(xiàng)選擇題（每題2分，共10題）1.下列函數(shù)中，在\((0,+\infty)\)上為增函數(shù)的是（）A.\(y=x^{2}\)B.\(y=\lnx\)C.\(y=2^{x}\)D.\(y=\frac{1}{x}\)答案：ABC2.已知\(\vec{a}=(1,1)\)，\(\vec=(-1,1)\)，則（）A.\(\vec{a}\parallel\vec\)B.\(\vec{a}\perp\vec\)C.\(\vert\vec{a}\vert=\vert\vec\vert\)D.\((\vec{a}+\vec)\perp(\vec{a}-\vec)\)答案：C3.若直線\(y=kx+1\)與圓\(x^{2}+y^{2}=1\)相交于\(A\)、\(B\)兩點(diǎn)，則（）A.\(d=\frac{1}{\sqrt{1+k^{2}}}\)（\(d\)為圓心到直線的距離）B.\(\vertAB\vert=2\sqrt{1-d^{2}}\)C.\(k\)的取值范圍是\((-\infty,+\infty)\)D.弦長(zhǎng)\(\vertAB\vert\)的取值范圍是\((0,2]\)答案：ABD4.下列關(guān)于數(shù)列的說法正確的是（）A.等差數(shù)列的前\(n\)項(xiàng)和\(S_{n}=An^{2}+Bn\)（\(A\)、\(B\)為常數(shù)）B.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，若\(a_{1}<0\)，\(q>1\)，則數(shù)列\(zhòng)(\{a_{n}\}\)單調(diào)遞減C.若數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=2n^{2}-3n+1\)，則數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列D.若數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{n+1}=a_{n}+d\)（\(d\)為常數(shù)），則數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列答案：ABD5.對(duì)于函數(shù)\(y=\sinx\)，下列說法正確的是（）A.它是奇函數(shù)B.它的圖象關(guān)于直線\(x=\frac{\pi}{2}\)對(duì)稱C.它的周期是\(2\pi\)D.在\([0,\frac{\pi}{2}]\)上單調(diào)遞增答案：ABCD6.已知雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0,b>0)\)，其漸近線方程為\(y=\pm\frac{a}x\)，則（）A.離心率\(e=\sqrt{1+\frac{b^{2}}{a^{2}}}\)B.若\(a=1\)，\(b=2\)，則雙曲線的焦點(diǎn)到漸近線的距離為\(2\)C.雙曲線與它的漸近線無限接近但不相交D.若雙曲線過點(diǎn)\((2,1)\)，則漸近線方程為\(y=\pm\frac{1}{2}x\)答案：AC7.下列命題正確的是（）A.若\(p\veeq\)為真命題，\(p\wedgeq\)為假命題，則\(p\)，\(q\)一真一假B.命題“若\(x^{2}-3x+2=0\)，則\(x=1\)或\(x=2\)”的逆否命題為“若\(x\neq1\)且\(x\neq2\)，則\(x^{2}-3x+2\neq0\)”C.命題“\(\forallx\inR\)，\(x^{2}+1>0\)”的否定是“\(\existsx\inR\)，\(x^{2}+1\leqslant0\)”D.若\(a>b\)，則\(ac^{2}>bc^{2}\)答案：ABC8.在空間中，下列命題正確的是（）A.平行于同一條直線的兩條直線互相平行B.垂直于同一條直線的兩條直線互相平行C.平行于同一個(gè)平面的兩條直線互相平行D.垂直于同一個(gè)平面的兩條直線互相平行答案：AD9.設(shè)\(x,y\inR\)，則\((x^{2}+y^{2})(x^{4}+y^{4})\)的最小值為（）A.\(0\)B.\(\frac{1}{4}\)C.\(\frac{1}{2}\)D.\(1\)答案：A10.若函數(shù)\(f(x)\)在\([a,b]\)上連續(xù)，在\((a,b)\)內(nèi)可導(dǎo)，且\(f(a)=f(b)\)，則（）A.至少存在一點(diǎn)\(\xi\in(a,b)\)，使得\(f'(\xi)=0\)B.\(f(x)\)在\([a,b]\)上為常數(shù)函數(shù)C.函數(shù)\(y=f(x)\)的圖象關(guān)于直線\(x=\frac{a+b}{2}\)對(duì)稱D.函數(shù)\(y=f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增或單調(diào)遞減答案：A三、判斷題（每題2分，共10題）1.空集是任何集合的子集。（）答案：對(duì)2.若\(a>b\)，則\(ac>bc\)。（）答案：錯(cuò)3.函數(shù)\(y=\tanx\)的定義域是\(\{x|x\neqk\pi+\frac{\pi}{2},k\inZ\}\)。（）答案：對(duì)4.兩個(gè)向量相等當(dāng)且僅當(dāng)它們的模相等且方向相同。（）答案：對(duì)5.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{n}=a_{1}q^{n-1}\)。（）答案：對(duì)6.在\(\triangleABC\)中，\(A+B+C=\pi\)。（）答案：對(duì)7.直線\(Ax+By+C=0\)（\(A\)、\(B\)不同時(shí)為\(0\)）的斜率\(k=-\frac{A}{B}\)。（）答案：錯(cuò)8.若函數(shù)\(y=f(x)\)是偶函數(shù)，則\(f(x)=f(-x)\)。（）答案：對(duì)9.若\(z=a+bi(a,b\inR)\)，則\(\vertz\vert=\sqrt{a^{2}+b^{2}}\)。（）答案：對(duì)10.若\(f(x)\)在\(x=x_{0}\)處可導(dǎo)，則\(f(x)\)在\(x=x_{0}\)處連續(xù)。（）答案：對(duì)四、簡(jiǎn)答題（每題5分，共4題）1.求函數(shù)\(y=\frac{1}{x-1}+x(x>1)\)的最小值。答案：\(y=\frac{1}{x-1}+x=\frac{1}{x-1}+(x-1)+1\)，因?yàn)閈(x>1\)，所以\(x-1>0\)。根據(jù)均值不等式\(a+b\geqslant2\sqrt{ab}\)，這里\(a=\frac{1}{x-1}\)，\(b=x-1\)，則\(y\geqslant2\sqrt{\frac{1}{x-1}\times(x-1)}+1=3\)，當(dāng)且僅當(dāng)\(\frac{1}{x-1}=x-1\)即\(x=2\)時(shí)取等號(hào)，所以最小值為\(3\)。2.已知數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=n^{2}+n\)，求數(shù)列\(zhòng)(\{a_{n}\}\)的通項(xiàng)公式。答案：當(dāng)\(n=1\)時(shí)，\(a_{1}=S_{1}=1^{2}+1=2\)；當(dāng)\(n\geqslant2\)時(shí)，\(a_{n}=S_{n}-S_{n-1}=n^{2}+n-\left[(n-1)^{2}+(n-1)\right]=2n\)。當(dāng)\(n=1\)時(shí)也滿足\(a_{n}=2n\)，所以\(a_{n}=2n\)。3.求過點(diǎn)\((1,2)\)且與直線\(2x-y+1=0\)垂直的直線方程。答案：直線\(2x-y+1=0\)的斜率為\(2\)，與其垂直的直線斜率為\(-\frac{1}{2}\)。設(shè)所求直線方程為\(y=-\frac{1}{2}x+b\)，把點(diǎn)\((1,2)\)代入得\(2=-\frac{1}{2}\times1+b\)，解得\(b=\frac{5}{2}\)，所以直線方程為\(y=-\frac{1}{2}x+\frac{5}{2}\)，即\(x+2y-5=0\)。4.設(shè)\(f(x)=\sinx+\cosx\)，求\(f(x)\)的最大值和最小值。答案：\(f(x)=\sinx+\cosx=\sqrt{2}(\frac{\sqrt{2}}{2}\sinx+\frac{\sqrt{2}}{2}\cosx)=\sqrt{2}\sin(x+\frac{\pi}{4})\)。因?yàn)閈(-1\leqslant\sin(x+\frac{\pi}{4})\leqslant1\)，所以\(-\sqrt{2}\leqslant\sqrt{2}\sin(x+\frac{\pi}{4})\leqslant\sqrt{2}\)，最大值為\(\sqrt{2}\)，最小值為\(-\sqrt{2}\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=x^{3}-3x^{2}-9x+5\)的單調(diào)性。答案：\(y'=3x^{2}-6x-9=3(x^{2}-2x-3)=3(x-3)(x+1)\)。令\(y'=0\)，得\(x=3\)或\(x=-1\)。當(dāng)\(x<-1\)或\(x>3\)時(shí)，\(y'>0\)，函數(shù)單調(diào)遞增；當(dāng)\(-1<x<3\)時(shí)，\(y'<0\)，函數(shù)單調(diào)遞減。2.討論等比數(shù)列\(zhòng)(\{a_{n}\}\)