河南高考理科試卷及答案

一、單項選擇題1.已知集合\(A=\{x|x^2-3x-4<0\}\)，\(B=\{x|2^x>8\}\)，則\(A\capB\)等于（）A.\((-1,3)\)B.\((3,4)\)C.\((-1,4)\)D.\((-\infty,-1)\cup(3,+\infty)\)答案：B2.設復數(shù)\(z\)滿足\((1+i)z=2i\)，則\(\vertz\vert\)等于（）A.\(\frac{1}{2}\)B.\(\frac{\sqrt{2}}{2}\)C.\(\sqrt{2}\)D.\(2\)答案：C3.已知向量\(\overrightarrow{a}=(1,m)\)，\(\overrightarrow=(3,-2)\)，且\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow\)，則\(m\)等于（）A.-8B.-6C.6D.8答案：D4.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\in(\frac{\pi}{2},\pi)\)，則\(\tan(\alpha+\frac{\pi}{4})\)等于（）A.\(\frac{1}{7}\)B.7C.-\frac{1}{7}\)D.-7答案：A5.執(zhí)行如圖所示的程序框圖，如果輸入的\(a=1\)，\(b=2\)，那么輸出的\(a\)的值為（）A.16B.8C.4D.2答案：B6.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項和為\(S_n\)，若\(a_3+a_4+a_5=12\)，則\(S_7\)等于（）A.28B.42C.56D.14答案：A7.已知\(f(x)\)是定義在\(R\)上的奇函數(shù)，當\(x\geqslant0\)時，\(f(x)=x^2-3x\)，則函數(shù)\(g(x)=f(x)-x+3\)的零點的集合為（）A.\(\{1,3\}\)B.\(\{-3,-1,1,3\}\)C.\(\{2-\sqrt{7},1,3\}\)D.\(\{-2-\sqrt{7},1,3\}\)答案：D8.已知橢圓\(C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的左、右焦點分別為\(F_1,F_2\)，離心率為\(\frac{\sqrt{3}}{3}\)，過\(F_2\)的直線\(l\)交\(C\)于\(A,B\)兩點，若\(\triangleAF_1B\)的周長為\(4\sqrt{3}\)，則\(C\)的方程為（）A.\(\frac{x^2}{3}+\frac{y^2}{2}=1\)B.\(\frac{x^2}{3}+y^2=1\)C.\(\frac{x^2}{12}+\frac{y^2}{8}=1\)D.\(\frac{x^2}{12}+\frac{y^2}{4}=1\)答案：A9.已知函數(shù)\(f(x)=\sin(\omegax+\varphi)(\omega>0,\vert\varphi\vert<\frac{\pi}{2})\)的最小正周期為\(4\pi\)，且對\(\forallx\inR\)，有\(zhòng)(f(x)\leqslantf(\frac{\pi}{3})\)成立，則\(f(x)\)圖象的一個對稱中心坐標是（）A.\((-\frac{2\pi}{3},0)\)B.\((-\frac{\pi}{3},0)\)C.\((\frac{2\pi}{3},0)\)D.\((\frac{5\pi}{3},0)\)答案：A10.已知函數(shù)\(f(x)=\begin{cases}x^2+1,x\geqslant0\\1,x<0\end{cases}\)，則滿足不等式\(f(1-x^2)>f(2x)\)的\(x\)的取值范圍是（）A.\((-1,\sqrt{2}-1)\)B.\((0,\sqrt{2}-1)\)C.\([0,\sqrt{2}-1)\)D.\([-1,\sqrt{2}-1]\)答案：A二、多項選擇題1.下列說法正確的是（）A.若\(a>b\)，\(c>d\)，則\(a-c>b-d\)B.若\(a>b\)，則\(ac^2>bc^2\)C.若\(a>b>0\)，則\(\frac{1}{a}<\frac{1}\)D.若\(a>b\)，\(c<0\)，則\(ac<bc\)答案：CD2.已知函數(shù)\(f(x)=2\sin(2x+\frac{\pi}{6})\)，則下列說法正確的是（）A.\(f(x)\)的最小正周期為\(\pi\)B.\(f(x)\)的圖象關于點\((\frac{5\pi}{12},0)\)對稱C.\(f(x)\)在區(qū)間\([-\frac{\pi}{6},\frac{\pi}{6}]\)上單調(diào)遞增D.\(f(x)\)的圖象可以由\(y=2\sin2x\)的圖象向左平移\(\frac{\pi}{6}\)個單位長度得到答案：ABC3.已知拋物線\(y^2=2px(p>0)\)的焦點為\(F\)，點\(M\)在拋物線上，且\(M\)的橫坐標為\(4\)，\(\vertMF\vert=5\)，則下列說法正確的是（）A.\(p=2\)B.準線方程為\(x=-1\)C.拋物線方程為\(y^2=4x\)D.焦點到準線的距離為\(1\)答案：ABC4.已知函數(shù)\(f(x)\)是定義在\(R\)上的偶函數(shù)，且在\([0,+\infty)\)上單調(diào)遞增，則下列結論正確的是（）A.\(f(-1)>f(0)>f(2)\)B.\(f(-1)>f(-2)>f(0)\)C.\(f(2)>f(0)>f(-1)\)D.\(f(2)>f(-1)>f(0)\)答案：BD5.已知\(\{a_n\}\)是等比數(shù)列，公比\(q\)滿足\(q^2=4\)，\(S_n\)是\(\{a_n\}\)的前\(n\)項和，則下列說法正確的是（）A.\(q=2\)B.\(q=-2\)C.\(\frac{S_4}{S_2}=5\)或\(\frac{S_4}{S_2}=-3\)D.\(\frac{S_4}{S_2}=5\)答案：BC6.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(-2,1)\)，則（）A.\(\overrightarrow{a}\parallel\overrightarrow\)B.\(\overrightarrow{a}\perp\overrightarrow\)C.\(\vert\overrightarrow{a}\vert=\vert\overrightarrow\vert\)D.\((\overrightarrow{a}+\overrightarrow)\perp(\overrightarrow{a}-\overrightarrow)\)答案：CD7.已知函數(shù)\(f(x)=\lnx+x^2\)，則（）A.\(f(x)\)在\((0,+\infty)\)上單調(diào)遞增B.\(f(x)\)在\((0,+\infty)\)上單調(diào)遞減C.\(f(x)\)有最小值D.\(f(x)\)有最大值答案：AC8.已知雙曲線\(C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)\)的漸近線方程為\(y=\pm\frac{a}x\)，離心率\(e=\frac{c}{a}\)（\(c\)為半焦距），則下列說法正確的是（）A.漸近線方程\(y=\pm\frac{a}x\)可表示為\(bx\pmay=0\)B.離心率\(e=\sqrt{1+\frac{b^2}{a^2}}\)C.當\(a=b\)時，雙曲線\(C\)是等軸雙曲線D.雙曲線\(C\)的漸近線與圓\((x-a)^2+y^2=b^2\)恒有兩個交點答案：ABC9.已知函數(shù)\(y=\cos(2x+\varphi)(-\pi\leqslant\varphi\leqslant\pi)\)的圖象向右平移\(\frac{\pi}{2}\)個單位長度后，與函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的圖象重合，則\(\varphi\)的值可以為（）A.\(\frac{5\pi}{6}\)B.\(-\frac{7\pi}{6}\)C.\(\frac{\pi}{6}\)D.\(-\frac{\pi}{6}\)答案：AB10.已知\(a,b\inR\)，且\(a+b=1\)，則下列結論正確的是（）A.\(a^2+b^2\geqslant\frac{1}{2}\)B.\(2^a+2^b\geqslant2\sqrt{2}\)C.\(\log_2a+\log_2b\leqslant-2\)D.\(\frac{1}{a}+\frac{1}\geqslant4\)答案：ABD三、判斷題1.若直線\(l\)與平面\(\alpha\)內(nèi)的無數(shù)條直線垂直，則\(l\perp\alpha\)。（×）2.若\(a>b\)，則\(a^3>b^3\)。（√）3.函數(shù)\(y=\sinx+\cosx\)的最大值為\(2\)。（×）4.若\(z\inC\)，且\(\vertz\vert=1\)，則\(\vertz-2i\vert\)的最大值為\(3\)。（√）5.等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項和為\(S_n\)，若\(a_1>0\)，\(S_3=S_11\)，則\(S_n\)的最大值是\(S_7\)。（√）6.若\(f(x)\)是奇函數(shù)，且\(f(x)\)在\(x=0\)處有定義，則\(f(0)=0\)。（√）7.橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)\)的離心率\(e\)越大，橢圓越圓。（×）8.已知\(a,b\inR\)，則“\(a^2+b^2=0\)”是“\(ab=0\)”的充分不必要條件。（√）9.函數(shù)\(y=\log_2(x^2-2x+3)\)的值域是\([1,+\infty)\)。（√）10.若\(x\in(0,\frac{\pi}{2})\)，則\(\sinx<x<\tanx\)。（√）四、簡答題1.已知函數(shù)\(f(x)=x^3-3x^2+2\)。求\(f(x)\)的單調(diào)區(qū)間和極值。答案：對\(f(x)=x^3-3x^2+2\)求導得\(f^\prime(x)=3x^2-6x=3x(x-2)\)。令\(f^\prime(x)=0\)，解得\(x=0\)或\(x=2\)。當\(x<0\)或\(x>2\)時，\(f^\prime(x)>0\)，\(f(x)\)單調(diào)遞增；當\(0<x<2\)時，\(f^\prime(x)<0\)，\(f(x)\)單調(diào)遞減。所以\(f(x)\)的極大值為\(f(0)=2\)，極小值為\(f(2)=-2\)。單調(diào)遞增區(qū)間為\((-\infty,0)\)和\((2,+\infty)\)，單調(diào)遞減區(qū)間為\((0,2)\)。2.在\(\triangleABC\)中，角\(A,B,C\)所對的邊分別為\(a,b,c\)，已知\(\cosA=\frac{3}{5}\)，\(b=2\)，\(\triangleABC\)的面積\(S=4\)。求\(c\)的值。答案：因為\(\cosA=\frac{3}{5}\)，\(A\in(0,\pi)\)，所以\(\sinA=\sqrt{1-\cos^{2}A}=\frac{4}{5}\)。又因為\(\triangleABC\)的面積\(S=\frac{1}{2}bc\sinA\)，已知\(b=2\)，\(S=4\)，則\(4=\frac{1}{2}\times2\timesc\times\frac{4}{5}\)，即\(4=\frac{4}{5}c\)，解得\(c=5\)。3.已知數(shù)列\(zhòng)(\{a_n\}\)滿足\(a_1=1\)，\(a_{n+1}=2a_n+1\)。求數(shù)