北師大版高中高二下冊(cè)數(shù)學(xué)期末試卷后附答案

一、填空題1.已知等差數(shù)列\(zhòng)(\{a_n\}\)中，\(a_3=5\)，\(a_7=13\)，則\(a_5=\)______。2.雙曲線(xiàn)\(\frac{x^2}{9}-\frac{y^2}{16}=1\)的漸近線(xiàn)方程為_(kāi)_____。3.若復(fù)數(shù)\(z=(m^2-1)+(m+1)i\)為純虛數(shù)（\(i\)為虛數(shù)單位），則實(shí)數(shù)\(m\)的值為_(kāi)_____。4.從\(5\)名男生和\(3\)名女生中任選\(2\)人參加活動(dòng)，至少有一名女生的選法有______種。5.函數(shù)\(y=x^3-3x^2+1\)的單調(diào)遞減區(qū)間是______。6.已知拋物線(xiàn)\(y^2=2px(p\gt0)\)的焦點(diǎn)坐標(biāo)為\((1,0)\)，則\(p=\)______。7.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(x,-4)\)，若\(\overrightarrow{a}\parallel\overrightarrow\)，則\(x=\)______。8.二項(xiàng)式\((2x-\frac{1}{x})^6\)的展開(kāi)式中常數(shù)項(xiàng)是______。9.已知\(\cos\alpha=\frac{3}{5}\)，\(\alpha\in(0,\frac{\pi}{2})\)，則\(\sin2\alpha=\)______。10.已知函數(shù)\(f(x)=x^2+2x+a\)，若\(f(x)\gt0\)對(duì)任意\(x\in[1,+\infty)\)恒成立，則實(shí)數(shù)\(a\)的取值范圍是______。二、單項(xiàng)選擇題1.已知\(a,b\inR\)，則“\(a\gtb\)”是“\(a^2\gtb^2\)”的（）A.充分不必要條件B.必要不充分條件C.充要條件D.既不充分也不必要條件2.若直線(xiàn)\(l_1:ax+2y+6=0\)與直線(xiàn)\(l_2:x+(a-1)y+a^2-1=0\)平行，則\(a\)的值為（）A.\(-1\)B.\(2\)C.\(-1\)或\(2\)D.\(1\)或\(-2\)3.已知\(\{a_n\}\)是等比數(shù)列，\(a_2=2\)，\(a_5=\frac{1}{4}\)，則公比\(q=\)（）A.\(-\frac{1}{2}\)B.\(-2\)C.\(2\)D.\(\frac{1}{2}\)4.函數(shù)\(f(x)=\frac{\lnx}{x}\)的最大值為（）A.\(e^{-1}\)B.\(e\)C.\(e^2\)D.\(\frac{10}{3}\)5.從\(1,2,3,4,5\)這\(5\)個(gè)數(shù)字中任取\(3\)個(gè)不同的數(shù)字，則這\(3\)個(gè)數(shù)字能構(gòu)成等差數(shù)列的概率為（）A.\(\frac{3}{10}\)B.\(\frac{2}{5}\)C.\(\frac{1}{2}\)D.\(\frac{3}{5}\)6.已知橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a\gtb\gt0)\)的離心率為\(\frac{\sqrt{3}}{2}\)，則雙曲線(xiàn)\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)的漸近線(xiàn)方程為（）A.\(y=\pm\frac{1}{2}x\)B.\(y=\pm2x\)C.\(y=\pm4x\)D.\(y=\pm\frac{1}{4}x\)7.已知函數(shù)\(f(x)=x^3-3x-1\)，若對(duì)于區(qū)間\([-3,2]\)上的任意\(x_1,x_2\)，都有\(zhòng)(\vertf(x_1)-f(x_2)\vert\leqt\)，則實(shí)數(shù)\(t\)的最小值是（）A.\(20\)B.\(18\)C.\(3\)D.\(0\)8.已知復(fù)數(shù)\(z=1+i\)，則\(\frac{z^2-2z}{z-1}=\)（）A.\(-2i\)B.\(2i\)C.\(-2\)D.\(2\)9.已知\(\overrightarrow{AB}=(2,3)\)，\(\overrightarrow{AC}=(3,t)\)，\(\vert\overrightarrow{BC}\vert=1\)，則\(\overrightarrow{AB}\cdot\overrightarrow{BC}=\)（）A.\(-3\)B.\(-2\)C.\(2\)D.\(3\)10.若\((1+x+x^2)^6=a_0+a_1x+a_2x^2+\cdots+a_{12}x^{12}\)，則\(a_2+a_4+\cdots+a_{12}=\)（）A.\(284\)B.\(356\)C.\(364\)D.\(378\)三、多項(xiàng)選擇題1.下列命題中正確的是（）A.若\(a\gtb\)，則\(ac^2\gtbc^2\)B.若\(a\gtb\)，\(c\gtd\)，則\(a-c\gtb-d\)C.若\(a\gtb\)，\(c\gtd\)，則\(ac\gtbd\)D.若\(ab\gt0\)，\(a\gtb\)，則\(\frac{1}{a}\lt\frac{1}\)2.已知直線(xiàn)\(l_1:y=k_1x+b_1\)，\(l_2:y=k_2x+b_2\)，則\(l_1\parallell_2\)的充要條件是（）A.\(k_1=k_2\)B.\(k_1=k_2\)且\(b_1\neqb_2\)C.\(k_1\neqk_2\)D.\(k_1\neqk_2\)且\(b_1\neqb_2\)3.已知等比數(shù)列\(zhòng)(\{a_n\}\)的公比為\(q\)，則下列命題正確的是（）A.若\(q\gt1\)，則\(\{a_n\}\)為遞增數(shù)列B.若\(0\ltq\lt1\)，則\(\{a_n\}\)為遞減數(shù)列C.若\(q\lt0\)，則\(\{a_n\}\)為擺動(dòng)數(shù)列D.若\(q=1\)，則\(\{a_n\}\)為常數(shù)列4.下列函數(shù)中，在區(qū)間\((0,+\infty)\)上單調(diào)遞增的是（）A.\(y=x^{\frac{1}{2}}\)B.\(y=2^{-x}\)C.\(y=\ln(x+1)\)D.\(y=\frac{1}{x}\)5.已知橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a\gtb\gt0)\)的左、右焦點(diǎn)分別為\(F_1,F_2\)，過(guò)\(F_1\)的直線(xiàn)交橢圓于\(A,B\)兩點(diǎn)，若\(\vertAF_2\vert+\vertBF_2\vert=10\)，\(\vertAB\vert=6\)，則橢圓的長(zhǎng)軸長(zhǎng)為（）A.\(8\)B.\(10\)C.\(16\)D.\(20\)6.已知函數(shù)\(f(x)=x^3-3x^2+2\)，則下列說(shuō)法正確的是（）A.\(f(x)\)的極大值為\(2\)B.\(f(x)\)的極小值為\(-2\)C.\(f(x)\)在區(qū)間\((-\infty,0)\)上單調(diào)遞增D.\(f(x)\)在區(qū)間\((2,+\infty)\)上單調(diào)遞增7.已知復(fù)數(shù)\(z_1=1+i\)，\(z_2=2-i\)，則下列說(shuō)法正確的是（）A.\(z_1+z_2=3\)B.\(z_1z_2=3+i\)C.\(\vertz_1\vert=\sqrt{2}\)D.\(\frac{z_1}{z_2}=\frac{1}{5}+\frac{3}{5}i\)8.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(2,-3)\)，則下列說(shuō)法正確的是（）A.\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為鈍角B.\(\overrightarrow{a}\)在\(\overrightarrow\)上的投影向量為\(-\frac{4}{13}\overrightarrow\)C.\((\overrightarrow{a}+\overrightarrow)\perp\overrightarrow{a}\)D.\(2\overrightarrow{a}-\overrightarrow=(0,7)\)9.已知二項(xiàng)式\((x+\frac{1}{x})^n\)的展開(kāi)式中第\(3\)項(xiàng)與第\(7\)項(xiàng)的二項(xiàng)式系數(shù)相等，則下列說(shuō)法正確的是（）A.\(n=8\)B.展開(kāi)式中常數(shù)項(xiàng)為\(70\)C.展開(kāi)式中各項(xiàng)系數(shù)之和為\(2^8\)D.展開(kāi)式中含\(x^2\)項(xiàng)的系數(shù)為\(56\)10.已知函數(shù)\(f(x)=\sin(\omegax+\varphi)(\omega\gt0,0\lt\varphi\lt\pi)\)的最小正周期為\(\pi\)，且\(f(\frac{\pi}{4})=0\)，則下列說(shuō)法正確的是（）A.\(\omega=2\)B.\(\varphi=\frac{\pi}{2}\)C.\(f(x)\)在區(qū)間\([0,\frac{\pi}{2}]\)上單調(diào)遞減D.\(f(x)\)的圖象關(guān)于點(diǎn)\((\frac{\pi}{4},0)\)對(duì)稱(chēng)四、判斷題1.若\(a\gtb\)，則\(a^3\gtb^3\)。（）2.直線(xiàn)\(y=kx+1\)與橢圓\(\frac{x^2}{m}+\frac{y^2}{5}=1\)恒有公共點(diǎn)，則\(m\)的取值范圍是\(m\geq1\)且\(m\neq5\)。（）3.若等比數(shù)列\(zhòng)(\{a_n\}\)的公比\(q\lt0\)，則\(\{a_n\}\)一定是遞減數(shù)列。（）4.函數(shù)\(f(x)=x^3-3x+1\)在區(qū)間\([-2,2]\)上的最大值為\(3\)。（）5.已知復(fù)數(shù)\(z=1+i\)，則\(z^2=2i\)。（）6.若向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(2,4)\)，則\(\overrightarrow{a}\)與\(\overrightarrow\)共線(xiàn)。（）7.二項(xiàng)式\((x-\frac{1}{x})^6\)的展開(kāi)式中含\(x^2\)項(xiàng)的系數(shù)為\(15\)。（）8.已知\(\cos\alpha=\frac{1}{3}\)，\(\alpha\in(0,\frac{\pi}{2})\)，則\(\sin2\alpha=\frac{4\sqrt{2}}{9}\)。（）9.若\(a\gtb\)，\(c\gtd\)，則\(ac\gtbd\)。（）10.已知函數(shù)\(f(x)=x^2+2x+a\)在區(qū)間\([-1,+\infty)\)上單調(diào)遞增，則\(a\)的取值范圍是\(a\geq-1\)。（）五、簡(jiǎn)答題1.求函數(shù)\(f(x)=x^3-3x^2-9x+5\)的單調(diào)區(qū)間。2.已知等差數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和為\(S_n\)，\(a_3=5\)，\(S_5=25\)，求數(shù)列\(zhòng)(\{a_n\}\)的通項(xiàng)公式。3.已知橢圓\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a\gtb\gt0)\)的離心率為\(\frac{\sqrt{3}}{2}\)，且過(guò)點(diǎn)\((\sqrt{3},\frac{1}{2})\)，求橢圓的標(biāo)準(zhǔn)方程。4.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(-3,4)\)，求\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角。六、討論題1.討論函數(shù)\(f(x)=x^3-3ax+1(a\inR)\)的單調(diào)性。2.已知等比數(shù)列\(zhòng)(\{a_n\}\)的公比\(q\gt0\)，\(a_1=1\)，\(3a_3=2a_2+a_4\)，討論數(shù)列\(zhòng)(\{a_n\}\)的前\(n\)項(xiàng)和\(S_n\)的變化情況。3.已知直線(xiàn)\(l:y=kx+1\)與橢圓\(C:\frac{x^2}{4}+\frac{y^2}{3}=1\)，討論直線(xiàn)\(l\)與橢圓\(C\)的位置關(guān)系。4.已知函數(shù)\(f(x)=x^2+2ax+1\)在區(qū)間\([-1,2]\)上的最大值為\(4\)，討論實(shí)數(shù)\(a\)的值。答案一、填空題1.\(9\)2.\(y=\pm\frac{4}{3}x\)3.\(1\)4.\(25\)5.\((0,2)\)6.\(2\)7.\(-2\)8.\(-160\)9.\(\frac{24}{25}\)10.\(a\gt-3\)二、單項(xiàng)選擇題1.D2.A3.D4.A5.A