高二數(shù)學(xué)期末考試題及答案下

一、單項(xiàng)選擇題（每題2分，共10題）1.若直線\(l\)的斜率\(k=-\sqrt{3}\)，則其傾斜角為（）A.\(30^{\circ}\)B.\(60^{\circ}\)C.\(120^{\circ}\)D.\(150^{\circ}\)2.橢圓\(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\)的焦距為（）A.\(2\sqrt{7}\)B.\(\sqrt{7}\)C.\(2\sqrt{25}\)D.\(\sqrt{25}\)3.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.\(-\frac{1}{2}\)4.命題“\(\forallx\inR\)，\(x^{2}+x+1\gt0\)”的否定是（）A.\(\forallx\inR\)，\(x^{2}+x+1\leq0\)B.\(\existsx\inR\)，\(x^{2}+x+1\leq0\)C.\(\existsx\inR\)，\(x^{2}+x+1\lt0\)D.\(\forallx\inR\)，\(x^{2}+x+1\lt0\)5.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{3}=5\)，\(a_{5}=9\)，則\(a_{7}\)的值為（）A.\(11\)B.\(12\)C.\(13\)D.\(14\)6.若\(x\gt0\)，\(y\gt0\)，且\(x+y=1\)，則\(\frac{1}{x}+\frac{1}{y}\)的最小值為（）A.\(2\)B.\(4\)C.\(6\)D.\(8\)7.拋物線\(y^{2}=8x\)的焦點(diǎn)坐標(biāo)是（）A.\((2,0)\)B.\((0,2)\)C.\((4,0)\)D.\((0,4)\)8.已知雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a\gt0,b\gt0)\)的漸近線方程為\(y=\pm\frac{3}{4}x\)，則該雙曲線的離心率為（）A.\(\frac{5}{4}\)B.\(\frac{5}{3}\)C.\(\frac{4}{3}\)D.\(\frac{4}{5}\)9.已知函數(shù)\(f(x)=x^{3}+ax^{2}+bx+c\)，若\(f^{\prime}(1)=0\)，則\(2a+b\)的值為（）A.\(-3\)B.\(-2\)C.\(-1\)D.\(0\)10.若曲線\(y=x^{3}\)在點(diǎn)\((a,a^{3})\)處的切線與直線\(x+3y+1=0\)垂直，則\(a\)的值為（）A.\(1\)B.\(-1\)C.\(\pm1\)D.\(\pm3\)二、多項(xiàng)選擇題（每題2分，共10題）1.下列說(shuō)法正確的是（）A.若\(a\gtb\)，\(c\gtd\)，則\(a-c\gtb-d\)B.若\(a\gtb\)，\(c\gtd\)，則\(ac\gtbd\)C.若\(a\gtb\)，則\(a^{2}\gtb^{2}\)D.若\(a\gtb\gt0\)，\(c\ltd\lt0\)，則\(ac\ltbd\)2.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\(F_{1}\)，\(F_{2}\)，點(diǎn)\(P\)在橢圓上，以下說(shuō)法正確的是（）A.\(\vertPF_{1}\vert+\vertPF_{2}\vert=2a\)B.離心率\(e=\frac{c}{a}\)（\(c\)為半焦距）C.當(dāng)點(diǎn)\(P\)為橢圓短軸端點(diǎn)時(shí)，\(\angleF_{1}PF_{2}\)最大D.橢圓的長(zhǎng)軸長(zhǎng)為\(2b\)3.已知直線\(l_{1}:A_{1}x+B_{1}y+C_{1}=0\)，\(l_{2}:A_{2}x+B_{2}y+C_{2}=0\)，下列說(shuō)法正確的是（）A.若\(l_{1}\parallell_{2}\)，則\(\frac{A_{1}}{A_{2}}=\frac{B_{1}}{B_{2}}\neq\frac{C_{1}}{C_{2}}\)B.若\(l_{1}\perpl_{2}\)，則\(A_{1}A_{2}+B_{1}B_{2}=0\)C.若\(A_{1}B_{2}-A_{2}B_{1}=0\)，則\(l_{1}\parallell_{2}\)D.若\(l_{1}\)與\(l_{2}\)重合，則\(A_{1}=A_{2}\)，\(B_{1}=B_{2}\)，\(C_{1}=C_{2}\)4.下列關(guān)于等差數(shù)列的說(shuō)法正確的是（）A.若\(\{a_{n}\}\)是等差數(shù)列，\(a_{m}+a_{n}=a_{p}+a_{q}\)，則\(m+n=p+q\)B.若\(\{a_{n}\}\)是等差數(shù)列，\(S_{n}\)為其前\(n\)項(xiàng)和，則\(S_{n}\)，\(S_{2n}-S_{n}\)，\(S_{3n}-S_{2n}\)仍成等差數(shù)列C.等差數(shù)列\(zhòng)(\{a_{n}\}\)的通項(xiàng)公式\(a_{n}\)是關(guān)于\(n\)的一次函數(shù)D.等差數(shù)列的前\(n\)項(xiàng)和公式\(S_{n}\)是關(guān)于\(n\)的二次函數(shù)（常數(shù)項(xiàng)可為\(0\)）5.已知向量\(\vec{a}=(x_{1},y_{1})\)，\(\vec=(x_{2},y_{2})\)，則下列結(jié)論正確的是（）A.\(\vec{a}\cdot\vec=x_{1}x_{2}+y_{1}y_{2}\)B.若\(\vec{a}\perp\vec\)，則\(x_{1}x_{2}+y_{1}y_{2}=0\)C.\(\vert\vec{a}\vert=\sqrt{x_{1}^{2}+y_{1}^{2}}\)D.若\(\vec{a}\parallel\vec\)，則\(x_{1}y_{2}-x_{2}y_{1}=0\)6.已知雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a\gt0,b\gt0)\)，以下說(shuō)法正確的是（）A.實(shí)軸長(zhǎng)為\(2a\)B.虛軸長(zhǎng)為\(2b\)C.漸近線方程為\(y=\pm\frac{a}x\)D.離心率\(e=\frac{c}{a}\)（\(c\)為半焦距，且\(c^{2}=a^{2}+b^{2}\)）7.已知函數(shù)\(y=f(x)\)，下列說(shuō)法正確的是（）A.\(f^{\prime}(x_{0})\)表示函數(shù)\(y=f(x)\)在\(x=x_{0}\)處的切線斜率B.函數(shù)\(y=f(x)\)的導(dǎo)數(shù)\(f^{\prime}(x)\)也是一個(gè)函數(shù)C.若\(f^{\prime}(x)\gt0\)在區(qū)間\((a,b)\)上恒成立，則\(y=f(x)\)在\((a,b)\)上單調(diào)遞增D.若\(f^{\prime}(x_{0})=0\)，則\(x=x_{0}\)是函數(shù)\(y=f(x)\)的極值點(diǎn)8.已知拋物線\(y^{2}=2px(p\gt0)\)，以下說(shuō)法正確的是（）A.焦點(diǎn)坐標(biāo)為\((\frac{p}{2},0)\)B.準(zhǔn)線方程為\(x=-\frac{p}{2}\)C.拋物線上一點(diǎn)\(M(x_{0},y_{0})\)到焦點(diǎn)的距離為\(x_{0}+\frac{p}{2}\)D.過(guò)焦點(diǎn)的弦長(zhǎng)為\(x_{1}+x_{2}+p\)（\(x_{1},x_{2}\)為弦端點(diǎn)橫坐標(biāo)）9.已知\(x\gt0\)，\(y\gt0\)，且\(x+2y=1\)，則（）A.\(xy\)的最大值為\(\frac{1}{8}\)B.\(\frac{1}{x}+\frac{1}{y}\)的最小值為\(3+2\sqrt{2}\)C.\(x^{2}+y^{2}\)的最小值為\(\frac{1}{5}\)D.\(y^{2}+2x\)的最大值為\(\frac{9}{8}\)10.已知函數(shù)\(f(x)=x^{3}-3x\)，以下說(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.\(f(x)\)的單調(diào)遞增區(qū)間為\((-\infty,-1)\)和\((1,+\infty)\)D.\(f(x)\)的單調(diào)遞減區(qū)間為\((-1,1)\)三、判斷題（每題2分，共10題）1.若\(a\gtb\)，則\(\frac{1}{a}\lt\frac{1}\)。（）2.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a\gtb\gt0)\)中，\(a\)一定大于\(b\)大于\(0\)。（）3.直線\(y=kx+b\)的斜率一定存在。（）4.若向量\(\vec{a}\)與向量\(\vec\)共線，則存在實(shí)數(shù)\(\lambda\)，使得\(\vec{a}=\lambda\vec\)。（）5.等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項(xiàng)和\(S_{n}=na_{1}+\frac{n(n-1)}{2}d\)（\(d\)為公差）。（）6.雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a\gt0,b\gt0)\)的漸近線互相垂直時(shí)，\(a=b\)。（）7.拋物線\(y^{2}=4x\)上一點(diǎn)到焦點(diǎn)的距離等于到準(zhǔn)線的距離。（）8.若函數(shù)\(f(x)\)在區(qū)間\((a,b)\)上的導(dǎo)數(shù)\(f^{\prime}(x)\lt0\)，則\(f(x)\)在\((a,b)\)上單調(diào)遞減。（）9.若\(x\gt0\)，\(y\gt0\)，且\(x+y=1\)，則\(xy\)的最大值為\(\frac{1}{4}\)。（）10.命題“若\(p\)，則\(q\)”的逆否命題是“若\(\negq\)，則\(\negp\)”。（）四、簡(jiǎn)答題（每題5分，共4題）1.求等差數(shù)列\(zhòng)(\{a_{n}\}\)：\(3\)，\(7\)，\(11\)，\(\cdots\)的通項(xiàng)公式與前\(n\)項(xiàng)和公式。答案：首項(xiàng)\(a_{1}=3\)，公差\(d=4\)。通項(xiàng)公式\(a_{n}=a_{1}+(n-1)d=3+4(n-1)=4n-1\)。前\(n\)項(xiàng)和公式\(S_{n}=na_{1}+\frac{n(n-1)}{2}d=3n+\frac{n(n-1)}{2}\times4=2n^{2}+n\)。2.已知橢圓方程\(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)，求其長(zhǎng)軸長(zhǎng)、短軸長(zhǎng)、焦距和離心率。答案：\(a^{2}=25\)，\(a=5\)，長(zhǎng)軸長(zhǎng)\(2a=10\)；\(b^{2}=16\)，\(b=4\)，短軸長(zhǎng)\(2b=8\)；\(c^{2}=a^{2}-b^{2}=9\)，\(c=3\)，焦距\(2c=6\)；離心率\(e=\frac{c}{a}=\frac{3}{5}\)。3.求過(guò)點(diǎn)\((1,2)\)且斜率為\(3\)的直線方程。答案：由點(diǎn)斜式\(y-y_{0}=k(x-x_{0})\)（\(x_{0}=1\)，\(y_{0}=2\)，\(k=3\)），可得\(y-2=3(x-1)\)，整理得\(3x-y-1=0\)。4.求函數(shù)\(f(x)=x^{2}-4x+3\)的單調(diào)區(qū)間。答案：對(duì)\(f(x)\)求導(dǎo)得\(f^{\prime}(x)=2x-4\)。令\(f^{\prime}(x)\gt0\)，即\(2x-4\gt0\)，解得\(x\gt2\)，單調(diào)遞增區(qū)間為\((2,+\infty)\)；令\(f^{\prime}(x)\lt0\)，即\(2x-4\lt0\)，解得\(x\lt2\)，單調(diào)遞減區(qū)間為\((-\infty,2)\)。五、討論題（每題5分，共4題）1.討論直線與橢圓的位置關(guān)系有哪些判斷方法。答案：可聯(lián)立直線與橢圓方程，消去一個(gè)變量得到一元二次方程。通過(guò)判別式\(\Delta\)判斷，\(\Delta\gt0\)時(shí)相交，\(\De