難度大的數(shù)學(xué)試卷及答案

一、單項選擇題1.若函數(shù)\(f(x)=\frac{1}{3}x^{3}-ax^{2}+(a^{2}-1)x\)在區(qū)間\((m,m+1)\)上單調(diào)遞減，則實數(shù)\(a\)的取值范圍是（）A.\([-1,2]\)B.\([-4,1]\)C.\([1,2]\)D.\([-1,1]\)答案：D2.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(x,-4)\)，若\(\overrightarrow{a}\parallel\overrightarrow\)，則\(x\)的值為（）A.\(-2\)B.\(-8\)C.\(2\)D.\(8\)答案：A3.設(shè)等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和為\(S_{n}\)，若\(S_{3}=9\)，\(S_{6}=36\)，則\(a_{7}+a_{8}+a_{9}\)等于（）A.\(63\)B.\(45\)C.\(36\)D.\(27\)答案：B4.已知雙曲線\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a\gt0,b\gt0)\)的一條漸近線方程是\(y=\sqrt{3}x\)，它的一個焦點在拋物線\(y^{2}=24x\)的準(zhǔn)線上，則雙曲線的方程為（）A.\(\frac{x^{2}}{36}-\frac{y^{2}}{108}=1\)B.\(\frac{x^{2}}{9}-\frac{y^{2}}{27}=1\)C.\(\frac{x^{2}}{108}-\frac{y^{2}}{36}=1\)D.\(\frac{x^{2}}{27}-\frac{y^{2}}{9}=1\)答案：B5.若\(\sin(\alpha+\frac{\pi}{6})=\frac{1}{3}\)，則\(\cos(2\alpha+\frac{\pi}{3})\)等于（）A.\(\frac{7}{9}\)B.\(\frac{2}{3}\)C.\(-\frac{7}{9}\)D.\(-\frac{2}{3}\)答案：C6.已知函數(shù)\(f(x)\)是定義在\(R\)上的奇函數(shù)，當(dāng)\(x\geq0\)時，\(f(x)=x^{2}-2x\)，則\(f(-1)\)的值為（）A.\(3\)B.\(-1\)C.\(-3\)D.\(1\)答案：C7.直線\(y=kx+3\)與圓\((x-1)^{2}+(y-2)^{2}=4\)相交于\(M\)，\(N\)兩點，若\(\vertMN\vert\geq2\sqrt{3}\)，則\(k\)的取值范圍是（）A.\([-\frac{3}{4},0]\)B.\([-\infty,-\frac{3}{4}]\cup[0,+\infty]\)C.\([-\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3}]\)D.\([-\frac{2}{3},0]\)答案：A8.已知\(a=\log_{3}2\)，\(b=\ln2\)，\(c=5^{-\frac{1}{2}}\)，則\(a\)，\(b\)，\(c\)的大小關(guān)系為（）A.\(a\ltb\ltc\)B.\(c\lta\ltb\)C.\(b\ltc\lta\)D.\(c\ltb\lta\)答案：B9.設(shè)\(x\)，\(y\)滿足約束條件\(\begin{cases}x+y\geq1\\x-y\leq1\\y\leq1\end{cases}\)，則\(z=2x-y\)的最大值為（）A.\(3\)B.\(2\)C.\(1\)D.\(-1\)答案：A10.已知函數(shù)\(f(x)\)的導(dǎo)函數(shù)為\(f^\prime(x)\)，且滿足\(f(x)=2xf^\prime(1)+\lnx\)，則\(f^\prime(1)\)等于（）A.\(-e\)B.\(-1\)C.\(1\)D.\(e\)答案：B二、多項選擇題1.下列說法正確的是（）A.若\(a\gtb\)，\(c\gtd\)，則\(ac\gtbd\)B.若\(a\gtb\)，則\(a^{3}\gtb^{3}\)C.若\(a\gtb\gt0\)，\(m\gt0\)，則\(\frac{a+m}{b+m}\gt\frac{a}\)D.若\(a\gtb\)，\(c\gtd\)，則\(a-c\gtb-d\)答案：BC2.已知函數(shù)\(f(x)=\sin(2x+\varphi)\)，\(0\lt\varphi\lt\frac{\pi}{2}\)，其圖象的一條對稱軸是\(x=\frac{\pi}{6}\)，則下列說法正確的是（）A.\(\varphi=\frac{\pi}{6}\)B.\(f(x)\)在\([0,\frac{\pi}{6}]\)上單調(diào)遞增C.\(f(x)\)的圖象關(guān)于點\((\frac{5\pi}{12},0)\)對稱D.\(f(x)\)在\([0,\frac{\pi}{2}]\)上的最大值為\(1\)答案：ABCD3.關(guān)于橢圓\(C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a\gtb\gt0)\)，以下說法正確的是（）A.離心率\(e=\frac{c}{a}\)（\(c\)為半焦距），且\(0\lte\lt1\)B.過橢圓上一點\(P(x_{0},y_{0})\)的切線方程為\(\frac{x_{0}x}{a^{2}}+\frac{y_{0}y}{b^{2}}=1\)C.長軸長為\(2a\)，短軸長為\(2b\)D.橢圓\(C\)的焦點三角形的面積為\(b^{2}\tan\frac{\theta}{2}\)（\(\theta\)為焦點三角形的頂角）答案：ABCD4.已知數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=1\)，\(a_{n+1}=2a_{n}+1\)，則下列說法正確的是（）A.數(shù)列\(zhòng)(\{a_{n}+1\}\)是等比數(shù)列B.\(a_{n}=2^{n}-1\)C.數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和\(S_{n}=2^{n+1}-n-2\)D.數(shù)列\(zhòng)(\{a_{n}\}\)是遞增數(shù)列答案：ABD5.已知向量\(\overrightarrow{a}=(x_{1},y_{1})\)，\(\overrightarrow=(x_{2},y_{2})\)，則下列說法正確的是（）A.若\(\overrightarrow{a}\parallel\overrightarrow\)，則\(x_{1}y_{2}-x_{2}y_{1}=0\)B.若\(\overrightarrow{a}\perp\overrightarrow\)，則\(x_{1}x_{2}+y_{1}y_{2}=0\)C.\(\vert\overrightarrow{a}\vert=\sqrt{x_{1}^{2}+y_{1}^{2}}\)D.若\(\overrightarrow{a}\)與\(\overrightarrow\)的夾角為\(\theta\)，則\(\cos\theta=\frac{\overrightarrow{a}\cdot\overrightarrow}{\vert\overrightarrow{a}\vert\vert\overrightarrow\vert}\)答案：ABCD6.已知函數(shù)\(f(x)=x^{3}-3x\)，則下列說法正確的是（）A.\(f(x)\)有兩個極值點B.\(f(x)\)在\((-\infty,-1)\)上單調(diào)遞增C.\(f(x)\)的圖象關(guān)于原點對稱D.\(f(x)\)在\(x=1\)處取得極大值答案：ABC7.已知圓\(C_{1}:x^{2}+y^{2}=1\)，圓\(C_{2}:(x-3)^{2}+(y-4)^{2}=16\)，則下列說法正確的是（）A.兩圓的圓心距為\(5\)B.兩圓外切C.兩圓的公切線有\(zhòng)(3\)條D.兩圓相交答案：ABC8.已知\(a\)，\(b\)為正實數(shù)，且\(a+b=1\)，則下列說法正確的是（）A.\(ab\leq\frac{1}{4}\)B.\(\frac{1}{a}+\frac{1}\geq4\)C.\(a^{2}+b^{2}\geq\frac{1}{2}\)D.\(\sqrt{a}+\sqrt\leq\sqrt{2}\)答案：ABCD9.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)遞減，則下列不等式成立的是（）A.\(f(-3)\gtf(2)\gtf(1)\)B.\(f(1)\gtf(2)\gtf(-3)\)C.\(f(-3)=f(3)\ltf(2)\)D.\(f(2)\ltf(-3)=f(3)\)答案：BC10.已知\(f(x)\)是定義在\(R\)上的周期為\(2\)的函數(shù)，當(dāng)\(x\in[0,2)\)時，\(f(x)=x^{2}-2x\)，則下列說法正確的是（）A.\(f(-1)=f(3)\)B.\(f(x)\)在\([0,2)\)上的最小值為\(-1\)C.\(f(x)\)的圖象關(guān)于直線\(x=1\)對稱D.\(f(x)\)在\([2,4)\)上的解析式為\(f(x)=(x-2)^{2}-2(x-2)\)答案：ABCD三、判斷題1.若\(a\gtb\)，則\(\frac{1}{a}\lt\frac{1}\)。（）答案：錯誤2.函數(shù)\(y=\sinx\)的圖象的一個對稱中心是\((\pi,0)\)。（）答案：正確3.若直線\(l_{1}:A_{1}x+B_{1}y+C_{1}=0\)與直線\(l_{2}:A_{2}x+B_{2}y+C_{2}=0\)垂直，則\(A_{1}A_{2}+B_{1}B_{2}=0\)。（）答案：正確4.數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和\(S_{n}=n^{2}+1\)，則\(a_{n}=2n-1\)。（）答案：錯誤5.橢圓\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a\gtb\gt0)\)的離心率\(e\)越大，橢圓越圓。（）答案：錯誤6.函數(shù)\(f(x)=x^{3}\)在\(R\)上是奇函數(shù)且單調(diào)遞增。（）答案：正確7.若向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow=(x,4)\)，且\(\overrightarrow{a}\)與\(\overrightarrow\)共線，則\(x=2\)。（）答案：正確8.已知函數(shù)\(f(x)\)的定義域為\([0,2]\)，則函數(shù)\(f(2x)\)的定義域為\([0,1]\)。（）答案：正確9.若\(\tan\alpha=2\)，則\(\sin2\alpha=\frac{4}{5}\)。（）答案：正確10.直線\(y=kx+1\)與圓\(x^{2}+y^{2}-2x-3=0\)恒有兩個交點。（）答案：正確四、簡答題1.求函數(shù)\(f(x)=x^{3}-3x^{2}+2\)在區(qū)間\([-1,3]\)上的最大值和最小值。答案：先對\(f(x)\)求導(dǎo)得\(f^\prime(x)=3x^{2}-6x=3x(x-2)\)。令\(f^\prime(x)=0\)，解得\(x=0\)或\(x=2\)。分別計算\(f(-1)=(-1)^{3}-3\times(-1)^{2}+2=-2\)，\(f(0)=2\)，\(f(2)=2^{3}-3\times2^{2}+2=-2\)，\(f(3)=3^{3}-3\times3^{2}+2=2\)。所以最大值為\(2\)，最小值為\(-2\)。2.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和為\(S_{n}\)，\(a_{3}=5\)，\(S_{4}=16\)，求數(shù)列\(zhòng)(\{a_{n}\}\)的通項公式。答案：設(shè)等差數(shù)列\(zhòng)(\{a_{n}\}\)的首項為\(a_{1}\)，公差為\(d\)。由\(a_{3}=5\)可得\(a_{1}+2d=5\)；由\(S_{4}=16\)可得\(4