历届高中数学试题和答案

历届高中数学试题和答案

一、单项选择题(每题2分,共10题)1.函数\(y=\sinx\)的最小正周期是(）A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)2.已知集合\(A=\{1,2,3\}\),\(B=\{2,3,4\}\),则\(A\capB\)=()A.\(\{1,2,3,4\}\)B.\(\{2,3\}\)C.\(\{1,4\}\)D.\(\varnothing\)3.直线\(y=2x+1\)的斜率是()A.\(-2\)B.\(-1\)C.\(1\)D.\(2\)4.若\(a\gtb\),则下列不等式成立的是()A.\(a^2\gtb^2\)B.\(ac\gtbc\)C.\(a+c\gtb+c\)D.\(\frac{1}{a}\lt\frac{1}{b}\)5.抛物线\(y^2=4x\)的焦点坐标是()A.\((1,0)\)B.\((0,1)\)C.\((2,0)\)D.\((0,2)\)6.已知向量\(\vec{a}=(1,2)\),\(\vec{b}=(3,4)\),则\(\vec{a}+\vec{b}\)=()A.\((4,6)\)B.\((2,2)\)C.\((-2,-2)\)D.\((-4,-6)\)7.\(\cos60^{\circ}\)的值为()A.\(\frac{1}{2}\)B.\(\frac{\sqrt{2}}{2}\)C.\(\frac{\sqrt{3}}{2}\)D.\(1\)8.等差数列\(\{a_n\}\)中,\(a_1=1\),\(d=2\),则\(a_5\)=()A.\(9\)B.\(11\)C.\(13\)D.\(15\)9.函数\(y=\log_2x\)的定义域是()A.\((0,+\infty)\)B.\((-\infty,0)\)C.\((0,1]\)D.\([1,+\infty)\)10.已知\(\sin\alpha=\frac{3}{5}\),且\(\alpha\)是第二象限角,则\(\cos\alpha\)=()A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(-\frac{3}{4}\)二、多项选择题(每题2分,共10题)1.下列函数中,是偶函数的有（）A.\(y=x^2\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=|x|\)2.以下哪些是直线的方程形式()A.点斜式B.斜截式C.两点式D.截距式3.关于椭圆\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)(\(a\gtb\gt0\)),正确的有()A.长轴长为\(2a\)B.短轴长为\(2b\)C.离心率\(e\lt1\)D.焦点在\(x\)轴4.以下属于基本初等函数的有()A.幂函数B.指数函数C.对数函数D.三角函数5.等差数列\(\{a_n\}\)的性质有()A.\(a_n=a_1+(n-1)d\)B.若\(m+n=p+q\),则\(a_m+a_n=a_p+a_q\)C.\(S_n=\frac{n(a_1+a_n)}{2}\)D.\(a_{n+1}-a_n=d\)6.已知向量\(\vec{a}=(x_1,y_1)\),\(\vec{b}=(x_2,y_2)\),则()A.\(\vec{a}\cdot\vec{b}=x_1x_2+y_1y_2\)B.若\(\vec{a}\parallel\vec{b}\),则\(x_1y_2-x_2y_1=0\)C.\(|\vec{a}|=\sqrt{x_1^2+y_1^2}\)D.\(\vec{a}+\vec{b}=(x_1+x_2,y_1+y_2)\)7.下列不等式中,正确的有()A.\(x^2+1\geq2x\)B.\(a^2+b^2\geq2ab\)C.\(\frac{a+b}{2}\geq\sqrt{ab}\)(\(a\gt0,b\gt0\))D.\(x^2-2x+3\gt0\)8.对于函数\(y=A\sin(\omegax+\varphi)\),影响其图象的因素有（)A.\(A\)(振幅)B.\(\omega\)(角速度)C.\(\varphi\)(初相)D.\(x\)的取值范围9.以下哪些是等比数列的性质()A.\(a_n=a_1q^{n-1}\)B.若\(m+n=p+q\),则\(a_m\cdota_n=a_p\cdota_q\)C.\(S_n=\frac{a_1(1-q^n)}{1-q}\)(\(q\neq1\))D.\(\frac{a_{n+1}}{a_n}=q\)10.下列命题中,真命题有（）A.垂直于同一条直线的两条直线平行B.平行于同一个平面的两条直线平行C.过直线外一点有且只有一条直线与已知直线平行D.两条平行直线确定一个平面三、判断题(每题2分,共10题)1.空集是任何集合的子集。()2.函数\(y=\frac{1}{x}\)在定义域内是单调递减函数。()3.若\(a\cdotb=0\),则\(\vec{a}\perp\vec{b}\)。（)4.直线\(Ax+By+C=0\)(\(A\)、\(B\)不同时为\(0\))的斜率为\(-\frac{A}{B}\)。()5.双曲线\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)的渐近线方程是\(y=\pm\frac{b}{a}x\)。()6.若\(\sin\alpha=\sin\beta\),则\(\alpha=\beta\)。(）7.数列\(\{a_n\}\)的前\(n\)项和\(S_n=n^2+1\),则\(a_n=2n-1\)。（)8.函数\(y=\cos^2x-\sin^2x\)的最小正周期是\(\pi\)。()9.两个平面垂直,则一个平面内垂直于交线的直线垂直于另一个平面。(）10.若\(a\gtb\gt0\),\(c\ltd\lt0\),则\(ac\ltbd\)。（）四、简答题(每题5分,共4题)1.求函数\(y=2x^2-4x+3\)的对称轴和顶点坐标。答案:对于二次函数\(y=ax^2+bx+c\),对称轴\(x=-\frac{b}{2a}\)。此函数\(a=2\),\(b=-4\),对称轴\(x=1\)。把\(x=1\)代入函数得\(y=1\),顶点坐标为\((1,1)\)。2.已知等差数列\(\{a_n\}\)中,\(a_3=5\),\(a_5=9\),求\(a_1\)和\(d\)。答案：由等差数列通项公式\(a_n=a_1+(n-1)d\)。\(a_3=a_1+2d=5\),\(a_5=a_1+4d=9\),两式相减得\(2d=4\),\(d=2\),再代入\(a_1+2d=5\)得\(a_1=1\)。3.求圆\((x-1)^2+(y+2)^2=9\)的圆心坐标和半径。答案:圆的标准方程\((x-a)^2+(y-b)^2=r^2\),圆心坐标为\((a,b)\),半径为\(r\)。此圆中\(a=1\),\(b=-2\),\(r=3\),圆心坐标\((1,-2)\),半径\(3\)。4.已知\(\tan\alpha=2\),求\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值。答案：将\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)分子分母同时除以\(\cos\alpha\)得\(\frac{\tan\alpha+1}{\tan\alpha-1}\),把\(\tan\alpha=2\)代入得\(\frac{2+1}{2-1}=3\)。五、讨论题(每题5分,共4题)1.讨论函数\(y=x^3\)的单调性和奇偶性。答案:对\(y=x^3\)求导得\(y^\prime=3x^2\geq0\),所以在\(R\)上单调递增。又\(f(-x)=(-x)^3=-x^3=-f(x)\),所以\(y=x^3\)是奇函数。2.讨论直线与圆的位置关系有哪些判断方法。答案:一是几何法,计算圆心到直线的距离\(d\),与半径\(r\)比较,\(d\gtr\)相离,\(d=r\)相切,\(d\ltr\)相交;二是代数法,联立直线与圆方程得方程组,消元后看判别式\(\Delta\),\(\Delta\gt0\)相交,\(\Delta=0\)相切,\(\Delta\lt0\)相离。3.讨论在等比数列中,当公比\(q\)满足不同条件时,数列的增减性情况。答案：当\(q\gt1\),\(a_1\gt0\)或\(0\ltq\lt1\),\(a_1\lt0\)时,数列递增；当\(q