必修四数学试题及答案

必修四数学试题及答案

一、单项选择题（每题2分，共20分）1.已知角\(\alpha\)的终边过点\((-3,4)\)，则\(\sin\alpha\)的值为（）A.\(\frac{3}{5}\)B.\(\frac{4}{5}\)C.\(-\frac{3}{5}\)D.\(-\frac{4}{5}\)2.函数\(y=\sinx\)的最小正周期是（）A.\(\frac{\pi}{2}\)B.\(\pi\)C.\(2\pi\)D.\(4\pi\)3.已知\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow{b}=(3,4)\)，则\(\overrightarrow{a}\cdot\overrightarrow{b}\)等于（）A.5B.11C.14D.164.化简\(\cos(\alpha+\pi)\)的结果是（）A.\(\cos\alpha\)B.\(-\cos\alpha\)C.\(\sin\alpha\)D.\(-\sin\alpha\)5.已知\(\tan\alpha=2\)，则\(\frac{\sin\alpha}{\cos\alpha}\)的值为（）A.\(\frac{1}{2}\)B.2C.\(\pm2\)D.\(\frac{2}{3}\)6.函数\(y=\cos(2x+\frac{\pi}{3})\)的对称轴方程是（）A.\(x=\frac{k\pi}{2}+\frac{\pi}{6}(k\inZ)\)B.\(x=\frac{k\pi}{2}-\frac{\pi}{6}(k\inZ)\)C.\(x=k\pi+\frac{\pi}{6}(k\inZ)\)D.\(x=k\pi-\frac{\pi}{6}(k\inZ)\)7.已知\(\overrightarrow{AB}=(2,3)\)，点\(A\)的坐标为\((1,2)\)，则点\(B\)的坐标为（）A.\((3,5)\)B.\((-1,-1)\)C.\((1,1)\)D.\((2,3)\)8.若\(\sin\alpha=\frac{1}{2}\)，且\(\alpha\)在第二象限，则\(\cos\alpha\)的值为（）A.\(\frac{\sqrt{3}}{2}\)B.\(-\frac{\sqrt{3}}{2}\)C.\(\frac{1}{2}\)D.\(-\frac{1}{2}\)9.函数\(y=A\sin(\omegax+\varphi)\)（\(A\gt0,\omega\gt0\)）的振幅是（）A.\(A\)B.\(\omega\)C.\(\varphi\)D.\(2\pi\)10.已知\(\overrightarrow{a}=(2,-1)\)，\(\overrightarrow{b}=(m,3)\)，若\(\overrightarrow{a}\perp\overrightarrow{b}\)，则\(m\)的值为（）A.\(\frac{3}{2}\)B.\(-\frac{3}{2}\)C.\(\frac{2}{3}\)D.\(-\frac{2}{3}\)答案：1.B2.C3.C4.B5.B6.B7.A8.B9.A10.A二、多项选择题（每题2分，共20分）1.下列函数中，是奇函数的有（）A.\(y=\sinx\)B.\(y=\cosx\)C.\(y=\tanx\)D.\(y=\sin2x\)2.已知向量\(\overrightarrow{a}=(1,2)\)，\(\overrightarrow{b}=(-1,3)\)，则（）A.\(\overrightarrow{a}+\overrightarrow{b}=(0,5)\)B.\(\overrightarrow{a}-\overrightarrow{b}=(2,-1)\)C.\(\overrightarrow{a}\cdot\overrightarrow{b}=5\)D.\(|\overrightarrow{a}|=\sqrt{5}\)3.以下哪些是\(\sin\alpha\)的诱导公式（）A.\(\sin(\alpha+2k\pi)=\sin\alpha(k\inZ)\)B.\(\sin(\pi+\alpha)=-\sin\alpha\)C.\(\sin(-\alpha)=-\sin\alpha\)D.\(\sin(\frac{\pi}{2}-\alpha)=\cos\alpha\)4.函数\(y=\tanx\)的性质有（）A.周期为\(\pi\)B.定义域为\(\{x|x\neqk\pi+\frac{\pi}{2},k\inZ\}\)C.是奇函数D.在\((-\frac{\pi}{2},\frac{\pi}{2})\)上单调递增5.已知\(\overrightarrow{a}\)与\(\overrightarrow{b}\)为非零向量，则下列说法正确的是（）A.若\(\overrightarrow{a}\parallel\overrightarrow{b}\)，则存在实数\(\lambda\)，使得\(\overrightarrow{b}=\lambda\overrightarrow{a}\)B.若\(\overrightarrow{a}\cdot\overrightarrow{b}=0\)，则\(\overrightarrow{a}\perp\overrightarrow{b}\)C.\(|\overrightarrow{a}+\overrightarrow{b}|\leq|\overrightarrow{a}|+|\overrightarrow{b}|\)D.\(|\overrightarrow{a}-\overrightarrow{b}|\geq|\overrightarrow{a}|-|\overrightarrow{b}|\)6.函数\(y=\sin(2x-\frac{\pi}{6})\)的单调递增区间是（）A.\([k\pi-\frac{\pi}{6},k\pi+\frac{\pi}{3}](k\inZ)\)B.\([k\pi+\frac{\pi}{3},k\pi+\frac{5\pi}{6}](k\inZ)\)C.\([2k\pi-\frac{\pi}{6},2k\pi+\frac{\pi}{3}](k\inZ)\)D.\([2k\pi+\frac{\pi}{3},2k\pi+\frac{5\pi}{6}](k\inZ)\)7.下列三角函数值相等的有（）A.\(\sin\frac{\pi}{4}\)与\(\cos\frac{\pi}{4}\)B.\(\sin\frac{5\pi}{6}\)与\(\sin\frac{\pi}{6}\)C.\(\cos\frac{7\pi}{6}\)与\(\cos\frac{\pi}{6}\)D.\(\tan\frac{\pi}{4}\)与\(\tan\frac{5\pi}{4}\)8.向量\(\overrightarrow{a}=(x_1,y_1)\)，\(\overrightarrow{b}=(x_2,y_2)\)，则（）A.\(\overrightarrow{a}\cdot\overrightarrow{b}=x_1x_2+y_1y_2\)B.\(|\overrightarrow{a}|=\sqrt{x_1^{2}+y_1^{2}}\)C.若\(\overrightarrow{a}\parallel\overrightarrow{b}\)，则\(x_1y_2-x_2y_1=0\)D.若\(\overrightarrow{a}\perp\overrightarrow{b}\)，则\(x_1x_2+y_1y_2=0\)9.函数\(y=A\sin(\omegax+\varphi)\)（\(A\gt0,\omega\gt0\)）的图象可以通过对\(y=\sinx\)的图象进行（）变换得到。A.相位变换B.周期变换C.振幅变换D.上下平移变换10.已知\(\sin\alpha=\frac{3}{5}\)，\(\alpha\)为第二象限角，则（）A.\(\cos\alpha=-\frac{4}{5}\)B.\(\tan\alpha=-\frac{3}{4}\)C.\(\sin2\alpha=-\frac{24}{25}\)D.\(\cos2\alpha=\frac{7}{25}\)答案：1.ACD2.ACD3.ABCD4.ABCD5.ABCD6.A7.ABD8.ABCD9.ABC10.ABCD三、判断题（每题2分，共20分）1.函数\(y=\cosx\)的图象关于\(y\)轴对称。（）2.若\(\overrightarrow{a}\cdot\overrightarrow{b}=0\)，则\(\overrightarrow{a}\)与\(\overrightarrow{b}\)中至少有一个为零向量。（）3.\(\sin(\alpha+\beta)=\sin\alpha+\sin\beta\)。（）4.向量\(\overrightarrow{a}=(1,2)\)与\(\overrightarrow{b}=(2,4)\)共线。（）5.函数\(y=\tanx\)在定义域内是单调递增函数。（）6.若\(\sin\alpha=\sin\beta\)，则\(\alpha=\beta\)。（）7.已知\(\overrightarrow{AB}=\overrightarrow{DC}\)，则\(A\)，\(B\)，\(C\)，\(D\)四点构成平行四边形。（）8.函数\(y=A\sin(\omegax+\varphi)\)（\(A\gt0,\omega\gt0\)）的最大值是\(A\)。（）9.\(\cos(\frac{\pi}{2}-\alpha)=\sin\alpha\)。（）10.若\(\overrightarrow{a}=(x_1,y_1)\)，\(\overrightarrow{b}=(x_2,y_2)\)，则\(\overrightarrow{a}\parallel\overrightarrow{b}\)的充要条件是\(\frac{x_1}{x_2}=\frac{y_1}{y_2}\)。（）答案：1.√2.×3.×4.√5.×6.×7.×8.×9.√10.×四、简答题（每题5分，共20分）1.已知\(\overrightarrow{a}=(3,-1)\)，\(\overrightarrow{b}=(1,2)\)，求\(\overrightarrow{a}+\overrightarrow{b}\)的坐标。答案：\(\overrightarrow{a}+\overrightarrow{b}=(3+1,-1+2)=(4,1)\)。2.求函数\(y=\sin(2x+\frac{\pi}{4})\)的最小正周期。答案：对于\(y=A\sin(\omegax+\varphi)\)，\(T=\frac{2\pi}{\omega}\)，此函数\(\omega=2\)，所以最小正周期\(T=\frac{2\pi}{2}=\pi\)。3.已知\(\tan\alpha=3\)，求\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}\)的值。答案：分子分母同时除以\(\cos\alpha\)，则\(\frac{\sin\alpha+\cos\alpha}{\sin\alpha-\cos\alpha}=\frac{\tan\alpha+1}{\tan\alpha-1}\)，把\(\tan\alpha=3\)代入得\(\frac{3+1}{3-1}=2\)。4.写出\(\cos\alpha\)的二倍角公式。答案：\(\cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha=2\cos^{2}\alpha-1=1-2\sin^{2}\alpha\)。五、讨论题（每题5分，共20分）1.讨论函数\(y=\sinx\)与\(y=\cosx\)在\([0,2\pi]\)上的单调性有何不同。答案：\(y=\sinx\)在\([0,\frac{\pi}{2}]\)递增，\([\frac{\pi}{