38届imo试题及答案
单项选择题(每题2分,共10题)
1.若集合\(A=\{x|x^2-5x+6=0\}\),\(B=\{2,3\}\),则\(A\)与\(B\)的关系是()
A.\(A\subsetneqqB\)B.\(A=B\)C.\(B\subsetneqqA\)D.\(A\capB=\varnothing\)
2.函数\(y=\sin(2x+\frac{\pi}{3})\)的最小正周期是()
A.\(\pi\)B.\(2\pi\)C.\(\frac{\pi}{2}\)D.\(4\pi\)
3.直线\(3x+4y-5=0\)与圆\(x^2+y^2=1\)的位置关系是()
A.相交B.相切C.相离D.无法确定
4.已知向量\(\overrightarrow{a}=(1,2)\),\(\overrightarrow{b}=(-2,m)\),若\(\overrightarrow{a}\parallel\overrightarrow{b}\),则\(m\)的值为()
A.\(4\)B.\(-4\)C.\(1\)D.\(-1\)
5.等差数列\(\{a_n\}\)中,\(a_1=1\),\(a_3+a_5=14\),其前\(n\)项和\(S_n=100\),则\(n\)等于()
A.\(9\)B.\(10\)C.\(11\)D.\(12\)
6.函数\(y=\log_2(x^2-1)\)的定义域是()
A.\((-1,1)\)B.\((-\infty,-1)\cup(1,+\infty)\)
C.\((-1,+\infty)\)D.\((1,+\infty)\)
7.若\(\alpha\)是第二象限角,\(\sin\alpha=\frac{3}{5}\),则\(\cos\alpha\)等于()
A.\(\frac{4}{5}\)B.\(-\frac{4}{5}\)C.\(\frac{3}{5}\)D.\(-\frac{3}{5}\)
8.抛物线\(y^2=8x\)的焦点坐标是()
A.\((2,0)\)B.\((0,2)\)C.\((-2,0)\)D.\((0,-2)\)
9.已知\(a=2^{0.3}\),\(b=0.3^2\),\(c=\log_2{0.3}\),则\(a\),\(b\),\(c\)的大小关系是()
A.\(a\gtb\gtc\)B.\(b\gta\gtc\)C.\(c\gta\gtb\)D.\(a\gtc\gtb\)
10.函数\(y=x^3-3x\)的极大值点是()
A.\(x=-1\)B.\(x=1\)C.\(x=0\)D.\(x=2\)
答案:1.B2.A3.B4.B5.B6.B7.B8.A9.A10.A
多项选择题(每题2分,共10题)
1.以下哪些是奇函数()
A.\(y=x^3\)B.\(y=\sinx\)C.\(y=x+1\)D.\(y=\frac{1}{x}\)
2.以下属于椭圆标准方程的是()
A.\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a\gtb\gt0)\)B.\(\frac{y^2}{a^2}+\frac{x^2}{b^2}=1(a\gtb\gt0)\)
C.\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)D.\(\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\)
3.下列关于数列极限的说法正确的是()
A.常数列一定有极限
B.若\(\lim\limits_{n\rightarrow\infty}a_n=A\),\(\lim\limits_{n\rightarrow\infty}b_n=B\),则\(\lim\limits_{n\rightarrow\infty}(a_n+b_n)=A+B\)
C.有界数列一定有极限
D.单调递增且有上界的数列一定有极限
4.以下哪些是基本不等式()
A.\(a^2+b^2\geq2ab\)(\(a,b\inR\))B.\(\fra