2026年教师资格证（学科知识与教学能力 数学）自测试题及答案

2026年教师资格证（学科知识与教学能力·数学）自测试题及答案

班级______姓名______（考试时间：90分钟满分100分）一、选择题（总共10题，每题3分，每题给出的选项中，只有一项符合题目要求，请将正确答案的序号填在括号内）1.函数$f(x)=\frac{1}{\ln(x-1)}$的定义域为（）A.$(1,+\infty)$B.$(1,2)\cup(2,+\infty)$C.$(2,+\infty)$D.$[2,+\infty)$2.已知向量$\vec{a}=(1,2)$，$\vec{b}=(3,4)$，则$\vec{a}$在$\vec{b}$方向上的投影为（）A.$\frac{\sqrt{5}}{5}$B.$\frac{2\sqrt{5}}{5}$C.$1$D.$\sqrt{5}$3.若双曲线$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$（$a\gt0$，$b\gt0$）的渐近线方程为$y=\pm\frac{3}{4}x$，则其离心率为（）A.$\frac{5}{4}$B.$\frac{4}{3}$C.$\frac{5}{3}$D.$\frac{\sqrt{7}}{4}$4.已知函数$f(x)$是定义在$R$上的奇函数，当$x\gt0$时，$f(x)=2^{x}-1$，则$f(-2)=$（）A.$-3$B.$3$C.$-\frac{3}{4}$D.$\frac{3}{4}$5.设等差数列$\{a_n\}$的前$n$项和为$S_n$，若$a_2=3$，$a_5=9$，则$S_5=$（）A.$15$B.$20$C.$25$D.$30$6.已知直线$l$过点$(1,0)$且倾斜角为$\frac{\pi}{4}$，则直线$l$的方程为（）A.$x-y-1=0$B.$x-y+1=0$C.$x+y-1=0$D.$x+y+1=0$7.函数$y=\sin(2x+\frac{\pi}{3})$的图象的一条对称轴方程为（）A.$x=\frac{\pi}{12}$B.$x=-\frac{\pi}{12}$C.$x=\frac{\pi}{6}$D.$x=-\frac{\pi}{6}$8.若$\log_a\frac{2}{3}\lt1$（$a\gt0$且$a\neq1$），则$a$的取值范围是（）A.$(0,\frac{2}{3})$B.$(\frac{2}{3},+\infty)$C.$(0,\frac{2}{3})\cup(1,+\infty)$D.$(\frac{2}{3},1)$9.已知圆$C$：$(x-1)^2+(y-2)^2=4$，则圆心坐标为（）A.$(1,2)$B.$(-1,-2)$C.$(2,1)$D.$(-2,-1)$10.已知$\triangleABC$中，角$A$，$B$，$C$所对的边分别为$a$，$b$，$c$，若$a=2$，$b=3$，$c=4$，则$\cosC=$（）A.$\frac{1}{4}$B.$-\frac{1}{4}$C.$\frac{1}{2}$D.$-\frac{1}{2}$二、填空题（总共5题，每题4分，请将答案填在题中横线上）1.已知函数$f(x)=x^3-3x$，则$f(x)$的单调递增区间为______。2.已知圆锥的底面半径为$1$，母线长为$3$，则该圆锥的体积为______。3.已知数列$\{a_n\}$满足$a_{n+1}=2a_n+1$，$a_1=1$，则$a_5=$______。4.已知向量$\vec{a}=(m,2)$，$\vec{b}=(1,-1)$，若$\vec{a}\perp\vec{b}$，则$m=$______。5.在区间$[0,2\pi]$上，函数$y=\sinx$与$y=\cosx$的图象交点的横坐标为______。三、解答题（总共每题10分，解答应写出文字说明、证明过程或演算步骤）1.已知函数$f(x)=x^2-2ax+1$，求函数$f(x)$在区间$[0,2]$上的最小值$g(a)$。\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_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