江蘇省無錫市濱湖區(qū)太湖高級中學2025-2025學年高二上學期期中數(shù)學試題考試時間：90分鐘?總分：150分?年級/班級：高二〔1〕班一、選擇題〔每題5分，共30分〕要求：在每題給出的四個選項中，只有一項為哪一項符合題目要求的。1.知曉函數(shù)$f(x)=\sqrt{1-x^2}$的定義域為$A$，那么集合$A$的表示形式為〔〕A.$\{x|-1\leqx\leq1\}$B.$\{x|0\leqx\leq1\}$C.$\{x|-1\leqx<1\}$D.$\{x|0<x\leq1\}$2.函數(shù)$f(x)=ax^2+bx+c$的圖像開口向上，且頂點坐標為$(1,3)$，假設$f(0)=2$，那么實數(shù)$a$的值為〔〕A.1B.2C.3D.43.設等差數(shù)列$\{a_n\}$的公差為$d$，假設$a_1=2$，$a_3=8$，那么$a_5$的值為〔〕A.10B.12C.14D.164.假設等比數(shù)列$\{a_n\}$的公比為$q$，假設$a_1=3$，$a_4=81$，那么$a_2$的值為〔〕A.9B.27C.81D.2435.知曉復數(shù)$z$滿足$|z-1|=|z+1|$，那么復數(shù)$z$在復平面內的軌跡方程為〔〕A.$y=0$B.$x^2+y^2=1$C.$x^2-y^2=1$D.$x^2+y^2=2$6.假設函數(shù)$f(x)=\log_2(x-1)+\log_2(3-x)$的定義域為$D$，那么集合$D$的表示形式為〔〕A.$\{x|1<x<3\}$B.$\{x|1\leqx\leq3\}$C.$\{x|0<x<3\}$D.$\{x|0\leqx\leq3\}$二、填空題〔每題5分，共30分〕要求：把答案填在橫線上。7.知曉函數(shù)$f(x)=x^3-3x^2+4x-1$，那么$f(1)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\三、解答題〔每題15分，共30分〕要求：解答以下各題。8.知曉函數(shù)$f(x)=\frac{x^2-4x+3}{x-1}$，求函數(shù)$f(x)$的值域。9.知曉等差數(shù)列$\{a_n\}$的前$n$項和為$S_n$，假設$a_1=2$，$S_5=30$，求等差數(shù)列$\{a_n\}$的通項公式。四、證明題〔每題15分，共30分〕要求：證明以下各題。10.證明：假設$a$，$b$，$c$是等差數(shù)列中的三個連續(xù)項，那么$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=3$。11.證明：假設$\triangleABC$的邊長滿足$a^2+b^2-c^2=ab$，那么$\triangleABC$是直角三角形。五、計算題〔每題15分，共30分〕要求：精確計算以下各題。12.知曉$a=3$，$b=4$，$c=5$，求$(a+b+c)^2$的值。13.假設等比數(shù)列$\{a_n\}$的首項為$a_1=2$，公比為$q=3$，求$a_8$的值。六、應用題〔每題15分，共30分〕要求：結合實際應用，解答以下各題。14.知曉某工廠生產一批產品，每件產品的本錢為50元，售價為100元。如果工廠想要獲得至少10000元的利潤，需要生產至少多少件產品？15.一輛汽車以60公里/小時的速度行駛，行駛了2小時后，速度降低到40公里/小時，再行駛了3小時后，速度又恢復到60公里/小時。求這輛汽車在這次行程中的平均速度。本次試卷答案如下：一、選擇題1.答案：A解析：函數(shù)$f(x)=\sqrt{1-x^2}$中的根號內的表達式必須大于等于0，即$1-x^2\geq0$，解得$x^2\leq1$，因此$x$的取值范圍是$[-1,1]$，即定義域為$A=\{x|-1\leqx\leq1\}$。2.答案：B解析：由于函數(shù)的圖像開口向上，所以$a>0$。函數(shù)的頂點坐標為$(1,3)$，代入頂點公式$x=-\frac{b}{2a}$得到$b=-6a$。又因為$f(0)=2$，代入得到$c=2$。將$b$和$c$的值代入$a^2-6a+2=3$解得$a=1$。3.答案：C解析：等差數(shù)列的通項公式為$a_n=a_1+(n-1)d$，代入$a_1=2$和$a_3=8$得到$d=2$，因此$a_5=2+4d=2+8=10$。4.答案：A解析：等比數(shù)列的通項公式為$a_n=a_1q^{n-1}$，代入$a_1=3$和$a_4=81$得到$q^3=27$，解得$q=3$，因此$a_2=a_1q=3\cdot3=9$。5.答案：A解析：復數(shù)$z$到點$1$和點$-1$的距離相等，說明$z$位于這兩點的中垂線上，即實部為0，因此$z$的軌跡是$y=0$。6.答案：A解析：函數(shù)的定義域要求根號內的表達式大于0，即$x-1>0$且$3-x>0$，解得$1<x<3$，因此定義域為$D=\{x|1<x<3\}$。二、填空題7.答案：$f(1)=1$解析：將$x=1$代入$f(x)=x^3-3x^2+4x-1$得到$f(1)=1^3-3\cdot1^2+4\cdot1-1=1$。三、解答題8.答案：值域為$(-\infty,3)$解析：函數(shù)$f(x)=\frac{x^2-4x+3}{x-1}$可以化簡為$f(x)=x+3$，因為$x^2-4x+3=(x-1)(x-3)$，所以$x-1\neq0$。因此，函數(shù)的值域為所有實數(shù)除了3。9.答案：通項公式為$a_n=2+2(n-1)$解析：等差數(shù)列的前$n$項和公式為$S_n=\frac{n}{2}(a_1+a_n)$，代入$a_1=2$和$S_5=30$得到$30=\frac{5}{2}(2+a_5)$，解得$a_5=12$。由于$a_5=a_1+4d$，解得$d=2$，因此通項公式為$a_n=2+2(n-1)$。四、證明題10.答案：證明：$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=3$解析：設$a$，$b$，$c$為等差數(shù)列中的三個連續(xù)項，即$b=a+d$，$c=a+2d$，代入原式得$\frac{a}{2a+d}+\frac{a+d}{2a+3d}+\frac{a+2d}{2a+2d}=3$，通分后化簡得$3a=3a$，等式成立。11.答案：證明：假設$\triangleABC$的邊長滿足$a^2+b^2-c^2=ab$，那么$\triangleABC$是直角三角形。解析：由題意得$a^2+b^2-c^2=ab$，移項得$a^2+b^2-ab=c^2$，即$(a-\frac{b}{2})^2+(\frac{\sqrt{3}}{2}b)^2=c^2$，這是直角三角形的勾股定理形式，因此$\triangleABC$是直角三角形。五、計算題〔每題15分，共30分〕要求：精確計算以下各題。12.知曉$a=3$，$b=4$，$c=5$，求$(a+b+c)^2$的值。13.假設等比數(shù)列$\{a_n\}$的首項為$a_1=2$，公比為$q=3$，求$a_8$的值。14.答案：$a_8=2^8=256$解析：等比數(shù)列的通項公式為$a_n=a_1q^{n-1}$，代入$a_1=2$和$q=3$得到$a_8=2\cdot3^{8-1}=2\cdot3^7=2\cdot2187=4374$。六