北京市東城區(qū)2015屆高三上學(xué)期期末考試（數(shù)學(xué)理）掃描版一、選擇題要求：本大題共10小題，每小題5分，共50分。在每小題給出的四個選項(xiàng)中，只有一項(xiàng)是符合題目要求的。1.已知函數(shù)$f(x)=\sqrt{1-x^2}$，其定義域?yàn)椋篈.$[-1,1]$B.$[-1,0]$C.$[0,1]$D.$[0,1]$2.若等差數(shù)列$\{a_n\}$的首項(xiàng)為$a_1$，公差為$d$，且$a_1+a_5=12$，$a_3+a_7=24$，則$a_1$的值為：A.6B.8C.10D.123.若函數(shù)$f(x)=ax^2+bx+c$的圖像開口向上，且頂點(diǎn)坐標(biāo)為$(1,2)$，則下列選項(xiàng)中正確的是：A.$a>0$，$b>0$，$c>0$B.$a>0$，$b<0$，$c>0$C.$a>0$，$b>0$，$c<0$D.$a>0$，$b<0$，$c<0$4.已知等比數(shù)列$\{a_n\}$的首項(xiàng)為$a_1$，公比為$q$，且$a_1+a_2+a_3=9$，$a_2+a_3+a_4=27$，則$q$的值為：A.2B.3C.6D.95.若函數(shù)$f(x)=x^3-3x^2+4x$的圖像與x軸的交點(diǎn)個數(shù)為：A.1B.2C.3D.46.已知等差數(shù)列$\{a_n\}$的首項(xiàng)為$a_1$，公差為$d$，且$a_1+a_4=12$，$a_2+a_5=18$，則$a_1$的值為：A.3B.6C.9D.127.若函數(shù)$f(x)=ax^2+bx+c$的圖像開口向下，且頂點(diǎn)坐標(biāo)為$(1,2)$，則下列選項(xiàng)中正確的是：A.$a<0$，$b>0$，$c>0$B.$a<0$，$b<0$，$c>0$C.$a<0$，$b>0$，$c<0$D.$a<0$，$b<0$，$c<0$8.已知等比數(shù)列$\{a_n\}$的首項(xiàng)為$a_1$，公比為$q$，且$a_1+a_2+a_3=3$，$a_2+a_3+a_4=9$，則$q$的值為：A.$\frac{1}{3}$B.$\frac{1}{2}$C.3D.99.若函數(shù)$f(x)=x^3-3x^2+4x$的圖像與x軸的交點(diǎn)個數(shù)為：A.1B.2C.3D.410.已知等差數(shù)列$\{a_n\}$的首項(xiàng)為$a_1$，公差為$d$，且$a_1+a_4=12$，$a_2+a_5=18$，則$a_1$的值為：A.3B.6C.9D.12二、填空題要求：本大題共5小題，每小題5分，共25分。把答案填在題中的橫線上。11.已知函數(shù)$f(x)=\frac{1}{x^2+1}$，則$f'(0)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\四、解答題要求：本大題共5小題，每小題10分，共50分。解答過程要求寫出必要的文字說明、證明過程或計(jì)算步驟。24.已知函數(shù)$f(x)=\frac{x^3-3x^2+4x}{x^2-1}$，求：（1）函數(shù)$f(x)$的定義域；（2）函數(shù)$f(x)$的奇偶性；（3）函數(shù)$f(x)$在定義域內(nèi)的單調(diào)性。25.設(shè)數(shù)列$\{a_n\}$是等差數(shù)列，若$a_1=3$，$a_4=13$，$a_n=2a_{n-1}-a_{n-2}$，求：（1）數(shù)列$\{a_n\}$的通項(xiàng)公式；（2）數(shù)列$\{a_n\}$的前$n$項(xiàng)和$S_n$。26.已知函數(shù)$f(x)=ax^2+bx+c$的圖像與x軸的交點(diǎn)坐標(biāo)為$(1,0)$和$(-3,0)$，且$f(x)$在$x=0$處的切線斜率為$2$，求函數(shù)$f(x)$的表達(dá)式。27.設(shè)數(shù)列$\{a_n\}$是等比數(shù)列，若$a_1=2$，$a_2+a_3=10$，$a_3+a_4=40$，求：（1）數(shù)列$\{a_n\}$的公比$q$；（2）數(shù)列$\{a_n\}$的前$n$項(xiàng)和$S_n$。28.已知函數(shù)$f(x)=\frac{1}{x^2+1}$，求證：（1）$f(x)$在定義域內(nèi)是連續(xù)的；（2）$f(x)$在定義域內(nèi)是單調(diào)遞減的。五、證明題要求：本大題共1小題，共10分。證明過程要求寫出必要的文字說明、證明過程或計(jì)算步驟。29.證明：若數(shù)列$\{a_n\}$是等比數(shù)列，公比為$q$，且$q\neq1$，則數(shù)列$\{a_n\}$的任意兩項(xiàng)之比也構(gòu)成等比數(shù)列。六、應(yīng)用題要求：本大題共1小題，共10分。解答過程要求寫出必要的文字說明、證明過程或計(jì)算步驟。30.已知某商品的成本為每件100元，根據(jù)市場調(diào)查，每增加1元售價，銷量將減少10件。設(shè)售價為$x$元，求：（1）總利潤$y$與售價$x$之間的函數(shù)關(guān)系式；（2）求售價$x$使得總利潤$y$最大，并求出最大利潤。本次試卷答案如下：一、選擇題1.A解析：函數(shù)$f(x)=\sqrt{1-x^2}$的定義域是使得根號內(nèi)的表達(dá)式非負(fù)的$x$的取值范圍，即$1-x^2\geq0$，解得$-1\leqx\leq1$，所以定義域?yàn)?[-1,1]$。2.B解析：由等差數(shù)列的性質(zhì)，$a_5=a_1+4d$，$a_3=a_1+2d$，$a_7=a_1+6d$，聯(lián)立方程組得到$a_1=8$，$d=2$。3.B解析：函數(shù)$f(x)=ax^2+bx+c$的圖像開口向上，說明$a>0$。頂點(diǎn)坐標(biāo)為$(1,2)$，代入得到$b=-2a$，$c=a+2$。4.B解析：由等比數(shù)列的性質(zhì)，$a_2=a_1q$，$a_3=a_1q^2$，$a_4=a_1q^3$，聯(lián)立方程組得到$q=3$。5.C解析：通過因式分解$f(x)=x^3-3x^2+4x=x(x-1)(x-4)$，得到與x軸的交點(diǎn)為$x=0$，$x=1$，$x=4$。6.D解析：由等差數(shù)列的性質(zhì)，$a_5=a_1+4d$，$a_2=a_1+d$，$a_5=a_2+3d$，聯(lián)立方程組得到$a_1=12$，$d=6$。7.B解析：函數(shù)$f(x)=ax^2+bx+c$的圖像開口向下，說明$a<0$。頂點(diǎn)坐標(biāo)為$(1,2)$，代入得到$b=-2a$，$c=a+2$。8.A解析：由等比數(shù)列的性質(zhì)，$a_2=a_1q$，$a_3=a_1q^2$，$a_4=a_1q^3$，聯(lián)立方程組得到$q=\frac{1}{3}$。9.C解析：通過因式分解$f(x)=x^3-3x^2+4x=x(x-1)(x-4)$，得到與x軸的交點(diǎn)為$x=0$，$x=1$，$x=4$。10.D解析：由等差數(shù)列的性質(zhì)，$a_5=a_1+4d$，$a_2=a_1+d$，$a_5=a_2+3d$，聯(lián)立方程組得到$a_1=12$，$d=6$。二、填空題11.$-1$解析：對函數(shù)$f(x)=\frac{1}{x^2+1}$求導(dǎo)，得到$f'(x)=\frac{-2x}{(x^2+1)^2}$，代入$x=0$得到$f'(0)=-1$。三、解答題24.解析：（1）函數(shù)$f(x)$的定義域?yàn)槭沟梅帜覆粸榱愕?x$的取值范圍，即$x^2-1\neq0$，解得$x\neq1$，$x\neq-1$，所以定義域?yàn)?\{x|x\neq1,x\neq-1\}$。（2）函數(shù)$f(x)$是奇函