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2019-2020学年北京市平谷区初三第一学期期末试题(含答案)

2019-2020学年北京市平谷区初三第一学期期末试题(含答案)
2019-2020学年北京市平谷区初三第一学期期末试题(含答案)

平谷区2019~2020学年度第一学期期末质量监控试卷

初 三 数 学 2020年1月

考生须知

1.试卷分为试题和答题卡两部分,所有试题均在答题卡上......

作答. 2.答题前,在答题卡上考生务必将学校、班级、准考证号、姓名填写清楚. 3.把选择题的所选选项填涂在答题卡上;作图题用2B 铅笔.

4.修改时,用塑料橡皮擦干净,不得使用涂改液.请保持卡面清洁,不要折叠.

下面各题均有四个选项,其中只有一个..

是符合题意的. 1.如果23x y =,那么x

y

的值为 (A)23; (B) 25; (C) 32; (D) 53

2.如图,已知AB ∥CD ∥EF ,:1:2BD DF =,那么:AC AE 的值是

(A )

13; (B )12; (C )2

3

; (D )2. 3.在Rt △ABC 中,∠C = 90°,∠A 、∠B 、∠C 所对的边分别为a 、b 、c ,下列等式中成立的是

(A )sin a A b =; (B )cos a B c =; (C )tan b B c =; (D )tan c

C b

=.

4.如果抛物线2

(2)y a x =+开口向下,那么a 的取值范围为

(A )2a >; (B )2a <; (C )2a >-; (D )2a <-. 5.如图,直角坐标平面内有一点P (2,4),如果OP 与x 轴正半轴的夹角为α,那么tan α的值是

(A )2; (B )2

1; (C )55; (D )5.

6.如图,P A 是⊙O 的切线,OP 交⊙O 于点B ,如果1

sin 2

P =,OB =1,

那么BP 的长是

(A )4; (B )2; (C )1; (D )3. 7.关于反比例函数4

y x

=-

,下列说法正确的是 (A )函数图象经过点(2,2); (B )函数图象位于第一、三象限;

(C )当0x >时,函数值y 随着x 的增大而增大; (D )当1x >时,4y <-.

8.二次函数y =kx 2+2x +1的部分图象如图所示,则k 的取值范围是

(A )k ≤1 (B )k ≥1 (C )k <1 (D )0

二、填空题(本题共16分,每小题2分) 9.函数1

1

y x =

-的自变量x 的取值范围是 .

10.在Rt △ABC 中,∠C =90°,如果cos B =2

3

,BC =4,那么AB 的长为 . 11.反比例函数x

k

y =

(0≠k )的图象经过点A (1,2),B (2,y 1),C (3,y 2),则y 1 y 2.(填“<,=,>”)

12.如图,点A ,B ,C 在⊙O 上,∠BOC =2∠AOB ,如果∠BAC =40°,那么∠ACB 的度数是 .

13.连结三角形各边中点,所得的三角形的周长与原三角形周长的比是 .

14.已知某二次函数图像的最高点是坐标原点,请写出一个符合要求的函数表达式_______. 15.两个函数y ax b =+和c

y x

=(abc ≠0)的图象如图所示,请直接写出关于x 的不等式c

ax b x

+>

的解集 .

16.我们将等腰三角形腰长与底边长的差的绝对值称为该三角形的“边长正度值” .若等腰三角形腰长为5,“边长正度值”为3,则这个等腰三角形底角的余弦值等于 .

三、解答题(本题共68分,第17~23题,每小题5分,第24题7分,第25,26题,每小题6分,第27,28题,每小题7分)解答应写出文字说明、演算步骤或证明过程.

17.计算:0

13122sin 302??

-+--+? ???

18.如图,在Rt △ABC 中,∠C =90°,点D 是AC 边上一点,DE ⊥AB 于点E .

(1)求证:△ABC ∽△ADE ;

(2)如果AC =8,BC =6,CD =3,求AE 的长.

19.二次函数2

y x bx =+上部分点的横坐标x 与纵坐标y 的对应值如下表: (1)直接写出此二次函数的对称轴 ;

(2)求b 的值;

(3)直接写出表中的m 值,m = ; (3)在平面直角坐标系xOy 中,画出此二次函数的图

x … 1-

12- 0 1 2 3 … y

3

54

1- 0

m …

y

x

–1

–2–3–4–5

1

234

5

–1–2–3–4–51234

5O

D

C

象.

20.如图,在△ABC 中,∠C =90°,AB 的垂直平分线分别交边AB 、BC 于点D 、E ,连结AE . (1)如果∠B =25°,求∠CAE 的度数; (2)如果CE =2,2

sin 3

CAE ∠=

,求tan B 的值.

A

21.如图,O e 是ABC ?的外接圆,圆心O 在ABC ?的外部,4AB AC

==,BC =,求O e 的半径.

22.如图,在平面直角坐标系xOy 中,曲线()0k

y x x

=>经过点A . (1)求曲线()0k

y x x

=

>的表达式; (2)直线y=ax +3(a ≠0)与曲线()0k

y x x

=>围成的封闭区域为图象G .

①当1a =-时,直接写出图象G 上的整数点个数是 ;(注:横,纵坐标均为整数的点称为整点,图象G 包含边界.)

②当图象G 内只有3个整数点时,直接写出a 的取值范围.

23.某无人机兴趣小组在操场上开展活动(如图),此时无人机在离地面30米的D处,无人机测得操控者A的俯角为37°,测得点C处的俯角为45°.又经过人工测量操控者A和教学楼BC距离为57米,求教学楼BC的高度.(注:点A,B,C,D都在同一平面上.参考数据:sin37°≈0.60,cos37°≈0.80,tan37°≈0.75)

?AB上一动点,连接AP,作∠APC=45°,交弦AB于点C.AB=6cm.24.如图,点P是

小元根据学习函数的经验,分别对线段AP,PC,AC的长度进行了测量.

下面是小元的探究过程,请补充完整:

?AB上的不同位置,画图、测量,得到线段AP,PC,AC长度的几组值,(1)下表是点P是

AP/cm 0 1.00 2.00 3.00 4.00 5.00 6.00

PC/cm0 1.21 2.09 2.69 m 2.82 0

AC /cm 0 0.87 1.57 2.20 2.83 3.61 6.00 ①经测量m 的值是 (保留一位小数).

②在AP ,PC ,AC 的长度这三个量中,确定______的长度是自变量,______的长度 和 的长度都是这个自变量的函数;

(2)在同一平面直角坐标系xOy 中,画出(1)中所确定的函数图象;

(3)结合函数图象,解决问题:当△ACP 为等腰三角形时,AP 的长度约为 cm (保留一位小数).

25.如图,△ABC 内接于⊙O ,AB 是⊙O 的直径,过点A 作AD 平分∠BAC ,交⊙O 于点D ,过点D 作DE ∥BC 交AC 的延长线于点E .

(1)依据题意,补全图形(尺规作图,保留痕迹); (2)判断并证明:直线DE 与⊙O 的位置关系; (3)若AB =10,BC =8,求CE 的长.

26.在平面直角坐标系xOy 中,抛物线()2

230y ax ax a =--?与y 轴交于点A .

(1)直接写出点A 的坐标;

(2)点A 、B 关于对称轴对称,求点B 的坐标;

(3)已知点(4,0)P ,1

(,0)Q a

-.若抛物线与线段PQ 恰有两个公共点,结合函数图象,

求a 的取值范围.

27.如图,正方形ABCD ,将边BC 绕点B 逆时针旋转60°,得到线段BE ,连接AE ,CE . (1)求∠BAE 的度数;

(2)连结BD ,延长AE 交BD 于点F .

①求证:DF=EF ;

②直接用等式表示线段AB ,CF ,EF 的数量关系.

28.在平面直角坐标系xOy 中,有任意三角形,当这个三角形的一条边上的中线等于这条边的一半时,称这个三角形叫“和谐三角形”,这条边叫“和谐边”,这条中线的长度叫“和谐距离”.

(1)已知A (2,0),B (0,4),C (1,2),D (4,1),这个点中,能与点O 组成“和谐三角形”的点是 ,“和谐距离”是 ;

(2)连接BD ,点M ,N 是BD 上任意两个动点(点M ,N 不重合),点E 是平面内任意一点,△EMN 是以MN 为“和谐边”的“和谐三角形”,求点E 的横坐标t 的取值范围;

(3)已知⊙O 的半径为2,点P 是⊙O 上的一动点,点Q 是平面内任意一点,△OPQ 是“和谐三角形”,且“和谐距离”是2,请描述出点Q 所在位置.

平谷区2019~2020学年度第一学期期末质量监控试卷评分标准

初 三 数 学 2020年1月

一、选择题(本题共16分,每小题2分)

二、填空题(本题共16分,每小题2分) 9.x ≠1;10.6;11.>;12.20°;13.

12

; 14.答案不唯一,如:2y x =;15.301x x -<<>或; 16.14

5

5

. 三、解答题(本题共68分,第17~23题,每小题5分,第24题7分,第25,26题,每小题6分,第27,28题,每小题7分)解答应写出文字说明、演算步骤或证明过程. 17.解:

1122

-?

(4)

=2-5 18.(1)证明:∵DE ⊥AB 于点E , ∴∠AED =∠C =90°. ...................................................................................... 1 ∵∠A =∠A ,

∴△ABC ∽△ADE . ......................................................................................... 2 (2)解: ∵AC =8,BC =6,

∴AB =10......................................................................................................... 3 ∵△ABC ∽△ADE , ∴

AE AD

AC AB

=

. .............................................................................................. 4 ∴AE =4.. (5)

19.解:(1)对称轴x =1. ........................................................................................................ 1 (2)∵二次函数2

y x bx =+的图象经过点(1,-1),

2

(3)3 (35

20.解:(1)∵DE 垂直平分AB ,

∴EA = EB , ......................................................................................................... 1 ∴∠EAB =∠B =25°. ∴∠CAE =40°. ...................................................................................................

2 (2)∵∠C =90°,

∴2

sin 3

CE CAE AE ∠=

=. ∵CE

=2,

∴AE =3. ............................................................................................................ 3 ∴AC (4)

∵EA = EB =3, ∴BC=5.

∴222AC CE AE +=, ∴tan AC B BC =

=. ..................................................................................... 5 A

21.解:连结AO ,交BC 于点D ,练结BO .

∵AB=AC ,

∴?

?AB AC =............................................................... 1 又∵AO 是半径,

∴AO ⊥BC ,BD=CD . (2)

∵BC =,

∴BD =. .............................................................. 3 在Rt ABD ?中,90ADB ∠=? ,

∵222BD AD AB +=,AB =4,

∴2AD =. ................................................................... 4 设O e 半径为r . 在 Rt BDO ?中, ∵222BD DO BO +=,

∴(()2

2

2+2r r -= ,

∴4r = . ................................................................... 5 ∴O e 的半径为4.

22.解:(1)∵A (1,1), ∴k =1. ......................................... 1 ∴()1

0y x x

=

>. ...................... 2 (2)①3; ............................................. 3 ②213

a -≤<-

. (5)

23.解:过点D 作DE ⊥AB 于点E ,过点C 作CF ⊥DE 于点F . ...................................... 1 由题意得,AB =57,DE =30,∠A =37°,∠DCF =45°. 在Rt △ADE 中,∠AED =90°, ∴tan37°=

DE

AE

≈0.75. ∴AE =40. ........................................... 2 ∵AB =57,

∴BE =17. ........................................... 3 ∵四边形BCFE 是矩形, ∴CF=BE =17.

在Rt △DCF 中,∠DFC =90°, ∴∠CDF =∠DCF =45°.

∴DF=CF =17. ................................... 4 ∴BC=EF =30-17=13. ...................... 5 答:教学楼BC 高约13米.

24.解:(1)①3.0; (1)

②AP 的长度是自变量,PC 的长度和AC 的长度都是这个自变量的函数;(答

案不唯一) (3)

(2)如图(答案不唯一,和(1)问相对应); (5)

(3)2.3或4.2 (7)

25.(1)如图. (1)

E

D

O

C

A

F

E

D

O

C

A

(2)判断:直线DE是⊙O的切线. (2)

证明:连结OD,交BC于F.

∵AD平分∠BAC,

∴∠BAD=∠CAD. (3)

??

CD BD

∴OD⊥BC于F.

∵DE∥BC,

∴OD⊥DE于D.

∴直线DE是⊙O的切线. (4)

(3)∵AB是⊙O的直径,

∴∠ACB=90°.

∵AB=10,BC=8,

∴AC=6. (5)

∵∠BOF=∠ACB=90°,

∴OD∥AC.

∵O是AB中点,

∴OF=

1

2

AC=3.

∵OD =

1

2

AB =5, ∴DF =2.

∵DE ∥BC ,OD ∥AC ,

∴四边形CFDE 是平行四边形. ∵∠ODE =90°,

∴平行四边形CFDE 是矩形.

∴CE=DF =2. (6)

26.解:(1)()0,3-; ............................................................................................................. 1 (2)∵212b a

x a a

-=-

=-=; ∴()2,3B -. .................................................................................................... 2 (3)当抛物线过点P (4,0)时,3

8

a =, ............................................................... 3 ∴8,03Q ??- ???

此时,抛物线与线段PQ 有两个公共点. 当抛物线过点1

(,0)Q a

-

时,a =1, 此时,抛物线与线段PQ 有两个公共点. ∵抛物线与线段PQ 恰有两个公共点,

3

18

a ≤≤.

........................................................................................................ 5 当抛物线开口向下时,3a <-. (6)

综上所述,当3

18

a ≤≤或3a <-时,抛物线与线段PQ 恰有两个公共点.

27.(1)解:∵AB=BE ,

∴∠BAE =∠BEA . .............................................................................................. 1 ∵∠ABE =90°-60°=30° ∴∠BAE =75°.. (2)

F

E

B C

A

D O F

E

B C A

D O

F

E

B

C

A

D

(2)证明:∴∠DAF =15°. (3)

连结CF .

由正方形的对称性可知,∠DAF =∠DCF =15°. ........................................... 4 ∵∠BCD =90°,∠BCE =60°, ∴∠DCF =∠ECF =∠DAF =15°. ∵BC=EC ,CF=CF , ∴△BCF ≌△ECF . .......................................................................................... 5 ∴BF=EF . .. (6)

(3)22AB EF CF =+. (7)

28.解:(1)A ,B ;5; .......................................................................................................... 3 (2)19

22

t -

≤≤; ...................................................................................................... 5 (3)点Q 在以点O 为圆心,4为半径的圆上;

或在以点O 为圆心,23为半径的圆上. (7)

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