一、选择题(本题有10小题,每小题3分,共30分.每小题只有一个选项是正确的,不选、多选、错选,均不给分)
-
1.
若
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
, 则
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
的值等于( )
-
2.
已知⊙
O的半径为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E5%3C%2Fmn%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmath%3E)
, 点
P到圆心
O的距离为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E4%3C%2Fmn%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmath%3E)
, 则点
P( )
A . 在圆内
B . 在圆上
C . 在圆外
D . 不能确定
-
3.
二次函数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmath%3E)
的图象与
y轴的交点坐标是( )
A . (-1,0)
B . (1,0)
C . (0,1)
D . (0,-1)
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4.
若两个三角形的相似比为1:3,则它们的面积比为( )
A . 1:3
B . 1:9
C . 3:1
D . 9:1
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5.
一个不透明的盒子里有6个除颜色外其他完全相同的小球,其中有3个红球,2个黄球和1个白球.从袋中任意摸出一个球,是白球的概率为( )
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6.
关于二次函数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E%28%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E4%3C%2Fmn%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmn%3E%29%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E6%3C%2Fmn%3E%3C%2Fmath%3E)
的最大值或最小值,下列说法正确的是( )
A . 有最大值4
B . 有最大值6
C . 有最小值4
D . 有最小值6
-
A . 110°
B . 120°
C . 130°
D . 140°
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8.
如图,有一块直角三角形余料
ABC , ∠
BAC=90°,
D是
AC的中点,现从中切出一条矩形纸条
DEFG , 其中
E ,
F在
BC上,点
G在
AB上,若
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmi%3EF%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmi%3EE%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmath%3E)
, 则矩形纸条DEFG的面积为( )
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3C%2Fmath%3E)
.
![](//tikupic.21cnjy.com/2023/12/15/b6/0a/b60a44d9a26954fefe82463d1e5479ac.png)
A . 3
B .
C . 19
D .
-
9.
二次函数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmath%3E)
与一次函数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmath%3E)
的图象交于点
A(2,5)和点
B(3,
m),要使
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ey%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3C%2Fmath%3E)
, 则
x的取值范围是( )
A . 2<x<3
B . x<3
C . x<2或x>3
D . x>2
-
10.
如图,在半圆
O中,直径
AB=2,
C是半圆上一点,将弧
AC沿弦
AC折叠交
AB于
D , 点
E是弧
AD的中点.连接
OE , 则
OE的最小值为( )
![](//tikupic.21cnjy.com/2023/12/15/10/ef/10effd5f7852406f67748ea7b3eac8b9_216x136.png)
二、填空题(本题有6小题.每小题4分,共24分)
-
11.
成语“守株待兔”反映的事件是事件(填必然、不可能或随机).
-
-
13.
若二次函数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmath%3E)
的图象与
x轴的一个交点是(3,0),则与
x轴的另一个交点坐标是
.
-
14.
鹦鹉螺曲线的每个半径和后一个半径的比都是黄金比例,是自然界最美的鬼斧神工,如图,
P是
AB的黄金分割点(
AP>
BP),若线段
AB的长为4
cm , 则
AP的长为
cm .
![](//tikupic.21cnjy.com/2023/12/15/be/4f/be4fa17f77b82486e9205304e83f57e6_225x166.png)
-
15.
如图,在Rt△
ABC中,∠
C=90°,点
D在
BC边上.连结
AD , 将△
ABD沿直线
AD翻折,点
B落在点
E处,
AE交
BC边于点
F . 已知
AC=1,
BC=2,若△
DEF为直角三角形,则△
DEF的面积为
.
![](//tikupic.21cnjy.com/2023/12/15/fa/1d/fa1d34fd28ec737326e2b4319cbc9476.png)
-
16.
如图,已知在⊙
O中,
AB是⊙
O的直径,
AC=4,
BC=3.若
D为⊙
O上一点,且△
ABD为等腰三角形,则弦
CD的长为
.
![](//tikupic.21cnjy.com/2023/12/15/89/41/8941aa6771eb750360422f3cd636f42e.png)
三、解答题(本题有8小题,第17~19题每题6分,第20、21题每题8分,第22、23题每题10分,第24题12分,共66分)
-
17.
一个不透明的袋子中装有红、白两种颜色的3个小球,这些球除颜色外都相同,若从中随机摸出一个球,这个球是白球的概率为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
.
-
-
(2)
随机摸出一个球后,放回并搅匀,再随机摸出一个球,求两次都摸到相同颜色的小球的概率.(请结合树状图或列表解答)
-
18.
如图是一位同学设计的用手电筒来测量某古城墙高度的示意图.点
P处放一水平的平面镜,光线从点
A出发经平面镜反射后刚好到古城墙
CD的顶端
C处,已知
AB⊥
BD ,
CD⊥
BD , 测得
AB=2米,
BP=3米,
PD=15米,求该古城墙的高度
CD .
![](//tikupic.21cnjy.com/2023/12/15/dc/2b/dc2b21085adf0c9ce41bd670847bc52d.png)
-
19.
如图,二次函数的图象的顶点坐标为(1,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E7%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3C%2Fmath%3E)
),现将等腰直角三角板直角顶点放在原点,一个锐角顶点A在此二次函数的图象上,而另一个锐角顶点B在第二象限,且点A的坐标为(2,1).
![](//tikupic.21cnjy.com/2023/12/15/fb/ee/fbeece3f5b28c645fc0c795956d42aff.png)
-
-
(2)
判断点B是否在此二次函数的图象上,并说明理由.
-
20.
如图,在Rt△
ABC中,∠
ACB=90°,
AC=6,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmi%3EC%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmsqrt%3E%3Cmrow%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsqrt%3E%3C%2Fmath%3E)
, 在线段
AC上取点
D , 使
AD=2
CD , 连接
BD并延长交△
ABC的外接圆于点
E .
![](//tikupic.21cnjy.com/2023/12/15/14/84/148427a30d13f79eb2ca8eeed4d7903c.png)
-
(1)
不添其他辅助线写出图中一对相似三角形,并说明理由;
-
-
21.
诸暨某百货商场购进一批单价为5元的日用商品.如果以单价7元销售,每天可售出140件,根据销售经验,销售单价每提高1元,销售量每天就相应减少10件,设这种商品的销售单价为x元(x≥7).
-
(1)
若该商场当天销售这种商品所获得的利润为600元,求x的值.
-
(2)
当商品的销售单价定为多少元时,该商店销售这种商品获得的利润最大?此时最大利润为多少?
-
22.
如图,隧道的截面由圆弧
AED和矩形
ABCD构成,矩形的长
BC为12
m , 宽
AB为3
m , 隧道的顶端
E(圆弧
AED的中点)高出道路(
BC)7
m .
![](//tikupic.21cnjy.com/2023/12/15/0d/16/0d165298a93c7bf3e21ccc77caa9f4a2.png)
-
-
(2)
如果该隧道内设双行道,现有一辆超高货运卡车高6m , 宽3.3m , 通过计算问这辆货运卡车能否通过该隧道,写出理由.
-
23.
已知二次函数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmsubsup%3E%3Cmrow%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmrow%3E%3Cmrow%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsubsup%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3Ec%3C%2Fmi%3E%3C%2Fmath%3E)
.
-
(1)
当
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E++%EF%BC%8C+%3C%2Fmn%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmath%3E)
时,
①求该函数图象的顶点坐标;
②当
≤x≤4时,求y的取值范围;
-
(2)
当
x≤0时,
y的最小值为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E8%3C%2Fmn%3E%3C%2Fmath%3E)
;当
x>0时,
y的最小值为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmn%3E9%3C%2Fmn%3E%3C%2Fmath%3E)
, 求二次函数的表达式.
-
24.
如图1所示,正方形
BEFG绕正方形
ABCD的顶点
B逆时针旋转α度(0°<α<45°),
GF与
AB交于点
H .
![](//tikupic.21cnjy.com/2023/12/15/63/df/63df334805468e761527d202826b35e2_490x168.png)
-
-
(2)
如图2,连接
DF ,
CE ,
BD;
①判断DF与CE的数量关系,并证明;
②当G , F , D三点共线时,延长BF交AD于点M ,
时,求BC的长.