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2024 年春学期江阴市初中学业水平调研测试
九年级数学参考答案及评分标准
一、选择题（本大题共 10小题，每小题 3分，共 30分）
1．B 2．C 3．D 4．D 5．A 6．D 7．B 8．C 9．C 10．D
二、选择题（本大题共 8小题，每小题 3分，共 24分）
11． 1.26×106 12．x(y－2) (y＋2) 13．x(x－2)＝0 （答案不唯一） 14．12
15． 45 5－45 16．3π 17．3 5 18．4＋ 10
三、解答题（本大题共 10 小题，共 96 分）
2
19．（1）解：原式＝ 2＋2× －1 ················································································· 3 分
2
＝ 2＋1 ···························································································· 4 分
（2）解：原式＝m2－2mn＋n2－m 2＋2mn ···································································· 2 分
＝ n2 ································································································· 4 分
x＋4＞－2x＋1， ①
20．（1）解方程：2x2－4x＋1＝0； （2）解不等式组： x x－1
－ ≤1． ② 2 3
解： Δ＝42－4×2＝8 ····················· 1 分 解：由①得： x＞－1； ············ ········· 1 分
4 8
x = ， ·························· 2 分 由②得： x≤4； ···························· 2 分
2 2
2 2
∴ x =1+ , x =1 . ··········· 4 分 所以解集为－1＜x≤4. ···················· 4 分 1 2
2 2
21．证明：（1）四边形 ABCD 是平行四边形，
∴AB∥CD，∴∠DCA＝∠CAB． ·············································································· 1 分
∵O 是 AC 中点，∴OA＝OC． ················································································· 2 分
在△AOF 和△COE 中
DCA＝ CAB

OA = OC

EOC＝ AOF
∴△AOF≌△COE（ASA）． ··················································································· 5 分
（2）由（1）知 CE＝AF，又∵DC∥AF，∴四边形 AFCE 是平行四边形． ························ 8 分
∵AC⊥EF∴四边形 AFCE 是菱形． ·········································································· 10 分
22.（1）100；126 ·········································································································· 4 分
（2）35（图略） ······································································································· 7 分
（3）900×5%＝45（名）
答：该校七年级大概有 45 名学生喜欢“围棋”社团课． ·············································· 10 分
1
23.（1） ； ··············································································································· 3 分
4
（2）画树状图如下：
结果：8 12 20 8 2 2 2 3 2 5 2 4 ······················································ 8 分
共有 8 种等可能的结果，其中两球数字乘积为有理数的情况有 4 种， ································ 9 分
4 1
∴两球数字乘积为有理数的概率为 P＝ ＝ ． ························································· 10 分
8 2
1
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24.（1）
······································································· 6 分
∴射线 BD 和点 E 即为所求． ···················································································· 7 分
2
（2） ··············································································································· 10 分
2
25.证明：（1）在△ABC 和△OBD 中 ∵∠A＝∠BOD，∠C＝∠D，∴∠ABD＝∠ABC ·············· 2 分
∴∠ABD＝∠ABC＝90° ∴AB⊥BC ∴CD 是⊙O 的切线. ·········································· 4 分
（2）连接 BF
∵在 Rt△OBD 中，OB＝3，BD＝4， ∴OD＝5． ······················································ 6 分
∵∠A＝∠BOD, ∠C＝∠D
∴△ABC∽△OBD
OB BD
∴ ＝ ．
AB BC
3 4
∴ ＝ ∴BC＝8 ····························································································· 8 分
6 BC
∵AB 为直径， ∴BF⊥AC，
CF CF
∴ cos∠C＝ ＝ ，
BC 8
4 32
∴cos∠C＝cos∠D＝ ∴CF＝ ． ·································································· 10 分
5 5
26.解：（1）x＝30 时，销售价为 60－0.5×30＝45 元，
销售量为 30＋20＝50（件）， ·················································································· 1 分
销售利润为（45－30）×50＝750（元）
答：第 30 天的销售利润为 750 元． ············································································ 2 分
（2）设日销售利润为 w 元，
当 1≤x≤40 时. w＝(60－0.5x－30)(x＋20)
1
化简得：w＝－ x2＋20x＋600 ·················································································· 4 分
2
∵对称轴为直线 x＝20，
1
又∵－ ＜0 ∴抛物线开口向下，
2
∴x＝20 时，w 取得最大值，w＝800 ·········································································· 5 分
1
当 41≤x≤60 时 w＝(40－30)(－ x＋80)
2
化简得：w＝－5x＋800 ···························································································· 7 分
∵－5＜0 ∴w 随 x 的增大而减小，
又∵销售量为整数
∴x＝42 时，w 取得最大值，w＝590. ········································································· 8 分
∵590＜800， ········································································································ 9 分
∴第 20 天时日销售利润最大，最大为 800 元. ···························································· 10 分
27. 解：（1）如图（1），在矩形 ABCD 中，M 是 AD 中点，AB＝1，AD＝2，
∴AM＝MD＝CD＝1，∴ΔMCD 是等腰直角三角形. ······················································ 1 分
∵点 M、A'、C 三点共线， ∴CA'＝MC－MA'＝ 2－1， ··············································· 2 分
∵∠MCD＝45°，∠MA'B'＝∠B'＝90°，
∴∠A'CP＝∠A'PC＝∠NPB'＝∠B'NP＝45°，
∴ ΔA'PC 与 ΔNPB'是等腰直角三角形， ····································································· 3 分
∴PC＝ 2A'C＝2－ 2，NP＝ 2x，∴ x＋ 2x＋2－ 2＝2，
2
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∴ x = 2 2 . ······································································································· 4 分
（ 图 1） （图 2） （图 3）
（2）当 0＜x＜1 时，如图（2），
过 M 作 BC 垂线 垂足为 H，假设 A'B'交 BC 于点 E，连接 ME，
可得 ΔMHE ≌ ΔMA'E，∴ 可设 A'E＝HE＝y，∴B'E＝1－y，
又∵ B'N＝BN＝x，
在 RtΔNB'E 中 由勾股定理知 B ' N 2 + B 'E2 = NE2 (1 x + y)2 = x2 + (1 y)2
x
整理得 y＝ ，
2－x
x2 x + 2
∴ S = . ·································································································· 7 分
4 2x
当 x＝1 时，S＝1，上式仍然成立；········································································· 8 分
当 1＜x＜2 时，如图（3）
设 NB'交 DC 于点 F，
∵ CN＝2－x，
由对称性可得 (2 x)
2 (2 x) + 2 x2 3x + 4
S = = . ······················································ 9 分
4 2(2 x) 2x
x2 x + 2
0＜x≤1
∴

4 2x
S = ． ················································································· 10 分
2
x 3x + 4 1＜x＜2
2x
k 2
28．解：（1）a = ； ······························································································ 2 分
2
（2）二次函数 y＝ax2＋2 图象的顶点为 B（0，2），
过 C 作 x 轴垂线，垂足为 H，
由点 C 横坐标为－5 可知，k＞0，
2
∴ A（ ，0）在 y 轴左侧，如图 4．
k
2 2
∴OH＝5，OA＝ ，AE＝5－ ，
k k
∵ ∠HCA＋∠CAH＝∠CAH＋∠BAO＝90°，
∴ ∠HCA＝∠BAO，
又 ∵ ∠CHA＝∠AOB＝90°，
∴ △HCA∽△OAB， ······························································································ 3 分
2
5
∴ AH CH k
2
= ，∴ =
OB AO 2 2
k
2k 2整理得： 5k + 2 = 0， ·················································································· 4 分
3
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1
解得： k = 2或 , ································································································· 5 分
2
1
∴a = 或 2. ····································································································· 6 分
8
（3）当 k 0时，如图 4，
过 D 作 x 轴垂线，垂足为 I，
由（2）得 △HCA∽△OAB，
∴ AH CH= ，
OB AO
2
又∵CH =OB = 2 , OA = ，
k
∴ AH = 2k ，
∵∠CHA＝∠AID ，∠CAH＝∠DAI，
∴ △HCA∽△IDA， （图 4）
∴ AH CH AC 16= = = . ··························································································· 7 分
AI ID AD 9
∵ AH = 2k ，CH = 2 .
9 9
∴ AI = k, DI = .
8 8
9 2 9
∴ D 坐标为 ( k , ). ······················································································ 8 分
8 k 8
k 2
代入二次函数 y = x2 ＋2 中，
2
9 k 2 9 2
得 = ( k )
2 + 2 ，
8 2 8 k
9
( k 2 2
25
整理得 2) = ，
8 4
∵ k 0，∴ k = 2， ····························································································· 9 分
同理，当 k 0时可得 k = 2 ，
综上所述： k = 2或2 . ·························································································· 10 分
4
{#{QQABAQKEggAoAIJAARhCQQVgCEAQkBECCKoOgBAAoAIBSRNABAA=}#}2024年春学期江阴市初三学业水平调研测试
20.(8分)
23.(10分)
九年级数学答题卡
(1)
(1)
姓名：
准考证号码
(2)
班级：
[0][01][0][0][0][0][0][0]
[1]
[1]
1
[1]
[1
[1
[1
学号：
[2]
[2]
「21
[2]
「21
[27
[2
(2)
3
3
3
3
3
[3]
[4]
[4]
47
[4]
[4]
[4]
[4
5
5
151
5
5
[6
6
6
l6
[6]
[6
[7
7
7
「7
「7
>
[
[8
[8
[9]
97
[9]
[9][9]
「91
[9]
[9]
注意事项
1.答题前请将姓名、班级和准考证号（或学号）填写清楚并填涂完整。
2
客观题答题必须使用2B铅笔填涂，修改时用橡皮擦干净。主观题答题必须
使用黑色签字笔书写。
3.请在规定区域内答题，超出答题区域书写无效
21.(10分)
D
、
(1)
选择题
1. 0四6.和I四
2.知B]四7.D]四
B
3.0000
8. m四
第21题图
4.D0四9.ABC0四
5. 0四10. BI00四
24.(10分)
(1)
(2)
二、填空题
11.
12.
13.
14.
15.
16.
17.
18.
三、解答题
第24题图(1)
22.(10分)
人数
19.(8分)计算：
(1)
(1)|-2+2sin45-(π-4)°；
(2)(m-n)2-m(m-2n).
(2)补全条形统计图
(画图并标注相应数据)；
(3)
8
(2)
0
类别
■
口口口
第1页
共6页
第2页
共6页
第3页
共6页
1
■
25.(10分)
27.(10分)
28.(10分)
(1)
(1)
(1)a=
少
(2)
A
B
第25题图
第27题图
(2)
第28题图
A
M
(3)
D
(2)
26.(10分》
(1)
备用图
备川图
(2)
■
■口
第4页
共6页
第5页共6页
第6页共6页
12024年春学期江阴市初中学业水平调研测试
九年级数学试题
本试卷分试题和答题卡两部分，所有答案一律写在答题卡上，考试时间为120分钟.试卷满分
150分.
注意事项：
1.答卷前，考生务必用0.5毫米黑色墨水签字笔将自己的姓名、准考证号填写在答题卡的相
应位置上，并认真核对条形码上的姓名、准考证号是否与本人的相符合
2.答选择题必须用2B铅笔将答题卡上对应题目中的选项标号涂黑.如需改动，请用橡皮擦干
净后，再选涂其他答案.答非选择题必须用0.5毫米黑色墨水签字笔作答，写在答题卡上各题目指
定区域内相应的位置，在其他位置答题一律无效.
3.作图必须用2B铅笔作答，并请加黑加粗，描写清楚
4.卷中除要求近似计算的结果取近似值外，其他均应给出精确结果，
一、选择题（本大题共10小题，每小题3分，共30分.在每小题所给出的四个选项中，只有一项
是正确的，请用2B铅笔把答题卡上相应的选项标号涂黑.)
1.2024的倒数是…
1
A.2024
B.2024
C.-2024
D
2024
2.若要使二次根式x一2有意义，则x的值可以为…
…(▲)
A.0
B.1
C.3
D.
1
3.下列运算正确的是…
A.(-2a3b)2=4ab2
B.a8÷a=a2
C.(a-b)2=a2-b2
D.2a2b-a2b=a2b
4,下列图案中，既是轴对称图形，又是中心对称图形的是…
A
5,方程21
xx于万的解为…
A.x=-2
B.x=2
C.x=-4
D.x=4
6.一组数据0、1、一1、1、一2的中位数和众数分别是…（▲）
A.-2、1
B.-2、1
C.1、1
D.0、1
7.《九章算术》中有一题：“今有大器五、小器一，容三斛：大器一、小器五，容二斛.问大、小
器各容几何？”译文：今有大容器5个，小容器1个，总容量为3斛（斛：古代容量单位）：大
容器1个，小容器5个，总容量为2斛，问大容器、小容器的容量各是多少斛？若大容器的容
量为x斛，小容器的容量为y斛，则可列方程组…（▲）
x+5y=3,
5x+y=3,
5x=y+3,
5x=y+2,
A.15x+y=2
B
C.
D.
x+5y=2
x=5y+2
2x+y=5
初三数学第1页（共6页）
8.如图，⊙O是△ABC的外接圆，AD是⊙O的直径，若∠ACB=70°，则∠BAD的度数是（▲）
A.30
B.40
C.20°
D.50
D
D
0
(第8题)》
(第10题)
9.已知不、八、z满足等式号针六-香，则下列结论不正确的是（4)
A,若x=y,则x=z
B.若z=4x,则y=4z:
C.若xD.若x10.如图，四边形ABCD是边长为4的菱形，∠A=60°，将△ABD沿着对角线BD平移到△A'B'D,
在移动过程中，A"B与AD交于点E,连接DE、CE、CD'.则下列结论：
①A'E=BB';
②当D'E⊥CE时，∠A'D'E-∠CEB=30°：
③当∠EDC=60时，BB'的长为N5:
④△CED的面积最大值为5V5.
其中正确的为…。
…(▲）
A.①③
B.②③
C.①②③
D.①②④
二、填空题（本大题共8小题，每小题3分，共24分，不需写出解答过程，只需把答案直接填写
在答题卡上相应的位置)
11.2023年我国国内生产总值约为1260000亿元，可将数字1260000用科学记数法表示为▲·
12.分解因式：y2-4x=▲
13.请写出一个一元二次方程，使其一个根为2，一个根为0：▲·
14.已知圆锥的母线长13cm,侧面积65πcm2,则这个圆锥的高是▲cm.
15.古筝是一种弹拨弦鸣乐器，又名汉筝、秦筝，是汉民族古老的民族乐器，流行于中国各地.若
古筝上有一根弦AB=90cm,支撑点C是靠近点A的一个黄金分割点，则BC=▲cm.(结
果保留根号)
16.如图，滑轮圆心为O,半径为6cm,若在力F作用下滑轮上一点A绕点O顺时针旋转90°，则
图中物块上升▲cm,(结果保留π)
初三数学第2页（共6页）

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