2011-2012学年南京市鼓楼区第二学初三年级
2023-2024学年江苏省南京市鼓楼区九年级(上)期中数学试卷(含解析)
2023-2024学年江苏省南京市鼓楼区九年级（上）期中数学试卷一、选择题（本大题共6小题，每小题2分，共12分.在每小题所给出的四个选项中，恰有一项是符合题目要求的，请将正确选项的序号填涂在答题卡相应位置上）1．（2分）下列方程中，是一元二次方程的是（）A．x﹣1＝0B．x2＝3C．4x﹣3y＝3D．x2﹣y2＝32．（2分）已知⊙O的半径为3，平面内点P到圆心O的距离为√5，则点P与⊙O的位置关系（）A．点P在⊙O外B．点P在⊙O上C．点P在⊙O内D．无法确定3．（2分）二次函数y＝2（x﹣3）2+2的图象的顶点坐标是（）A．（3，2）B．（﹣3，2）C．（3，﹣2）D．（﹣3，﹣2）4．（2分）如图，在长40m、宽22m的矩形地面内，修筑两条同样宽且互相垂直的道路，余下的部分铺上草坪，要使草坪的面积达到760m2，设道路的宽为x m，则根据题意，可列出方程（）A．（40﹣x）（22﹣x）＝760B．（40+x）（22+x）＝760C．40×22﹣40x﹣22x＝760D．40×22﹣40x﹣22x﹣x2＝7605．（2分）二次函数y＝ax2+bx+c的图象上部分点的横坐标x与纵坐标y的对应值如下表所示．下列说法中，正确的是（）A．图象的开口向上B．图象经过点（4，0）C．图象与x轴只有一个公共点D．点（2，3）右边的图象呈下降趋势6．（2分）如图，在⊙O中，动弦AB与直径CD相交于点E且总有∠BED＝45°，则AE2+BE2的值（）A．随着OE的增大而增大B．随着OE的增大而减小C．随着OE的增大先增大后减小D．保持不变二、填空题（本大题共10小题，每小题2分，共20分.不需写出解答过程，请把答案直接填写在答题7．（2分）写出一个一元二次方程，使它的两个根分别是3、﹣2．8．（2分）已知圆锥的底面半径为6，母线长为10，则此圆锥的侧面积为．9．（2分）若点（1，y1）与（2，y2）都在函数y＝﹣x2+3的图象上，则y1，y2的大小关系是．10．（2分）若扇形的半径为3，圆心角120°，为则此扇形的弧长是．11．（2分）若一元二次方程x2+mx﹣3＝0（m为常数）的一个根是x＝1，则另一个根是．12．（2分）如图是函数y＝ax2+bx+c的部分图象，则该函数图象与x轴负半轴的交点坐标是．13．（2分）已知⊙O的半径等于4cm，AB为⊙O的弦，其长为4√2cm，则弦AB所对的圆周角的度数为．14．（2分）五边形ABCDE为圆的内接五边形，其中AB＝AE，∠A＝100°，则∠B+∠D＝°．15．（2分）如图，AB是⊙O的直径，若∠E＝25°，∠CAD＝45°，则∠CDA的度数为°．16．（2分）等边△ABC的边长是4√3，直线l经过等边△ABC的外心O，过B作BD⊥l，垂足为D，连接CD，则CD的最小值是．三、解答题（本大题共11小题，共88分.请在答题卡指定区域内作答，解答时应写出文字说明、证明过程或演算步骤）17．（8分）解下列方程．（1）（x﹣1）2﹣16＝0；（2）x2﹣3x+1＝0．18．（7分）已知二次函数y＝x2﹣mx+1的图象经过点（2，3）．（1）求m的值；（2）该二次函数的图象是否经过点（﹣2，6）？判断并说明理由．19．（7分）如图，在以点O为圆心的两个同心圆中，大圆的弦AB与小圆相切于点P．（1）求证AP＝BP；（2）已知图中阴影部分面积为10π，求弦AB的长．20．（8分）一个直角三角形的两条直角边长度之和是10cm，面积是12cm2，求斜边的长．21．（7分）二次函数图象的顶点为（﹣1，2），图象经过（0，1）．（1）求该二次函数的表达式；（2）结合图象，直接写出当﹣2≤x≤3时y的取值范围．22．（8分）尺规作图：作已知圆的一条直径．要求：①保留作图痕迹；②用两种不同方法作图．23．（8分）已知关于x的方程x2﹣4x+4﹣k2＝0（k为常数）．（1）求证：不论k取何值时，该方程总有实数根．（2）方程的两个实数根可能都是负根吗？判断并说明理由．24．（8分）如图是二次函数y＝x2﹣2x﹣3的大致图象．（1）求该图象顶点的坐标；（2）该图象经过怎样的平移可以得到函数y＝x2的图象？（3）将该图象绕原点旋转180°直接写出所得图象对应的表达式．25．（8分）某商店八月份的销售额为30万元，九月份因经营不善销售额有明显下降，商店积极改进，十月份的销售额恢复到30万元，已知十月份销售额的增长率是九月份销售额的下降率的1.5倍．求九月份的销售额．26．（10分）如图，AB是⊙O的直径，∠BAC＝90°，BC与⊙O相交于点E，D是AC的中点，直线DE 与直线AB相交于点F．（1）求证：DF是⊙O的切线．（2）已知AB＝4，当AC长度变化时，AF的长也随之变化．①当AC＝时，AF＝6；②在整个变化过程中，AF的长是否存在最大值？判断并说明理由．27．（9分）小敏在查阅资料时得知：已知一个四边形各边长均为定值，当它的四个顶点在同一个圆上时，四边形的面积最大．从特殊验证已知四边形ABCD的各边长依次为7，15，20，24，求它的面积S何时最大？易得S＝⋯Ⅱ．∠B为钝角易证当∠ABC为钝角时，∠ADCⅢ．∠B为锐角同理可得Ⅱ中结论综上所述，S的最大值为…．（1）探索情形Ⅰ：①求证：点A，B，C，D在同一个圆上．②S的值为．（2）探索情形Ⅱ：说明此时S的值小于情形Ⅰ中S的值．向一般进发（3）已知四边形ABCD的各边长依次为6，8，8，12，借助已有结论对它展开探索，求它的面积S的最大值．2023-2024学年江苏省南京市鼓楼区九年级（上）期中数学试卷参考答案与试题解析一、选择题（本大题共6小题，每小题2分，共12分.在每小题所给出的四个选项中，恰有一项是符合题目要求的，请将正确选项的序号填涂在答题卡相应位置上）1．（2分）下列方程中，是一元二次方程的是（）A．x﹣1＝0B．x2＝3C．4x﹣3y＝3D．x2﹣y2＝3【分析】根据一元二次方程的定义逐个判断即可．【解答】解：A．方程x﹣1＝0是一元一次方程，不是一元二次方程，故本选项不符合题意；B．方程x2＝3是一元二次方程，故本选项符合题意；C．方程4x﹣3y＝3是二元一次方程，不是一元二次方程，故本选项不符合题意；D．方程x2﹣y2＝3是二元二次方程，不是一元二次方程，故本选项不符合题意．故选：B．2．（2分）已知⊙O的半径为3，平面内点P到圆心O的距离为√5，则点P与⊙O的位置关系（）A．点P在⊙O外B．点P在⊙O上C．点P在⊙O内D．无法确定【分析】点与圆的位置关系有3种．设⊙O的半径为r，点P到圆心的距离OP＝d，则有：点P在圆外⇔d＞r，点P在圆上⇔d＝r，点P在圆内⇔d＜r，由此即可判断．【解答】解：∵圆的半径r＝3，点P到O的距离d=√5，∴d＜r，∴点P在圆内，故选：C．3．（2分）二次函数y＝2（x﹣3）2+2的图象的顶点坐标是（）A．（3，2）B．（﹣3，2）C．（3，﹣2）D．（﹣3，﹣2）【分析】由于二次函数y＝a（x﹣b）2+c的顶点坐标为（b，c），由此即可求出抛物线的顶点坐标．【解答】解：∵二次函数y＝2（x﹣3）2+2，∴其图象的顶点坐标为（3，2）．故选：A．4．（2分）如图，在长40m、宽22m的矩形地面内，修筑两条同样宽且互相垂直的道路，余下的部分铺上草坪，要使草坪的面积达到760m2，设道路的宽为x m，则根据题意，可列出方程（）A．（40﹣x）（22﹣x）＝760B．（40+x）（22+x）＝760C．40×22﹣40x﹣22x＝760D．40×22﹣40x﹣22x﹣x2＝760【分析】把四块耕地拼到一起正好构成一个矩形，矩形的长和宽分别是（40﹣x）和（22﹣x），根据矩形的面积公式，列出关于道路宽的方程．【解答】解：设道路的宽应为x米．依题意得：（40﹣x）（22﹣x）＝760，故选：A．5．（2分）二次函数y＝ax2+bx+c的图象上部分点的横坐标x与纵坐标y的对应值如下表所示．下列说法中，正确的是（）A．图象的开口向上B．图象经过点（4，0）C．图象与x轴只有一个公共点D．点（2，3）右边的图象呈下降趋势【分析】利用抛物线的对称性得到抛物线的顶点坐标为（1，4），则设顶点式y＝a（x﹣1）2+4，再把（0，3）代入求出a得到抛物线解析式为y＝﹣（x﹣1）2+4，于是根据二次函数的性质可对选项进行判断；通过解方程﹣（x﹣1）2+4＝0得到抛物线与x轴的交点坐标为（﹣1，0），（3，0），则可对B选项和C 选项进行判断；根据二次函数的性质，当x＞1时，y随x的增大而减小，从而可对D选项进行判断．【解答】解：∵抛物线经过点（0，3）、（2，3），∴抛物线的顶点坐标为（1，4），设抛物线解析式为y＝a（x﹣1）2+4，把（0，3）代入得3＝a×（0﹣1）2+4，解得a＝﹣1，∴抛物线解析式为y＝﹣（x﹣1）2+4，∵a＜0，∴抛物线开口向下，所以A选项不符合题意；当y＝0时，﹣（x﹣1）2+4＝0，解得x1＝﹣1，x2＝3，∴抛物线与x轴的交点坐标为（﹣1，0），（3，0），所以B选项不符合题意，C选项不符合题意；∵抛物线的对称轴为直线x＝1，∴当x＞1时，y随x的增大而减小，∴点（2，3）右边的图象呈下降趋势，所以D选项符合题意．故选：D．6．（2分）如图，在⊙O中，动弦AB与直径CD相交于点E且总有∠BED＝45°，则AE2+BE2的值（）A．随着OE的增大而增大B．随着OE的增大而减小C．随着OE的增大先增大后减小D．保持不变【分析】作OH⊥AB于点H，连接OA，OB，设半径为R，在直角三角形OAH和OBH中，利用勾股定理整理化简即可．【解答】解：作OH⊥AB于点H，连接OA，OB．设半径为R，∵∠BED＝45°，∴OH＝HE，∴AE2+BE2＝（AH+HE）2+（BH﹣HE）2＝AH2+HE2+2AH•HE+BH2+HE2﹣2BH•HE＝R2+R2+2HE（AH﹣BH）＝2R2∴AE2+BE2的值保持不变．故选：D．二、填空题（本大题共10小题，每小题2分，共20分.不需写出解答过程，请把答案直接填写在答题7．（2分）写出一个一元二次方程答案不唯一，正确即可，如：x2﹣x﹣6＝0，使它的两个根分别是3、﹣2．【分析】以3、﹣2为根分方程为（x﹣3）（x+2）＝0，即x2﹣x﹣6＝0．【解答】解：由题意可知方程为（x﹣3）（x+2）＝0，即x2﹣x﹣6＝0．故答案为x2﹣x﹣6＝0．8．（2分）已知圆锥的底面半径为6，母线长为10，则此圆锥的侧面积为60π．【分析】利用圆锥的底面半径为6，母线长为10，直接利用圆锥的侧面积公式求出即可．【解答】解：依题意知母线长＝10，底面半径r＝6，则由圆锥的侧面积公式得S＝πrl＝π×10×6＝60π．故答案为：60π．9．（2分）若点（1，y1）与（2，y2）都在函数y＝﹣x2+3的图象上，则y1，y2的大小关系是y1＞y2．【分析】首先求得抛物线的开口方向和对称轴，然后根据二次函数的性质判断即可．【解答】解：∵二次函数y＝﹣x2+3的开口向下，对称轴是y轴，∴在y轴的右面y随x的增大而减小，∵点点（1，y1）与（2，y2）都在函数y＝﹣x2+3的图象上，且0＜1＜2，∴y1＞y2．故选：y1＞y2．10．（2分）若扇形的半径为3，圆心角120°，为则此扇形的弧长是2π．【分析】根据弧长的公式l=nπr180进行计算即可．【解答】解：∵扇形的半径为3，圆心角为120°，∴此扇形的弧长=120π×3180=2π．故答案为：2π11．（2分）若一元二次方程x2+mx﹣3＝0（m为常数）的一个根是x＝1，则另一个根是﹣3．【分析】法一：利用方程的根，求出m的值，再解方程即可；法二：利用一元二次方程的根与系数的关系，进行求解即可．【解答】解：法一：由题意得：12+m﹣3＝0，解得：m＝2，∴方程为：x2+2x﹣3＝0，解得：x1＝1，x2＝﹣3；法二：设方程的根为：x1＝1，x2，由根与系数的关系得：x1x2＝﹣3，∴1×x2＝﹣3，∴x2＝﹣3．故答案为：﹣3．12．（2分）如图是函数y＝ax2+bx+c的部分图象，则该函数图象与x轴负半轴的交点坐标是（﹣1，0）．【分析】根据二次函数的性质，抛物线的对称轴为直线x＝2，然后写出点（5，0）关于直线x＝2的对称点即可．【解答】解：∵抛物线的对称轴为直线x＝2，抛物线与x轴的一个交点坐标为（5，0），∴抛物线与x轴的另一个交点坐标为（﹣1，0）．故答案为：（﹣1，0）．13．（2分）已知⊙O的半径等于4cm，AB为⊙O的弦，其长为4√2cm，则弦AB所对的圆周角的度数为45°或135°．【分析】连接OA、OB，∠ACB与∠ADB为弦所对的圆周角作OH⊥AB于H，根据垂径定理得AH=12 AB＝2√2，在Rt△OAH中利用勾股定理计算出OH＝2√2，于是可判断△OAH为等腰直角三角形，得到∠AOH＝45°，则∠AOB＝90°，根据圆周角定理得∠ACB=12∠AOB＝45°，根据圆内接四边形性质得∠ADB＝135°．【解答】解：连接OA、OB，∠ACB与∠ADB为弦所对的圆周角，作OH⊥AB于H，则AH＝BH=12AB＝2√2，在Rt△OAH中，∵OA＝4，AH＝2√2，∴OH=√OA2−AH2=2√2，∴△OAH为等腰直角三角形，∴∠AOH＝45°，∴∠AOB＝90°，∴∠ACB=12∠AOB＝45°，∴∠ADB＝180°﹣∠ACB＝135°，即弦AB所对的圆周角的度数为45°或135°．故答案为45°或135°．14．（2分）五边形ABCDE为圆的内接五边形，其中AB＝AE，∠A＝100°，则∠B+∠D＝220°．【分析】连接BE、BD、AD，先由等腰三角形的性质和三角形内角和定理得∠ABE＝40°，再由圆周角定理得∠ADE＝40°，然后由圆内接四边形的性质得出∠ABC+∠ADC＝180°，即可解决问题．【解答】解：如图，连接BE、BD、AD，∵AB ＝AE ，∠BAE ＝100°，∴∠ABE ＝∠AEB =12（180°﹣100°）＝40°，∴∠ADE ＝∠ABE ＝40°，∵∠ABC +∠ADC ＝180°，∴∠ABC +∠CDE ＝∠ABC +∠ADC +∠ADE ＝180°+40°＝220°，故答案为：220．15．（2分）如图，AB 是⊙O 的直径，若∠E ＝25°，∠CAD ＝45°，则∠CDA 的度数为 35 °．【分析】连接BC ，由圆周角定理得∠ACB ＝90°，∠BAD ＝∠BCD ，再证∠CAB +∠ABC ＝70°+2∠BAD ＝90°，解得∠BAD ＝10°，即可得出结论．【解答】解：如图，连接BC ，∵AB 是⊙O 的直径，∴∠ACB ＝90°，∴∠CAB +∠ABC ＝90°，∵∠CAB ＝∠CAD +∠BAD ，∠ABC ＝∠BCD +∠E ，∠BAD ＝∠BCD ，∴∠CAB +∠ABC ＝∠CAD +∠BAD +∠BAD +∠E ＝45°+2∠BAD +25°＝70°+2∠BAD ＝90°，解得：∠BAD＝10°，∴∠CDA＝∠BAD+∠E＝10°+25°＝35°，故答案为：35．16．（2分）等边△ABC的边长是4√3，直线l经过等边△ABC的外心O，过B作BD⊥l，垂足为D，连接CD，则CD的最小值是2√7−2．【分析】当直线l过点A时，CD的值最小，即点D为BC的中点，根据等边三角形的性质即可得到结论．【解答】解：∵BD⊥l，垂足为D，连接CD，∵BD+CD≥BC，∴当BD+CD＝BC时，点D在以BO为直径的圆上运动，设BO的中点为M，当D在MC上时，CD取最小值＝2√7−2，故答案为：2√7−2．三、解答题（本大题共11小题，共88分.请在答题卡指定区域内作答，解答时应写出文字说明、证明过程或演算步骤）17．（8分）解下列方程．（1）（x﹣1）2﹣16＝0；（2）x2﹣3x+1＝0．【分析】（1）移项后开方，即可得出两个一元一次方程，再求出方程的解即可；（2）先求出b2﹣4ac的值，再代入公式求出答案即可．【解答】解：（1）（x﹣1）2﹣16＝0，（x﹣1）2＝16，开方，得x﹣1＝±4，解得：x1＝5，x2＝﹣3；（2）x2﹣3x+1＝0，这里a＝1，b＝﹣3，c＝1，∵b2﹣4ac＝（﹣3）2﹣4×1×1＝5＞0，∴x=−b±√b2−4ac2a=3±√52×1，∴x1=3+√52，x2=3−√52．18．（7分）已知二次函数y＝x2﹣mx+1的图象经过点（2，3）．（1）求m的值；（2）该二次函数的图象是否经过点（﹣2，6）？判断并说明理由．【分析】（1）将（2，3）代入二次函数即可求解；（2）把x＝﹣2代入二次函数计算，若y＝7，则点P（﹣2，6）不在二次函数图象上．【解答】解：（1）将（2，3）代入二次函数y＝x2﹣mx+1得，4﹣2m+1＝3，解得m＝1，∴m的值是1．（2）由（1）得二次函数的解析式为y＝x2﹣x+1，当x＝﹣2时，y＝4﹣（﹣2）+1＝7≠6，∴点P（﹣2，6）不在这个二次函数图象上．19．（7分）如图，在以点O为圆心的两个同心圆中，大圆的弦AB与小圆相切于点P．（1）求证AP＝BP；（2）已知图中阴影部分面积为10π，求弦AB的长．【分析】（1）连接OP，利用圆的切线的性质定理和垂径定理解答即可；（2）连接OA，利用圆的面积公式和垂径定理解答即可．【解答】（1）证明：连接OP ，如图，∵大圆的弦AB 与小圆相切于点P ，∴OP ⊥AB ，∴P A ＝PB =12AB ，∴AP ＝BP ；（2）连接OA ，∵图中阴影部分面积为10π，∴π•OA 2﹣π•OP 2＝10π，∴OA 2﹣OP 2＝10．∵OP ⊥AB ，∴AP 2＝OA 2﹣OP 2＝10，∴AP =√10，∴AB ＝2AP ＝2√10．20．（8分）一个直角三角形的两条直角边长度之和是10cm ，面积是12cm 2，求斜边的长．【分析】设这个直角三角形的两直角边为a 、b ，斜边为c ，根据题意得a +b ＝14，12ab ＝12，由c 2＝a 2+b 2＝（a +b ）2﹣2ab ，代值开平方求斜边长．【解答】解：设这个直角三角形的两直角边为a 、b ，斜边为c ，根据题意得a +b ＝14，12ab ＝12，即ab ＝24， ∴c 2＝a 2+b 2＝（a +b ）2﹣2ab ＝102﹣2×24＝52，开平方，得c =2√13，即斜边长为2√13cm ．21．（7分）二次函数图象的顶点为（﹣1，2），图象经过（0，1）．（1）求该二次函数的表达式；（2）结合图象，直接写出当﹣2≤x ≤3时y 的取值范围．【分析】（1）设顶点式y＝a（x+1）2+2，然后把已知点的坐标代入求出a即可；（2）先利用（1）中的解析式计算出自变量为﹣2和3所对应的自变量的范围，再根据二次函数的性质得到x＝﹣1时，y有最大值2，然后结合图象求解．【解答】解：（1）设抛物线解析式为y＝a（x+1）2+2，把（0，1）代入得1＝a×（0+1）2+2，解得a＝﹣1，∴抛物线解析式为y＝﹣（x+1）2+2；（2）当x＝﹣2，y＝﹣（﹣2+1）2+2＝1，当x＝3，y＝﹣（3+1）2+2＝﹣14，而x＝﹣1时，y有最大值2，∴﹣2≤x≤3时，﹣14≤y≤2．22．（8分）尺规作图：作已知圆的一条直径．要求：①保留作图痕迹；②用两种不同方法作图．【分析】方法1：在圆上任取弦AB，作线段AB的垂直平分线，与圆分别交于点C，D，连接CD，根据垂径定理可知CD为已知圆的一条直径；方法2：在圆上任取弦EF，过点F作EF的垂线，交圆于点G，连接EG，由圆周角定理可知EG为已知圆的一条直径．【解答】解：方法1：如图，在圆上任取弦AB，作线段AB的垂直平分线，与圆分别交于点C，D，连接CD，则CD即为已知圆的一条直径．方法2：如图，在圆上任取弦EF，过点F作EF的垂线，交圆于点G，连接EG，则EG即为已知圆的一条直径．23．（8分）已知关于x的方程x2﹣4x+4﹣k2＝0（k为常数）．（1）求证：不论k取何值时，该方程总有实数根．（2）方程的两个实数根可能都是负根吗？判断并说明理由．【分析】（1）证明判别式大于等于0即可；（2）根据两根之和是正数，判断即可．【解答】（1）证明：∵Δ＝（﹣4）2﹣4（4﹣k2）＝4k2≥0，∴方程总有实数根；（2）解：不可能．理由：∵两根之和＝4＞0，∴两个根不可能都是负数．24．（8分）如图是二次函数y＝x2﹣2x﹣3的大致图象．（1）求该图象顶点的坐标；（2）该图象经过怎样的平移可以得到函数y＝x2的图象？（3）将该图象绕原点旋转180°直接写出所得图象对应的表达式．【分析】（1）把解析式化为顶点式，即可求出顶点坐标；（2）按照平移的规律：左加右减，上加下减即可；（3）根据旋转180°关于原点对称，可得答案．【解答】解：（1）y＝x2﹣2x﹣3＝（x﹣1）2﹣4，∴该图象顶点的坐标为（1，﹣4）；（2）将该图象向左平移1个单位，再向上平移4个单位可得函数y＝x2的图象；（3）该图象绕原点旋转180°直接写出所得图象对应的表达式为﹣y＝（﹣x）2﹣2（﹣x）﹣3，即y＝﹣x2﹣2x+3．25．（8分）某商店八月份的销售额为30万元，九月份因经营不善销售额有明显下降，商店积极改进，十月份的销售额恢复到30万元，已知十月份销售额的增长率是九月份销售额的下降率的1.5倍．求九月份的销售额．【分析】设九月份销售额的下降率为x，则十月份销售额的增长率为1.5x，根据某商店八月份的销售额为30万元，九月份因经营不善销售额有明显下降，十月份的销售额恢复到30万元，列出一元二次方程，解之取其正值，即可解决问题．【解答】解：设九月份销售额的下降率为x，则十月份销售额的增长率为1.5x，由题意得：30（1﹣x）（1+1.5x）＝30，整理得：3x2﹣x＝0，解得：x1=13，x2＝0（不符合题意，舍去），∴30（1﹣x ）＝30×（1−13）＝30×23=20， 答：九月份的销售额为20万元． 26．（10分）如图，AB 是⊙O 的直径，∠BAC ＝90°，BC 与⊙O 相交于点E ，D 是AC 的中点，直线DE 与直线AB 相交于点F ．（1）求证：DF 是⊙O 的切线．（2）已知AB ＝4，当AC 长度变化时，AF 的长也随之变化．①当AC ＝ 4√3 时，AF ＝6；②在整个变化过程中，AF 的长是否存在最大值？判断并说明理由．【分析】（1）由等腰三角形的性质可得∠EDA ＝∠EAD ，可得∠ODA ＝∠OAD ，由余角的性质可求∠OED ＝90°，可得结论；（2）①通过证明△ABE ∽△CBA ，可得AB AC =BE AE ，通过证明△FEB ∽△FAE ，可得BE AE =BF EF 即可求解；②利用①中结论得出AC 和AF 的关系，可判断AF 的长度的变化．【解答】（1）证明：如图1，连接OE ，AE ，∵AB 是⊙O 的直径，∴∠AEB ＝90°，∴∠AEC ＝90°，∵D 是AC 的中点，∴AD ＝DE ＝CD ，∴∠EAD ＝∠AED ，∵OE ＝OA ，∴∠OAE ＝∠OEA ，∵∠BAC ＝∠OAE +∠EAD ＝90°，∴∠OED ＝∠OEA +∠AED ＝90°，∵OE 是⊙O 的半径，∴DF 是⊙O 的切线．（2）解：①∵∠BAC ＝∠AEC ＝90°，∴∠C ＝∠BAE ，又∵∠ABE ＝∠CBA ，∴△ABE ∽△CBA ，∴AB AC =BE AE ，∵∠FEB +∠BEO ＝∠BEO +∠AEO ＝90°，∴∠FEB ＝∠AEO ＝∠OAE ，∵∠F ＝∠F ，∴△FEB ∽△F AE ，∴BE AE =BF EF =EF AF ， ∴AB AC =BF EF ，EF 2＝AF •BF ，∴AB •EF ＝AC •BF ，如图1，∵AB ＝4，AF ＝6，∴BF ＝2，EF ＝2√3，∴4×2√3=2AC ，即AC ＝4√3；如图2，∵AB ＝4，AF ＝6，∴BF ＝10，EF ＝2√15，∴4×2√15=10AC ，即AC =4√155； 故答案为：4√3或4√155；②AF 不存在最大值，理由如下：如图1，设AC ＝x ，AF ＝y ，∴EF 2＝y （y ﹣4），∴4EF ＝x （y ﹣4），整理得，y＝4+64x2−16，当x无限接近4时，y的值无限大，即当DE和AB接近平行时，此时AF无限大．27．（9分）小敏在查阅资料时得知：已知一个四边形各边长均为定值，当它的四个顶点在同一个圆上时，四边形的面积最大．从特殊验证已知四边形ABCD的各边长依次为7，15，20，24，求它的面积S何时最大？易得S＝⋯Ⅱ．∠B为钝角易证当∠ABC为钝角时，∠ADCⅢ．∠B为锐角同理可得Ⅱ中结论综上所述，S的最大值为…．（1）探索情形Ⅰ：①求证：点A，B，C，D在同一个圆上．②S的值为234．（2）探索情形Ⅱ：说明此时S的值小于情形Ⅰ中S的值．向一般进发（3）已知四边形ABCD的各边长依次为6，8，8，12，借助已有结论对它展开探索，求它的面积S的最大值．【分析】（1）①连接AC，取AC的中点O，连接OB，OD，证出∠ADC＝90°，由直角三角形的性质及圆的定义可得出结论；②由三角形面积公式可得出答案；（2）由三角形面积公式可得出答案；（3）设AB＝6，BC＝8，CD＝8，AD＝12，连接AC，过点C分别作CE⊥AB于点E，CF⊥AD于点F，证明Rt△BCE≌Rt△DCF（HL），△AEC≌△AFC，得出AE＝AF，BE＝FD，求出BE＝3，则可得出答案．【解答】（1）①证明：连接AC，取AC的中点O，连接OB，OD，∵∠ABC＝90°，∴AC2＝BC2+AB2＝625，又∵AD2+CD2＝225+400＝625，∴AD2+CD2＝AC2，∴∠ADC＝90°，∵O为AC的中点，∴AO＝OB＝OC＝OD，∴点A，B，C，D在同一个圆上；②解：∵∠ADC＝∠ABC＝90°，∴S＝S△ADC+S△ABC=12AD⋅DC+12AB⋅BC=12×15×20+12×7×24＝150+84＝234．故答案为：234；（2）解：∵AE＝x，AF＝y，∴S =12×20⋅x +12×24⋅y =10x +12y ， 在Rt △AED 中，x ＜15，在Rt △AFB 中，y ＜7，∴S ＝10x +12y ＜10×15+12×7，即S ＜234；（3）解：由题意可知，当四边形ABCD 四顶点共圆时，它的面积最大，设AB ＝6，BC ＝8，CD ＝8，AD ＝12，连接AC ，过点C 分别作CE ⊥AB 于点E ，CF ⊥AD 于点F ， ∵BC ＝CD ，∴BĈ=CD ̂， ∴∠BAC ＝∠DAC ，∵CE ⊥AB ，CF ⊥AD ，∴CE ＝CF ，∵BC ＝CD ，∴Rt △BCE ≌Rt △DCF （HL ），同理可证△AEC ≌△AFC ，∴AE ＝AF ，BE ＝FD ，∴AD ﹣BE ＝AB +BE ，∴BE ＝3，∵∠BEC ＝90°，∴CE =√BC 2−BE 2=√82−32=55，∴CF =√55，∴S 四边形ABCD ＝S △ABC +S △ACD =12AB ⋅EC +12AD ⋅CF =9√55，即四边形ABCD 面积S 的最大值为9√55．。
2011年江苏省南京市中考数学试卷及答案(解析版)
南京市2011年初中毕业生学业考试数学数学注意事项：1．本试卷共6页，全卷满分120分，考试时间为120分钟，考生答题全部答在答题卡上，答在本试卷上无效．2．请认真核对监考教师在答题卡上所有粘贴条形码的姓名、考试证号是否与本人相符合，再将自己的姓名、准考证号用0.5毫米的黑色墨水签字笔填写在答题卡及本试卷上．3．答选择题必须用2B铅笔将答题卡上对应的答案标号涂黑．如需要改动，请用橡皮擦干净后，再选涂其他答案，答非选择题必须0.5毫米黑色墨水签字笔写在答题卡上指定位置，在其他位置答题一律无效．4．作图必须用2B铅笔作答，并请加黑加粗，描写清楚．一、选择题（本大题共6小题，每小题2分，共12分，在每小题所给出的四个选项中，恰有一项是符合题目要求的，请将正确的选项的字母代号填涂在答题卡相应位置．．．．．．．上）1A．3B．－3C．±3D．【答案】A．【考点】算术平方根。
【分析】利用算术平方根的定义，直接得出结果2．下列运算正确的是A．a2＋a3=a5 B．a2•a3=a6C．a3÷a2=a D．(a2)3=a8【答案】C．【考点】指数运算法则。
【分析】a3÷a2=a= a3-2= a3．在第六次全国人口普查中，南京市常住人口约为800万人，其中65岁及以上人口占9.2%．则该市65岁及以上人口用科学记数法表示约为A．0.736×106人B．7.36×104人C．7.36×105人D．7.36×106 人【答案】C．【考点】科学记数法。
【分析】利用科学记数法的定义，直接得出结果:8000000×9.2%=736000=7.36×105.4．为了解某初中学校学生的视力情况，需要抽取部分学生进行调查，下列抽取学生的方法最合适的是A．随机抽取该校一个班级的学生B．随机抽取该校一个年级的学生C．随机抽取该校一部分男生D．分别从该校初一、初二、初三年级中各班随机抽取10%的学生【答案】D．【考点】随机抽样样本的抽取。
2023-2024学年江苏省南京市鼓楼区九年级(上)期末数学试卷+答案解析
一、选择题：本题共6小题，每小题3分，共18分。
在每小题给出的选项中，只有一项是符合题目要求的。
1.平面内，的半径为3，若点P 在外，则OP 的长可能为( )A. 4B. 3C. 2D. 12.一元二次方程的根的情况是2023-2024学年江苏省南京市鼓楼区九年级（上）期末数学试卷( )A. 没有实数根B. 有一个实数根C. 有两个不相等的实数根D. 有两个相等的实数根3.下列实际问题中的y 与x 之间的函数表达式是二次函数的是( )A. 正方体集装箱的体积，棱长xmB. 高为14m 的圆柱形储油罐的体积，底面圆半径xmC. 妈妈买烤鸭花费86元，烤鸭的重量y 斤，单价为x 元/斤D. 小莉驾车以的速度从南京出发到上海，行驶xh ，距上海ykm 4.如图，D ，E 分别是的边AB ，AC 上的点，，，若的周长为6，则的周长等于( )A. 24B. 18C. 12D. 95.在地球上同一地点，不同质量的物体从同一高度同时下落，如果除地球引力外不考虑其他外力的作用，那么它们的落地时间相同．物体的下落距离与下落时间之间的函数表达式为其中g 取值为小莉进行自由落体实验，她从某建筑物抛下一个小球，经过4s 后落地，则该建筑物的高度约为( )A. 98mB.C. 49mD. 6.平面直角坐标系内，已知点，，当时，若最大，则t 的值为( )A.B.C.D.二、填空题：本题共10小题，每小题3分，共30分。
7.若，则为__________.8.点C是线段AB的黄金分割点，若，则__________9.某汽车厂商经过两次增产，将汽车年产量由万辆提升至6万辆，设平均每次增产的百分率是x，可列方程为__________.10.一个圆锥的底面圆半径是1，母线长是3，沿着一条母线将圆锥侧面剪开并展平，得到一个扇形，则这个扇形的圆心角度数为__________11.将二次函数的图象向左平移1个单位，再向上平移一个单位，得到的新图象函数的表达式为__________.12.有3个样本如图所示，关于它们的离散程度有下列几种说法：①样本1与样本3的离散程度相同；②样本2的离散程度最小；③三组数据的离散程度从小到大依次为：样本2、样本3、样本正确的序号为__________.13.如图，AB是的直径，弦于点E，若，，则OA长为__________.14.分别以等边的三个顶点为圆心，边长为半径画弧得到的曲边三角形叫莱洛三角形．如图，等边的边长为2cm，则图中阴影部分的面积为__________15.如图，夜晚路灯下，小莉在点D处测得自己影长，在点G处测得自己影长，E、D、G、B在同一条直线上．已知小莉身高为，则灯杆AB的高度为__________16.中，，，点I是的内心，点O是的外心，则__________.三、计算题：本大题共1小题，共6分。
江苏省南京市鼓楼区近三年(2021-2023)九年级上学期期末英语试题分类汇编：阅读填空
南京市鼓楼区近三年（2021-2023）九年级上学期期末英语试题分类汇编阅读填空江苏省南京鼓楼区2022-2023学年九年级上学期期末英语试卷六、阅读填空（共20小题；每小题1分，满分20分）A）请认真阅读下列短文，根据所读内容在文章后表格中第76——85小题的空格里填入一个最恰当的单词，并将答案填写在答题卡标号为76——85的相应横线上。
注意：每个空格只填1个单词。
Famous actress Qin Yi passed away at age 100 in Shanghai on May 9, 2022. It is a huge loss to the Chinese film industry. Born on Feb 4, 1922, Qin Yi, named Qin Dehe at birth,was one of eight daughters in the family. Qin, a film lover, turned her passion for the silverscreen into a lifelong career.In 1939 Qin Yi first appeared in the film A Good Husband. In 1947, she had her first lead role in Far Away Love, which made her a famous film star. In 1949, she was named with three other actresses as the “Top Four Actresses of China”.After 1949, Qin Yi entered the Shanghai Film Studio, where she created some of the most memorable characters of Chinese women on the movie and TV screens, including Railway Guerrillas, Woman Basketball Player No. 5, Lin Zexu, Song of the Youth and Loyal Overseas Chinese.With the rise of television in China in the 1980s, Qin Yi also appeared in major roles in a variety of TV dramas, winning more popularity. She was presented with the Golden Eagle Award in 1983 for Best Actress for Under the Roofs of Shanghai. From 2008 to 2009 she received three lifetime achievement awards. Qin Yi is the only actor in Chinese history to be given the titles of “People’s Artist” (2019) and “Most Beautiful Fighter” (2019).Qin Yi was a strong person, though her personal life is challenging. Qin took care of her sick husband Jin Yan, who spent the rest of his life fighting against a disease for more than 20 years. Her son, Jin Jie, had a serious mental problem and could not live on his own. Qin looked after him for over 40 years before he died in 2007. She had saved 200,000 yuan for the care of her son, then she donated the mony to the victims of 2008 Wenchuan earthquake. Qin Yi is considered as a“national treasure”with her contribution to the Chinese film industry. Qin is not only an excellent actress but also a moral hero who has a great influence on Chinese people.A famous actress——Qin YiIntroduction > Qin Yi was born in 1922 and died in 2022.> Her_____76_____ is a great loss to the Chinese film industry. > She _____77_____ most of her lifetime to the silver screen.Acting_____78_____ > In 1939, Qin Yi made her first _____79_____ in the film A Good Husband. > In 1947, she became famous for her first lead role in Far Away Love.> In 1949, her name was on the _____80_____ of “Top Four Actresses of China”.> After 1949, she became a _____81_____of the Shanghai Film Studio. > In 1983, Qin _____82_____the Golden Eagle Award for Best Actress. > In 2019, Qin was awarded the title of People’s Artist.Personal life > Though Qin Yi’s life is full of _____83_____, she is a strong person.> Qin spent years looking after her sick husband and son _____84_____ the end of their lives.Praise > Qin Yi is considered as a “national treasure”.> Her spirits _____85_____ Chinese people a lot.B）请根据短文内容及首字母提示，填写所缺单词，并将答案填写在答题卡标号为86—95的相应位置上。
鼓楼区2011-2012学年第一学期七年级数学期末试卷
如图，已知线段AB＝4cm，直线AB上有一点C，且AC＝1cm，点O是线段BC的中点，求线段OB的长度.
五、探究与思考（共3大题，满分23分）
24．（本题满分6分）
一件夹克衫按成本价提高40%标价，又以8折（即按标价的80%）出售，结果获利16.8元，求这件夹克衫的成本.（请列方程解答）
A．9B．18C．27D．36
二、填空题（每空2分，共22分）
9．比较大小：－0.5 （填“＜”、“＝”或“＞”）．
10．化简5a–(2a–4b)的结果为．
11．单项式－ 的系数是，次数是．
12．地球平均每年发生的雷电次数约为16 000 000次，可用科学记数法表示为次．
13．如图，直线AB、CD相交于O点，∠AOE＝∠COF=90°，则与∠AOF相等的角为．
3．下列各个式子化简后结果为正的是
A．（–3）2B．–32C．（–3）3D．–（–3）2
4．“迎青奥绿色行”六个汉字分别在正方体的六个面上，展开后如图．那么与“迎”相对面上的汉字是
A．色B．绿C．青D．奥
5．一副三角板以如图所示的方式摆放，则∠ACD的度数为
A．150°B．135°C．120°D．105°
6．如图，数轴的单位长度为1．如果点B、C表示的数的绝对值相等，那么点A表示的数是
A．－6B．－5C．－4D．－2
7．下列说法错误的是
A．过一点有且只有一条直线与已知直线平行
B．过一点有且只有一条直线与已知直线垂直
C．两点之间的所有连线中，线段最短
D．如果 ∥ ， ∥ ，那么 ∥ .
8．如图，在一个８８的方格棋盘的A格里放一枚棋子，如果规定棋子每步只能向上、下或左、右走一格，那么这枚棋子走如下的步数后能到达B格的是
【试卷】南京市鼓楼区2012-2013学年度第一学期期末测试
江苏省南京市2012-2013学年度第一学期七年级期末测试一、听力（共10小题；每小题1分，满分10分）第一部分 听对话选择正确图片 每段对话听两遍。
1. What does Sandy want to eat for breakfast today?A. B. C .2. What does Daniel plan to buy this afternoon?A. B. C.3. When is Tom ’s birthday?A. B. C.4. What is the girl going to wear for tonight’s party? A.B. C.5. How much is the Teddy bear now?A. B. C. 第二部分 听独白回答问题 你将听到一段独白，听两遍。
6. When is Mother’s Day?A. On the second Sunday of May.B. On the second Saturday of May.C. Always on May 12th .7. Who gets up earliest(最早) on Mother ’s Day?A. Mother.B. Father.C. The writer.8. What does the writer do on Mother ’s Day?A. He cooks breakfast for his mother.B. He does some cleaning for his mother.C. He wakes his mother up.9. What present does the writer give his mother?A. A card with good wishes.B. A nice cooking book.C. A large bucket of flowers.10. How does the writer ’s mother feel on that day?A. Funny.B. Happy.C. Sad.二、选择填空(共10小题；每小题1分，满分10分)11. — Sandy, _______ apple on the table is for you. Have it before you go out.27th28th26th 29th—Thanks, Mum. _____ apple a day keeps the doctor away.A. the; AnB. an; AnC. an; TheD. the; The12. Mary’s aunt is trying to keep slim. It’s not good for her to have ________ too often.A. fruitB. ice creamC. vegetablesD. water13. —May I have ________ fruit before supper?—Sorry. There isn’t _______ fruit at home now.A. some; someB. any; anyC. some; anyD. any; some14. I_______ watch Happy Goat and Big Big Wolf because I think it’s so boring.A. oftenB. alwaysC. usuallyD. seldom15. —I sometimes don’t have enough time ______ with my parents.—You will if you spend more time ______ with them at home.A. to chat; to stayB. to chat; stayingC. chatting; to stayD. chatting; staying16. —________ do you visit your grandparents?—Three or four times a year. Usually at Mid-Autumn festival and Spring Festival.A. How longB. How oftenC. How muchD. How many17. —The price of an iphone 4s will be more than 5000 yuan.—Yes. Good things always ________ much.A. payB. spendC. takeD. cost18. Do your homework and you can play games.A. then; firstB. first; thenC. next; afterwordsD. next; finally19. I like the trainers very much. Can I ?A. try it onB. try on itC. try them onD. try on them20. —Oh, dear! I don’t take any money with me. I can’t buy the CD.—_________. I have enough. You can borrow some from me.A. I am sorryB. You are welcomeC. Never mindD. Of course not三、完形填空（共10小题；每小题1分，满分10分）Do you like shopping on the Internet? I like to buy books online(在线的). Usually, the price isn’t very 21 . You don’t have to pay a lot. When I want to know something about a book, I write an e-mail to the online shop for 22 . I am shy. It is difficult for me to talk to the shopkeepers in real(真实的) shops.Sometimes online shops have pictures of the book. They can show you some pages of the book, too. I think this is very 23 . I don’t like to read books in shops because the shopkeepers may come and talk to me. I don’t know 24 to say to them.I can shop at midnight 25 I want to. Real shops close at nine o’clock in the evening. Some people may worry that the postman may lose or damage(损坏) the things they 26 from online shops, but I don’t worry about that. I think it is hard to damage a book and most of the postmen are very careful.Some people say that shopping in real shops is good because you can go with your friends and 27 many different things. They call this window shopping. I don’t have any friends and I don’t like to talk too much—it makes me 28 .I am 29 my grandmother how to shop online. She cannot walk very far so it will make shopping easy for her if she shops online. My brother 30 wants to learn how to do it. He is very busy most of the time. When he has a little free time, he can shop online.21. A. expensive B. cheap C. high D. much22. A. a model B. an album C. a question D. an answer23. A. careful B. helpful C. colourful D. beautiful24. A. what B. how C. where D. why25. A. and B. but C. if D. when26. A. have B. borrow C. buy D. sell27. A. look after B. look like C. look for D. look at28. A. tired B. sad C. happy D. smart29. A. asking B. studying C. teaching D. giving30. A. again B. also C. too D. never四、阅读理解（共15小题；每小题1分，满分15分）阅读下列材料，从每题所给的四个选项（A、B、C、D）中，选出最佳选项。
2023-2024学年江苏省南京市鼓楼区九年级上学期期中化学质量检测模拟试卷(含答案)
2023-2024学年江苏省南京市鼓楼区九年级上学期期中化学质量检测模拟试卷可能用到的相对原子质量：H-1C-12N-14O-16S-32Cl-35.5一、选择题（本题共15小题，每小题只有一个选项符合题意。
每小题2分，共30分）1.地壳中含量排第四位的元素是A.铝B.氧C.铁D.硅2.下列物质和颜色不对应的是A.氧化汞——灰色B.二氧化硫——无色C.氧化镁——白色D.二氧化锰——黑色3.过滤操作不需要用到的仪器是A.烧杯B.玻璃棒C.漏斗D.酒精灯4.下列化学用语中表示“2个氢分子”的是A.2HB.2H2C.H2D.2H+5.下列物质中属于混合物的是A.水银B.蒸馏水C.矿泉水D.冰水混合物6.今年世界环境日的中国主题为“共建清洁美丽世界”，下列做法不符合这一理念的是A.分类回收垃圾B.露天焚烧秸秆C.合理施用化肥D.倡导绿色出行7.2022年北京冬奥会上冰壶机器人使用的是质量轻且能提供充足能量的锂电池。
锂元素在元素周期表中的信息如图所示，下列说法不正确的是A.锂属于非金属元素B.锂的原子序数是3C.锂的相对原子质量是6.941D.锂原子中的质子数为38.下图中表示“国家节水标志”的是A B C D9.下列实验现象描述正确的是A.木炭在氧气中燃烧，发出白光，生成二氧化碳B.铁丝伸入氧气中火星四射，生成黑色固体C.红磷在空气中燃烧产生大量白烟D.硫在空气中燃烧产生明亮的蓝紫色火焰10.1909年，卢瑟福用带正电的α粒子束轰击金箔后发现：大多数α粒子不改变原来的运动方向，一部分α粒子发生偏转，极少数α粒子偏转几乎达到180°。
由该实验现象可知A.原子是实心球体B.原子核带负电C.原子核外有电子D.原子核在原子中所占体积很小11.下列物质的用途与其性质对应关系正确的是A.氮气无色无味，可用作保护气B.氢气具有可燃性，可用作燃料C.稀有气体很不活泼，可制成电光源D.氧气密度比空气略大，可缓解高原反应阅读下列材料，回答第12～13题：通常情况下，氨（NH3）是一种无色、有刺激性气味的气体。
江苏省南京市鼓楼区十二初级中学2022-2023学年九年级上学期英语期初测试(含答案)
全卷满分100分测试时间75分钟第I卷（选择题）（总分40分）一、单项选择（每小题1分；共15小题，满分15分）1.From the following pictures, we know that _________________.A. the boy can skate on the ice as it is thick enoughB. the boy can skate on the ice as there is enough iceC. the boy can’t skate on the ice as it isn’t thick enoughD. the boy can’t skate on the ice as there isn’t enough ice2. ----How do you like _________ mascot (吉祥物) for the Beijing Winter Olympics, BingDwen Dwen?----Well. I’ve never seen _________ more exciting one.A. a; aB. the; theC. the; aD. a; the3. ----Simon, we can save energy by riding shared bicycles.----That’s true. _________________, using air conditioners less is also a good way.A.HoweverB. OtherwiseC. AnywayD. Moreover4.----The oil on the earth will _________________ one day.----I think so. We should make good use of it.A. turn offB. use upC. depend onD. run out5. ----Jim, turn down the music. The baby is sleeping.----Don’t worry. He _________________ for half an hour.A. woke upB. has woken upC. was awakeD. has been awake6. It’s not _________________ him to help these children with intellectual disabilitiesbecause he has received little training so far.A. simple forB. simple ofC. hard forD. hard of7. Sadly, the task is done _________________ to make the boss satisfied. Poor thing!A. too carelesslyB. too carelessC. carelessly enoughD. careless enough8. The China International Search and Rescue Team has brought help and hope to people in disasters around the world since it _________________ 18 years ago.A. set upB. is set upC. was set upD. will be set up9. Read the following text. What kind of way is used to organize the idea?I like reading Treasure Island and I’ve learned a lot from it: Jim, the main character inthe book, is very brave. His story gave me lots of confidence. After reading the book, I’m willing to try new things. I also decide to learn to write my own stories.A. Time order.B. Space order.C. Specific to general.D. General to specific.10. I joined in the dancing to show that I am _________________ to do something I’m not good at.A. brave enoughB. shy enoughC. too braveD. too shy11. ----How do we start the online tour?----I _________________ you. Weren’t you listening?A.tellB. am tellingC. will tellD. have told12.According to the new traffic rules, people _________________ wear a helmet whenriding an e-bike.A. mightB. mayC. canD. must13. A baby’s first month birthday is a special event in our hometown and_________________ with a special party.A. is celebratingB. celebratesC. is celebratedD. will celebrate14. ----Song of Autumn by Liu Yuxi describes autumn in a different way.----I agree. Most poets (诗人) describe this season to _________________ a feeling of sadness, but in Liu’s eyes autumn is full of life and hope.A. excuseB. expressC. expectD. explain15. ----I promise I will work harder this term, Miss Wang.----Well, just as the saying goes, “_________________” I do hope that you will act soon.A. Many hands make light work.B. Actions speak louder than words.C. It never rains but it pours.D. A friend in need is a friend indeed.二、完形填空（每小题1分；共10小题，满分10分）When we see well, we do not think about our eyes very often.It is only when we cannot see perfectly that we ____16____ how important our eyes are.People who are near-sighted can only see things that are very close to their eyes.Everything else seems ____17____. A person ____18____ does a lot of work, such as writing, reading and sewing becomes near-sighted then. People who are far-sighted suffer from just the ____19____ problem. They can see things that are far away, but they have ____20____ in reading a book unless they hold it at arm’s length. If they want to do much ____21____, they must wear glasses, too.Other people do not see clearly because their eyes are not exactly the right shape. They have what is called astigmatism (散光). This, too, can be corrected by glasses. Some people’s eyes become ____22____ because of cataracts (白内障). Long ago these people often became blind. Now, however, it is possible to operate on the cataracts and remove them.Having two good eyes is important for judging distances. ____23____ eye sees things from a slightly different angle (角度). To prove this to ____24____, look at an object out of one eye; then look at the same object out of your other eye. You will find the object’s relation to the background and other things around it has changed. The difference between these two different eye views ____25____ us to judge (判断) how far away an object is. People who have only one eye cannot judge distance as people with two eyes do.16.A. remind B. receive C. review D. realize17.A. unhappy B. unpleasant C. unclear D. unfair18.A. what B. who C. when D. which19.A. right B. same C. wrong D. opposite20.A. difficulty B. fun C. time D. experience21.A. shopping B. cleaning C. reading D. fishing22.A. cloudy B. sunny C. windy D. rainy23.A. Each B. Every C. Both D. All24.A. yourself B. himself C. herself D. myself25.A. helps B. makes C. encourages D. agrees三、阅读理解（每小题1分；共15小题，满分15分）AWhile the mobile Internet is changing the way we live, more and more old people in China are getting in on the digital world, although they will have some trouble.Yang Su is among them. He is both a pianist of Shenzhen Symphony Orchestra (交响乐团), a 61-year-old professional vlogger (视频博主). With more than 2.3 million followers on China’s famous short video app “Douyin”, his videos have been watched by over 1 million times.Three years ago, my son put a short video of me playing the piano at home on ‘douyin’, and hours later it was played over 1 million times”, Yang remembered.However, it’s not easy for him to learn how to film a video. He needs to play through the music since he doesn’t know how to edit (编辑) and cut clips. His child helps him a lot later. “The Internet has opened the door to a new world” Yang said. “I always keep myself busy and feel very young at my age.”Do other old people get used to the digital world like Yang? Ding Qiulin is a 73-year-old man from Hunan. “I got my smartphone more than one year ago but I only use WeChat for video calls,” he said. Ding finds it hard to learn to use other apps.____________▲_____________ Luo Xu, a post-90s volunteer, joined a non-profit organization in Beijing nine years ago to teach old people how to use the Internet. “While I am helping people aged between 50 and 90 years old, I have a strong feeling of pleasure,” Luo said.26.Why is it difficult for Yang Su to film a video?A.Because he can only use WeChat for video calls.B.Because nobody can help him.C.Because he doesn’t know how to edit or cut clips.D.Because he has just got his smartphone.27.Which of the following sentences is an opinion (观点)?A.Ding Qiulin is a 73-year-old man from Hunan.B.It’s hard for Ding Qiulin to learn to use other apps.C.More and more old people in China are getting in on the digital world.D.Yang Su is both a pianist of the Shenzhen Symphony Orchestra and a 61-year-oldprofessional vlogger.28.Which of the following can we put in “____________▲_____________” ?A. In fact, many other old people have the same feeling as Ding.B. However, many young people find it hard to teach the old to use the Internet.C. As a result, old people have to spend much time learning to use the smartphone.D. Luckily, younger people are building a bridge between the digital world and the old.29. Why did the writer write the passage?A. To tell us we should learn to make short videos like YangB. To explain why the mobile Internet is popular in our country.C. To show the mobile Internet makes a great difference to the old.D. To ask young people to spend more time caring for their parents.BLee put the body of a 2-year-old cat on a small bed surrounded by flowers. Then he let the cat’s owner say a last goodbye. Lee, 26, is a pet undertaker. “We send off 400 to 500 pets each year, and there’s a great need of pet funerals (葬礼),” he told China News.Keeping pets has become a fashion for young people in China and people are more than willing to pay for whatever their pets need. A report says, from 2019 to 2021 China’s pet market grew by almost 18 percent and hit 348.8 billion yuan in value in 2021.In addition to basic services, pet owners are increasingly paying for many new services, such as training: pet funerals, insurance, photography and entertainment. The growing pet market has also given birth to a large number of new pet-related jobs.Liu Wei, 33, is a famous pet detective in Shanghai. He and his team work on finding families’ lost pets. “We have successfully helped find more than 1000 pets. Besides cats and dogs, there are other pets like birds, lizards and hamsters.” Liu said.According to Líu, being a pet detective requires a sense of responsibility (责任), kindness and love for animals. Pet detectives also need to be clear-eyed. They also make good use of high-tech products, such as infrared thermal imaging devices (红外热成像仪).“Searching for pets needs lots of technical knowledge, for example when we look for cats, knowing their habits is a must” said Liu “In many big cities cats and dogs have become an important part of many families and are taken good care of. What we look for are not just pets, but family members.”Apart from pet undertakers and detectives, some people work as pet food testers, pet sitters and pet trainers. A study says that the booming (繁荣的) pet market shows people’s improved living standards.30. Lee’s work is mentioned in Paragraph 1 to show that _______________.A.the number of dead pets is big each yearB.people need to take good care of their petsC.cats are pets that are not easy to keepD.many new services for pets are offered now31.According to the passage, a good pet detective should _______________.① know the habits of pets well ① be rich enough to support pets① have a sense of responsibility① be an animal-loverA.①①①B. ①①①C. ①①①D. ①①①32.According to the passage, we can know that _______________.A.keeping pets is popular among the oldB.pet detective is a basic but fashion serviceC.searching for pets is always easy and funD.people’s living standards get higher than before33. What mainly leads to the development of China’s pet market?A. The great need of new pet-related jobs.B. More important roles that pets play in families.C. The animal-lovers who can’t look after pets well.D. The use of high-tech products for looking for pets.CSince the COVID-19 outbreak, many people have been forced to stay at home for long periods of time to protect themselves from the virus. This has given people more free time to learn new skills and find different ways to entertain themselves.Some of these activities include things like singing, learning to cook and ordering fresh food online. All of these things can be done in the palm of your hand with mobile apps.Before, young people would go outside and meet friends at karaoke bars. Now, friends can meet and sing on the mobile karaoke app Changba, which translates to Singing Bar “The Changba app not only gives me the chance to sing whatever I like at home, but also lets me keep in touch with friends by sharing our songs,” said an app user.Some people have also taken up cooking as a new hobby to pass the time. You don’t need to attend culinary (烹饪的) school, thanks to apps like Xiachufang and Ecook, which make it easier for those who want to learn how to cook. These apps provide a platform for users to look up different recipes (菜谱) and to share their own recipes with others. “Using this app, I have learned many new and healthy dishes which help me eat better,” an app user commented on the app’s website.To help people avoid crowded places like grocery stores, apps that deliver goods right to your door have also become very popular. “It is noteworthy that the epidemic has attracted new groups of consumers (消费者), such as elderly people, who originally didn’t belong to our target user group,” Zhang Yi, an analyst (分析师) from the market research company iiMedia Research, said.Indeed, during this special time, these apps have opened up a whole new world of opportunities for people of all ages and backgrounds.34. What did the author tell us about Changba in Paragraph 3?A. It is more user-friendly than karaoke bars.B. It helps users to discover their singing talent.C. It provides opportunities to communicate with friends.D. It allows users to improve their singing skills fast.35. What does the article tell us about food-delivery apps?A. They have lost many young customers.B. They have opened culinary classes for their users.C. They have more elderly users than before.D. They have caused many grocery stores to close.36. Which of the following shows the structure of the article? (①①①①①①=Paragraphs1-6)DIf you are a recent social science graduate who has had to listen to jokes about unemployment (失业) from your computer major (专业课) classmates, you may have had the last laugh. There are many advantages for the social science major because this high-tech “Information Age” requires people who are flexible (灵活的) and who have good communication skills.There are many social science majors in large companies who fill important positions. For example, a number of research studies found that social science majors had achieved greater managerial (管理的) success than those who had technical training or pre-professional courses. Studies show that social science majors are most suitable for change, which is the leading feature (特点) of the kind of high-speed, high-pressure, high-tech world we now live in.Social science majors are not only experiencing success in their long-term company jobs, but they are also finding jobs more easily. A study showed that many companies had filled a large percentage of their entry-level (入门的) positions with social science graduates. The study also showed that the most important quality in a person who was looking for a job was communication skills, noted as “very important”, by 92 percent of the companies. Social science majors have these skills, often without knowing how important they are, It is probably because of these skills that they have been offered different kinds of positions.Finally, although some social science majors may still find it more difficult than their technically trained classmates to land the first job, recent graduates report that they don’t regret their choice of study.37.By saying that “you may have had the last laugh” in the first paragraph, the author means that you may have _______________.A.shared the jokes with computer majorsB.achieved as much as computer majorsC.stopped joking about computer majorsD.found jobs more easily than computer majorspared with graduates of other subjects, social science graduates _______________.A.are ready to change when situations changeB.are better able to deal with difficultiesC.are equally good at computer skillsD.are likely to give others pressure (压力)39. The underlined word “land” in the last paragraph probably means _______________.A. keep for some timeB. successfully getC. start at onceD. lose regretfully40. According to the text, what has made it easy for social science graduates to find jobs?A. Willingness to take low-paid jobs.B. Readiness to get high-tech knowledge.C. Skills in expressing themselves.D. Part-time work experience.第II卷(非选择题) (总分60分)四、A) 根据汉语提示填空。
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注意事项： 1．本试 卷共 6 页．全卷满分 120 分．考试时间为 120 分钟．考生答题全部答在答题卡上，答在本试 卷上无效． 2．请认真核对监考教师在答题卡上所粘贴条形码的姓名、考试证号是否与本人相符合，再将自己的姓 名、准考证号用 0.5 毫米黑色墨水签字笔填写在答题 卡及本试卷上． 3．答选择题必须用 2B 铅笔将答题卡上对应的答案标号涂黑．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写在答题卡上的指定位置，在其他位置答题一律无 效． 4．作图必须用 2B 铅笔作答，并请加黑加粗，描写清楚．
考点：二次函数的性质
分析：看到二次函数 y=ax2+bx+c，首先需要弄清楚 C 的意义以及二次函数的对称轴问题。 那么通过题目，我们发现此函数恒过（0,3）点，并且，想到对称轴，我们就要对此函数 化简为 y=a（x+b）2+c，根据函数的性质，我们知道函数恒过（-b，c） 。此题难度不 大，主要是考察函数的一般形式，与对称轴形式的转化。
考点：随机事件的运用 分析：根据题意，即可得随机事件,注意此题答案不唯一，只要满足条件即可。
13．学习了 “幂的运算”后，课本提出了一个问题； “根据负整数指数幂的意义，你能用同底数幂的乘 法性质（am·an＝am+n，其中 m、n 是整数）推导出同底数幂除法的性质（am÷an＝am－n，其中 m、
2 1 b a
7．
的相反数是
▲
．
（第 8 题）
考点：相反数 分析：求一个数的相反数就是在这个数前面添上“-”号。本题考查了相反数的意义，一个 数的相反数就是在这个数前面添上“-”号；一个正数的相反数是负数，一个负数的相反 数是正数，0 的相反数是 0
8．如图，直线 a∥b，若∠1＝40°，则∠2＝
▲
考点：数轴的性质 分析：根据题意得：先向负方向移动 3 个单位长度表示为（-3） ，向正方向移动 2 个单位长 度表示为（+2） ，然后求和即可求得答案．此题考查了数轴的性质．此题比较简单，注意正 负数的表示方法是解题的关键．
2．已知⊙O1 的半径为 2，⊙O2 的半径为 5，若⊙O1 和⊙O2 有 2 个公共点，则圆心距 O1O2 的长度可以 是 A．3 B．5 C．7 D．9
考点：圆与圆的位置关系． 分析：半径不相等的两个圆有 2 个公共点，则两圆相交，根据两圆相交时圆心距 d 满足的条 件：R-r＜d＜R+r（d 表示圆心距，R，r 分别表示两圆的半径，其中 R＞r）即可求解．本题 考查了圆与圆的位置关系．圆和圆的位置与两圆的圆心距、半径的数量之间的关系（d 表示 圆心距，R，r 分别表示两圆的半径） ： ①两圆外离⇔d＞R+r； ②两圆外切⇔d=R+r； ③两圆相交⇔R-r＜d＜R+r（R≥r） ； ④两圆内切⇔d=R-r（R＞r） ； ⑤两圆内含⇔d＜R-r（R＞r） ．
15．把两个相同的矩形按如图所示的方式叠合起来，若它们的长与宽分别为 48cm 与 36cm，则重叠部 D 分的面积为 ▲
36cm A 48cm B ① C
cm2．
（第 15 题）
（第 16 题）
②
考点：相似三角形的判定与性质；矩形的性质． 分析：先求出 AD 的长，再证明△ADE∽△ABC，得出 S△ADE= 1、16S△ABC，故重叠部分的面积= 15/16S△ABC． 解：在 RT△ABC 中，由勾股定理，得 AC=√AB2+BC2=60cm， AD=AC-CD=60-48=12． 在△ADE 与△ABC 中，∵∠DAE=∠BAC，∠ADE=∠B=90°， ∴△ADE∽△ABC， ∴S△ADE /S△ABC =（AD/AB）2=（12/48）2=1/16， ∴S△ADE=1/16S△ABC， ∴重叠部分的面积=5/16S△ABC=5/16×1/2×48×36=810（cm2）．
A E D
B
F （第 20 题）
C
考点：平行四边形的判定与性质；全等三角形的判定与性质． 分析： （1）首先由平行四边形的性质可得 AB∥CD，AB=CD；∠A=∠C，∠ABC=∠CDA，再由条件∠ABC 的平分线交 AD 于 E，∠CDA 的平分线交 BC 于 F 可得∠ABE＝ ∠ABE=∠CDF，再利用 ASA 定理可判定△ABE≌△CDF； （2）首先根据△ABE≌△CDF 可得 AE=CF，再根据平行四边形的性质可得 AD=CB，AD∥BC，进而得到 DE=BF 且 DE∥BF，根据一组对边平行且相等的四边形是平行四边可证出四边形 BFDE 是平行四边形，再 根据平行四边形对角线互相平分可证出结论． 此题主要考查了全等三角形的判定与性质， 平行四边形的判定与性质， 关键是熟练掌握平行四边形的性质： 平行四边形对应边相等，对应角相等，对角线互相平分． 1 2 ∠ABC，∠CDF＝ 1 2 ∠CDA．进而得到
三、解答题（本大题共 12 小题，共 88 分．请在答题卡指定区域 内作答，解答时应写出文字说明、证明 ．．．．．．． 过程或演算步骤） 5 12
17． （6 分）计算（
+2
3）×
15．
考点：有理数中根式的计算 分析： （ 5 12 5 12 × ＋2 3）× 15
＝
15＋2
3×
15
＝ ＝ 5 2
10．国务院总理温家宝在政府工作报告中指出，我国 2011 年国内生产总值 47.2 万亿元．47.2 万亿元用 科学计数法表示为： ▲ 元．
考点：科学技术法表示方法。 分析：科学计数法表示形式为 a×10n（1≤a<10） ，此题还需注意单位问题，难度不大，考生
容易粗心少数或多数 0
11．写出一个含 x 的分式，使得当 x＝2 时，分式的值是 3．这个分式可以是：
考点：二元一次方程组的应用． 分析：根据等量关系：上有三十五头，下有九十四足，可分别得出方程，联立求解即可得出 答案． 此题考查了二元一次方程的知识， 解答本题的关键是仔细审题， 根据等量关系得出方程组， 难度一般．
20． （7 分）已知：如图，□ABCD 中，∠ABC 的平分线交 AD 于 E，∠CDA 的平分线交 BC 于 F． （1）求证：△ABE≌△CDF； （2）连接 EF、BD，求证：EF 与 BD 互相平分．
反比例函数的解析式， 即可得到关于 a，b 的方程组求得 A 的坐标，从而求得∠AOC 的度数， 进而得到∠AOB 的度数，利用弧长的计算公式即可求解．本题是反比例函数与三角函数、弧 长的计算的综合题，正确求得圆周角的度数是关键．
二、填空题（本大题共 10 小题，每小题 2 分，共 20 分．不需写出解答过程，请把答案直接填写在答题 ．． 卡相应位置 上） ．．．．． 1 2
°．
考点：平行线的性质 分析：由直线 a∥b，∠1=40°，根据两直线平行，同位角相等，即可求得∠2 的度数．
9．分解因式：2x2y－8y＝
▲
．
考点：提公因式法与公式法的综合运用． 分析：先提取公因式 2y，再利用平方差公式，进行二次分解。本题考查了提公因式法，公式 法分解因式，利用平方差公式进行二次分解因式是解本题的难点，也是关键．
4．如图，关于∠α 与∠β 的同一种三角函数值，有三个结论：① tanα ＞tanβ，② sinα ＞sinβ， ③ cosα ＞cosβ ．正确的结论为 A．①② B．②③ C．①③
A B A β α B （第 4 题） D （第 5 题） C O （第 6 题）
D．①②③
y
x
考点：锐角三角函数的增减性 分析：首先根据图形可得：∠α ＞∠β，然后根据各锐角函数的增减性，即可求得答案．此题 考查了锐角函数的增减性与三角形外角的性质．注意当角度在 0°～90°间变化时， ①正弦值随着角度的增大（或减小）而增大（或减小） ； ②余弦值随着角度的增大（或减小）而减小（或增大） ； ③正切值随着角度的增大（或减小）而增大（或减小） ．
6．如图，以 O 为圆心，半径为 2 的圆与反比例函数 y＝ 度为 4 A． π 3 B．π 2 C． π 3 1 D． π 3
3
x
⌒ (x＞0)的图象：反比例函数综合题． 分析：作 AC⊥x 轴，设 A 的坐标是： （a，b） ，在直角△OAC 中，利用勾股定理以及 A 满足
5．如图，△ABC 中，∠ABC＝45°，AC＝10，对折使点 B 与点 A 重合，折痕与 BC 交于点 D，BD：DC=4： 3，则 DC 的长为 A．4 B．6 C．8 D．10
考点：翻折变换（折叠问题） ． 分析：根据翻折变换的性质得出∠B=∠BAD=45°，AD=BD，进而得出∠ADB=90°，即可得 出 AD2+CD2=AC2，再利用 BD：DC=4：3，即可求出 DC 的长．本题考查了折叠的性质， 属于基础题，比较简单，掌握折叠前后对应边相等是解答本题的关键，另外同学们要学会等 线段间的代换，这对以后的解题很有帮助．
▲
．
考点：分式的值 分析：根据题意可得分式，注意此题答案不唯一，只要满足条件即可。此题考查了分式的值 的问题．此题属于开放题，难度不大，注意掌握分式的值与分式的定义．
12．在 1 个不透明的口袋里装了 2 个红球和 3 个白球，每个球除颜色外都相同，将球摇匀．据此，请你 设计一个摸球的随机事件： ▲ ．
一、选择题（本大题共 6 小题，每小题 2 分，共 12 分．在每小题所给出的四个选项中，恰有一项是符合 题目要求的，请将正确选项前的字母代号填涂在答题 卡 相应位置 上） ．． ． ．．．．
1．把笔尖放在数轴的原点处，先向负方向移动 3 个单位长度，再向正方向移动 2 个单位长度，这时笔尖 的位置表示什么数？用算式表示以上过程及结果为 A． （+3）+（+2）=+5 C． （－3）－（+2）=－5 B． （+3）+（－2）=+1 D． （－3）+（+2）=－1