广东省高中阶段学校招生模拟考试(一)
2024届广东省大亚湾区普通高中毕业高三第一次联合模拟考试(一)英语试卷(2)
2024届广东省大亚湾区普通高中毕业高三第一次联合模拟考试（一）英语试卷(2)一、听力选择题1. Why is the man late?A．The trains were delayed.B．He couldn't find a bus stop.C．His friend's motorbike didn't work.2. What happened to the woman last night?A．Her house fell down.B．Her car was badly damaged.C．Her electricity kept shutting off.3. What does the woman want to be?A．A businesswoman.B．A university teacher.C．A university student.4.A．Parker has refused to come to the party.B．Parker probably won’t like the party.C．The party is not held in Parker’s honor.D．The party is open to young people only.5. What does the woman want to do?A．Have a rest.B．Dance with the man.C．Practice dancing for another hour.二、听力选择题6. 听下面一段较长对话，回答以下小题。
1. What is the man satisfied with about his new phone?A．Its size.B．Its camera.C．Its apps.2. What are the speakers mainly talking about?A．The brands of phones.B．Some ways to chat.C．The man’s new phone.7. 听下面一段较长对话，回答以下小题。
2024年广东省普通高中学业水平合格性考试模拟(一)数学试题
一、单选题1. 庄严美丽的国旗和国徽上的五角星是革命和光明的象征．正五角星是一个非常优美的几何图形，且与黄金分割有着密切的联系，在如图所示的正五角星中，以A，B，C，D，E 为顶点的多边形为正五边形，且＝.下列关系中正确的是（）A．B．C．D．2.函数的反函数是（）A．B．C．D．.3.已知函数满足，且（为的导函数），若对于任意的，不等式恒成立，则的取值范围为（）A．B．C．D．4. 设，函数的导函数是奇函数，若曲线的一条切线的斜率是，则切点的横坐标为（）A．B．C．D．5. 某校组织甲、乙两个班的学生到“农耕村”参加社会实践活动，某天安排有酿酒、油坊、陶艺、打铁、纺织、竹编制作共六项活动可供选择，每个班上午、下午各安排一项活动（不重复），且同一时间内每项活动都只允许一个班参加，则活动安排方案的种数为（）A．126B．360C．600D．6306. 在中，、、的对边分别为、、，其中，且，则其最小角的余弦值为A．B．C．D．7. “对任意正整数，不等式都成立”的一个必要不充分条件是（ ）A．B．C．D．8. 若a，b均为实数，则“”是“”的（）A．充分不必要条件B．必要不充分条件C．充分必要条件D．既不充分也不必要条件9. 从800件产品中抽取6件进行质检，利用随机数表法抽取样本时，先将800件产品按001，002，…，800进行编号．如果从随机数表第8行第8列的数开始往右读数（随机数表第7行至第9行的数如下），则抽取的6件产品的编号的75%分位数是（）……2024年广东省普通高中学业水平合格性考试模拟（一）数学试题二、多选题8442175331 5724550688 77047447672176335025 83921206766301637859 1695566711 69105671751286735807 44395238793321123429 7864560782 52420744381551001342 9966027954A ．105B ．556C ．671D ．16910. 已知，则下列结论错误的是（ ）A．是周期函数B．在区间上是增函数C ．的值域为D ．关于对称11. 已知，则的值为（ ）A．B．C．D．12.设等差数列的前项和为，若，则（ ）A ．9B ．15C ．18D ．3613.如图，已知圆锥的轴与母线所成的角为，过的平面与圆锥的轴所成的角为，该平面截这个圆锥所得的截面为椭圆，椭圆的长轴为,短轴为,长半轴长为，短半轴长为，椭圆的中心为,再以为弦且垂直于的圆截面，记该圆与直线交于，与直线交于，则下列说法正确的是（）A ．当时，平面截这个圆锥所得的截面也为椭圆B．C．平面截这个圆锥所得椭圆的离心率D．平面截这个圆锥所得椭圆的离心率14. 若将函数f (x )=cos(2x +)的图象向左平移个单位长度，得到函数g (x )的图象，则下列说法正确的是（ ）A ．g (x )的最小正周期为πB ．g (x )在区间[0，]上单调递减C ．x =是函数g (x )的对称轴D ．g (x )在[﹣，]上的最小值为﹣15. 已知函数f (x )=|sin x |﹣|sin(﹣x )|(π=3.14159……)，则下列说法中正确的是（ ）A ．π是f (x )的周期B ．f (x )的值域为[﹣，]C ．f (x )在(，5π)内单调递减D ．f (x )在[﹣2021，2021]中的零点个数不超过2574个16.是边长为2的等边三角形，已知向量，满足，，则（ ）三、填空题四、填空题五、解答题六、解答题七、解答题A．B．C．D．17. 函数的值域是______．18. 设点，，若直线关于对称的直线与圆相切，则________．19.已知正方体的棱长为1，E ，F ，G 分别是，，的中点.下列命题正确的是___________（写出所有正确命题的编号）.①以正方形的顶点为顶点的三棱锥的四个面最多只有三个面是直角三角形；②P在直线上运动时，；③Q 在直线上运动时，三棱锥的体积不变；④M 是正方体的面内到点D 和距离相等的点，则M 点的轨迹是一条线段.20.已知函数的定义域为R ，且满足，在区间上，的解析式为，则________，________.21. 黎曼猜想由数学家波恩哈德∙黎曼于1859年提出，是至今仍未解决的世界难题．黎曼猜想研究的是无穷级数，我们经常从无穷级数的部分和入手．已知正项数列的前n项和为﹐且满足，则__________，__________．（其中表示不超过x 的最大整数）22. 已知函数.(1)当时，讨论函数的单调性；(2)若不等式在上恒成立，求实数的取值范围.23. （1）求曲线和曲线围成图形的面积；（2）化简求值：．24. 已知函数（其中，，均为常数，，，）．在用五点法作出函数在某一个周期的图像时，列表并填入了部分数据，如表所示：(1)求函数的解析式，并直接写出函数的单调递增区间；(2)已知函数满足，若当函数的定义域为（）时，其值域为，求的最大值与最小值．25. 如图，水平面上摆放了两个棱长为的正四面体和．八、解答题九、解答题(1)求证：；(2)求二面角的余弦值．26. 已知a ，b ，c 为三角形的三边．(1)求证：；(2)若，求证：．27. 某中学为了响应国家双减政策，开展了校园娱乐活动．在一次五子棋比赛活动中，甲、乙两位同学每赛一局，胜者得1分，对方得0分，没有平局．规定当一人比另一人多得5分或进行完10局比赛时，活动结束．假设甲、乙两位同学获胜的概率都为，且两人各局胜负分别相互独立．已知现在已经进行了3局比赛，甲得2分，乙得1分，在此基础上继续比赛．(1)只有当一人比另一人多得5分时，得分高者才能获得比赛奖品，求甲获得比赛奖品的概率；(2)设X 表示该活动结束时所进行的比赛的总轮数，求X 的分布列及数学期望．28. 已知函数（为自然对数的底数），.（Ⅰ）求的极值；（Ⅱ）若，求的最大值.。
2024年广东省普通高中学业水平合格性考试模拟(一)数学试题
一、单选题二、多选题1. 命题“对任意直线l ，有平面与其垂直”的否定是A ．对任意直线l ，没有平面与其垂直B ．对任意直线l ，没有平面与其不垂直C ．存在直线，有平面与其不垂直D ．存在直线，没有平面与其不垂直2. 已知椭圆C：的左、右焦点分别为，，A 是C上一点，，则的最大值为（ ）A ．7B ．8C ．9D ．113. 在(1－x )5－(1－x )6的展开式中，含x 3的项的系数是（ ）A ．－5B ．5C ．－10D ．104.如图，直三棱柱中，，点分别是棱的中点，点在棱上，且，截面内的动点满足，则的最小值是（）A．B．C．D ．25.设集合，则（ ）A．B．C．D．6. 在长方体中，，与平面所成的角为，则该长方体的体积为A．B．C．D．7. 某公司为了促进技术部门之间良好的竞争风气，公司决定进行一次信息化技术比赛，三个技术部门分别为麒麟部，龙吟部，鹰隼部，比赛规则如下：①每场比赛有两个部门参加，并决出胜负；②每场比赛获胜的部门与未参加此场比赛的部门进行下一场的比赛；③在比赛中，若有一个部门首先获胜两场，则本次比赛结束，该部门就获得此次信息化比赛的“优胜部门”．已知在每场比赛中，麒麟部胜龙吟部的概率为，麒麟部胜鹰隼部的概率为，龙吟部胜鹰隼部的概率为．当麒麟部与龙吟部进行首场比赛时，麒麟部获得“优胜部门”的概率是（ ）A．B．C．D．8. 在下列条件中，可判断平面与平行的是（ ）A ．都垂直于平面B ．内存在不共线的三点到的距离相等C ．l ，m 是内两条直线，且D ．l ，m是两条异面直线，且9.已知三棱柱为正三棱柱，且，，是的中点，点是线段上的动点，则下列结论正确的是（ ）A．正三棱柱外接球的表面积为B ．若直线与底面所成角为，则的取值范围为C ．若，则异面直线与所成的角为2024年广东省普通高中学业水平合格性考试模拟（一）数学试题三、填空题四、填空题五、填空题六、解答题七、解答题D ．若过且与垂直的截面与交于点，则三棱锥的体积的最小值为10. 已知函数（，，）的部分图象如图所示，则下列结论正确的是（）A ．，B．C ．点为函数图象的一个对称中心D ．函数在上单调递减11. 已知，且，则（ ）A．B．C．D．12.设函数，若存在实数m ，使得关于x 的方程有4个不相等的实根，且这4个根的平方和存在最小值，则实数a 的取值范围是______.13. 计算:___________.14.在中，已知，若点为的中点，且，则__________．15.展开式中的系数为___；所有项的系数和为____．16.已知数列的前项和为，且，记，则________；若数列满足，则的最小值是________．17. 直线与轴交于点，交圆于，两点，过点作圆的切线，轴上方的切点为，则__________；的面积为__________．18.已知（1）求的值；（2）若是第三象限的角，化简三角式，并求值.19. 如图所示，在四棱锥中，平面平面，，且，设平面与平面的交线为．八、解答题九、解答题十、解答题(1)作出交线（写出作图步骤），并证明平面；(2)记与平面的交点为，点S 在交线上，且，当二面角的余弦值为，求的值．20. 已知函数，是的导函数，且有两个零点.（1）讨论的单调性；（2）若，求证：.21. 随着科技发展的日新月异，人工智能融入了各个行业，促进了社会的快速发展．其中利用人工智能生成的虚拟角色因为拥有更低的人工成本，正逐步取代传统的真人直播带货．某公司使用虚拟角色直播带货销售金额得到逐步提升，以下为该公司自2023年8月使用虚拟角色直播带货后的销售金额情况统计．年月2023年8月2023年9月2023年10月2023年11月2023年12月2024年1月月份编号123456销售金额／万元15.425.435.485.4155.4195.4若与的相关关系拟用线性回归模型表示，回答如下问题：(1)试求变量与的样本相关系数（结果精确到0.01）；(2)试求关于的经验回归方程，并据此预测2024年2月份该公司的销售金额．附：经验回归方程，其中，，样本相关系数；参考数据：，．22. 在中，角，，的对边分别为，，，且．(1)求；(2)若为锐角三角形，求的取值范围．。
2024年广东省普通高中学业水平合格性考试模拟一数学试题
一、单选题1. 新冠肺炎疫情发生以来，医用口罩成为抗疫急需物资．某医用口罩生产厂家生产A 、B 、C 三种不同型号的N 95口罩，A 、B 、C 三种型号的口罩产量之比为．为了提高这三种口罩的质量，用分层抽样的方法抽取一个容量为n 的样本．在样本中B 种口罩数量比A 种口罩数量多40只，比C 种口罩数量多80只，则n =（ ）A ．240B ．280C ．320D ．3602. 设全集，，，则（ ）A．B．C．D．3. 已知各项都为正数的等比数列，满足，若存在两项，，使得，则最小值为（ ）A ．2B．C．D ．14. 六氟化硫，化学式为，在常压下是一种无色、无臭、无毒、不燃的稳定气体，有良好的绝缘性，在电器工业方面具有广泛用途．六氟化硫结构为正八面体结构，如图所示，硫原子位于正八面体的中心，6个氟原子分别位于正八面体的6个顶点，若相邻两个氟原子之间的距离为m ，则该正八面体结构的内切球表面积为（）A．B．C．D．5.数列满足，，现求得的通项公式为，，若表示不超过的最大整数，则的值为（ ）A ．43B ．44C ．45D ．466. 在第十三届女排世界杯赛中，中国女排以不败战绩夺得冠军，女排精神一直激励着全国人民在各行各业为祖国的腾飞而努力拼搏.在女排世界杯赛闭幕后，某收视调查机构对某社区内名居民收看比赛的情况用随机抽样方式进行调查，样本容量为，将数据分组整理后，列表如下：观看场数01234567891011观看人数占调查人数的百分比2%2%4%6%m %12%8%10%12%16%12%10%从表中可以得出正确的结论为（ ）A ．表中的值为B ．估计观看比赛不低于场的人数是人C．估计观看比赛场数的众数为D ．估计观看比赛不高于场的人数是人7. 已知复数满足（其中为虚数单位），则（ ）A ．5B ．1C ．D．8. 函数（且）的图象恒过定点，若对任意正数、都有，则的最小值是（ ）A．B．C．D．9. 在正方形ABCD 中，O 为BD 中点，将平面ABD 沿对角线BD 翻折，使得平面平面BCD ，则直线AB 与CD 所成角的大小为（ ）2024年广东省普通高中学业水平合格性考试模拟一数学试题二、多选题A ．30°B ．45°C ．60°D ．90°10. 已知双曲线的一条渐近线方程是，它的一个焦点坐标为，则双曲线的方程为（ ）A．B．C．D．11. 已知直线和平面满足,则（ ）A．B ．或C．D ．或12. 已知集合，则（ ）A．B．C．D．13. 已知直线l ：过抛物线C ：的焦点F ，且与抛物线C 交于A ，B 两点（点A 在第一象限），则下列结论正确的有（ ）A ．抛物线C的方程为B ．线段AB 的长度为8C ．以AF 为直径的圆和抛物线的准线相切D．14.已知正方体的棱长为，点是的中点，点是侧面内的动点，且满足，下列选项正确的是（）A ．动点轨迹的长度是B ．三角形在正方体内运动形成几何体的体积是C ．直线与所成的角为，则的最小值是D ．存在某个位置，使得直线与平面所成的角为15. 下列说法正确的是（ ）A．函数的单调增区间为B ．函数为奇函数C ．幂函数是减函数D ．图像关于点成中心对称16. 已知中，，，为边上的高，且，沿将折起至的位置，使得，则（ ）三、填空题四、填空题五、解答题六、解答题A ．平面平面B ．三棱锥的体积为8C．D．三棱锥外接球的表面积为17. 函数，若关于x的不等式的解集为，则实数a 的取值范围为__________．18. 已知直线与直线垂直，则实数a 的值为____________．19. 已知的展开式的第7项为常数项，则正整数的值为_________.20. 曲率在数学上是表明曲线在某一点的弯曲程度的数值.对于半径为的圆，定义其曲率.对于一般曲线，我们可通过曲线上某点处的密切圆半径来描述该点的曲率，其中对于曲线在点处的密切圆半径计算公式为.已知函数，椭圆：，则曲线在点处的曲率为____________；上任一点处曲率的最大值为____________.21. 在我国古代，是数字之极，代表尊贵之意，所以中国古代皇家建筑中包含许多与相关的设计.例如，北京天坛丘的地面由扇环形的石板铺成，如图，最高一层的中心是一块天心石，围绕它的第一圈有块石板，从第二圈开始，每一圈比前一圈多块，共圈，则第圈的石板数为___________，前圈的石板总数为___________.22. 求值.（1）；（2）.23. 化简求值：（1）（2）已知，，求的值；24. 把函数的图象向左平移个单位，得到函数的图象，函数的图象关于直线对称，记函数.七、解答题八、解答题九、解答题（1）求函数的最小正周期和单调增区间；（2）画出函数在区间上的大致图象.25. 已知数列{}满足∈N *，为该数列的前n 项和.(1)求证：数列{}为递增数列；(2)求证：.26.小明同学参加综合实践活动，设计了一个封闭的包装盒，包装盒如图所示：底面是边长为8（单位：）的正方形，均为正三角形，且它们所在的平面都与平面垂直．(1)证明：平面；(2)求该包装盒的容积（不计包装盒材料的厚度）．27. 为丰富社区群众的文化生活，某社区利用周末举办羽毛球比赛．经过抽签，甲乙两人进行比赛，比赛实行三局两胜制（若某人胜了两局则为获胜方，比赛结束）．根据以往数据统计，甲乙两人比赛时，甲每局获胜的概率为，每局比赛相互独立．(1)求甲获胜的概率；(2)比赛规则规定：比赛实行积分制，胜一局得3分，负一局得1分；若连胜两局，则还可获得5分的加分．用X 表示甲乙比赛结束后甲获得的积分,求X 的分布列和数学期望．28. 某企业从生产的一批产品中抽取个作为样本，测量这些产品的一项质量指标值，由测量结果制成如图所示的频率分布直方图．(1)求这件产品质量指标值的样本平均数（同一组数据用该区间的中点值作代表）和中位数；(2)已知某用户从该企业购买了件该产品，用表示这件产品中质量指标值位于内的产品件数，用频率代替概率，求的分布列和数学期望．。
广东省肇庆市2025届高中毕业班第一次模拟考试化学试题(含答案)
广东省肇庆市2025届高中毕业班第一次模拟考试化学试题本试题共8页，考试时间75分钟，满分100分2024.11注意事项：1. 答题前，考生先将自己的信息填写清楚、准确，将条形码准确粘贴在条形码粘贴处。
2. 请按照题号顺序在答题卡各题目的答题区域内作答，超出答题区域书写的答案无效。
3. 答题时请按要求用笔，保持卡面清洁，不要折叠，不要弄破、弄皱，不得使用涂改液、修正带、刮纸刀。
考试结束后，请将本试题及答题卡交回。
可能用到的相对原子质量: H1B11C12O16Na23Cl35.5Ni59Zn65As75一、选择题: 本题共16小题, 共44分。
第1~10小题, 每小题2分; 第11~16小题,每小题4分。
在每小题给出的四个选项中，只有一项是符合题目要求的。
1. 下列物质使用的材料属于有机高分子材料的是A. 航空发动机——碳化硅纤维B. 人造骨骼——镍钛合金C. 量子计算机——铁基超导材料D. 滑雪服面料——聚酰胺纤维2. 下列化学用语或图示表达不正确的是A. Na₂O₂的电子式:B. 丙烯的结构简式：CH₃CHCH₂C. 3p:电子云轮廓图:D. 基态Cl⁻的价层电子轨道表示式：3. 粤剧被誉为“南国的红豆”，是世界非物质文化遗产之一。
下列说法不正确的是A. 粤剧头饰“凤冠” 上的金丝银线属于金属材料B. 粤剧演奏乐器“椰胡”的琴筒由椰壳制成，椰壳的主要成分为多糖C. 粤剧彩绘脸谱使用的红色油彩主要成分为Fe₃O₄D. 粤剧头饰“翎子”(也称“野鸡翎”“雉毛翎”)，其主要成分为天然纤维高三·化学第1页 (共8页)4. 化学与科技、生产、生活息息相关。
下列说法不正确的是A. “深海勇士”号潜水艇使用的锂离子电池是二次电池B. 保温杯内胆使用316不锈钢，不锈钢的含碳量大于生铁C. 月球新矿物“嫦娥石”是一种磷酸盐矿物 (含镁和铁)，它不易溶于水D. 甲流病毒主要由RNA和蛋白质组成，可采用高温、紫外线、75%乙醇等有效灭活5. 利用电化学原理既能输出电能，又能合成有机物。
2023届广东省普通高等学校招生全国统一考试模拟测试一英语试题及参考答案
2023年普通高等学校招生全国统一考试模拟测试（一）英语试题本试卷满分120分。
考试用时120分钟。
第二部分阅读（共两节，满分50分）第一节（共15小题；每小题2.5分，满分37.5分）阅读下列短文，从每题所给的A、B、C、D四个选项中选出最佳选项。
ACheck out what’s coming soon，and what’s in development. Playbill will update these listings when new information is made available.THE COLLABORATION at Samuel J. Friedman TheatreThe play tells a true story in New York. Fifty-six-year-old Warhol’s star is falling. Jean is the new wonder-kid taking the art world by storm. When Jean agrees to work together with Warhol on a new exhibition, it soon becomes the talk of the city. The two artists set foot on a shared journey, both artistic and deeply personal, which redraws both their worlds.ALMOST FAMOUS at Bernard B. Jacobs TheatreBased on the 2,000 film of the same name, the musical features Pulitzer winner Kitt and a book by Crowe, who earned an Oscar award for penning the original film based on his own teenage experience. The coming-of-age story tracks a 15-year-old music fan named William who follows the emerging band Stillwater on tour.THE PIANO LESSON at Ethel Barrymnore TheatreThe play is the fourth in Wilson’s Century Cycle, which digs into the Black experience in every decade of the 20th century. Set in Pittsburgh’s Hill District in 1936, it centers around a brother and a sister involved in a battle over a piano carved with the faces of their ancestors.KIMBERLY AKIMBO at Booth TheatreKim is a bright and funny high school girl, who happens to look like a 72-year-old lady. And yet her aging disease may be the least of her problems. Forced to deal with family secrets, and possible crime charges, Kim is determined to explore happiness in a world where not even time is on her side.21. What kind of play is THE COLLABORATION?A. A romance.B. A sci-fi story.C. A fairy tale.D. A life story.22. Which theatre should you visit if you want to learn about racial issues?A. Samuel J. Friedman Theatre.B. Bernard B. Jacobs Theatre.C. Ethel Barrymore Theatre.D. Booth Theatre.23. What can you see in both ALMOST FAMOUS and KIMBERLY AKIMBO?A. Struggles against crime charges.B. Cutting-edge film technology.C. Award-winning performances.D. Teenagers’ growth experiences.BFor some people, October is October. On October 5, Katmai National Park and Preserve in southern Alaska kicked off Fat Bear Week, when fans compare before and after photos of bears to vote for the one they think has gained the most weight before they bed down for the long freezing winter. Bear 747, also known as “Bear Force One”, took the first-place spot with 68,105 votes. It weighed around 1,400 pounds.Fat Bear Week started in 2014. At first, it was just Fat Bear Tuesday. Mike Fitz, a former park keeper at Katmai, noticed that live webcams showing the bears generated a lot of online comments, so he let people vote for their favorite fat bear on the Internet. The one-day event attracted 1,700 votes. In 2015, it was extended to a week. Last year, nearly 800,000 people got involved, turning it into a global campaign.The contest serves two main purposes. First, some of America’s national parks are in remote and undeveloped areas, which are difficult or costly to reach. Fat Bear Week brings Alaska’s wilderness to fans’ computer screens.“It’s not limited to the fortunate few who can go to the river anymore.”Fitz said.Second, Fat Bear Week brings attention to the bears and the ecosystems they live in. The 2,200 bears of Katmai are strong for feeding on some of the healthiest salmon （三文鱼）in the world，said Sara Wolman, a former park keeper. Salmon in the Pacific Northwest are suffering from overfishing, dam construction and warming rivers due to climate change. Luckily, Brooks River, which Katmai’s bears like to fish in, has been free from these threats so far.Fat Bear Week has spread like viruses. Fans and creators say it’s because it shows a conservation success story. For one week, people don’t have to focus on climate change or habitat loss. “Maybe things aren’t super great all the time in the world,”said Felicia Jimenez, who is a current Katmai keeper. “But there are some really fat bears in Alaska.”24. Why was Bear 747 the winner?A. It was the heaviest.B. It was the most famous.C. It gained the most pounds.D. It presented the best photos.25. What is the purpose of paragraph 2?A. To praise Mike Fitz for his work.B. To review the history of the event.C. To show the popularity of fat bears.D. To explain the role of live webcams.26. What can we learn about Katmai’s bears?A. They bring tourists to the park.B. They have a stable food source.C. They remain difficult to be seen.D. They face a population decline.27. Which can be the best title for the text?A. Fat Bear Week: Conservation Is JoyfulB. Brown Bears: Winter Survival Is CrucialC. Bear 747: A Fat Yet Healthy ChampionD. Katmai’s Bears: A Special Beauty ContestCA new study has found that a child's reading speed can be improved by simply increasing the space between letters within a piece of text. The research, led by Dr Steven Stagg of Anglia Ruskin University（ARU）, examined the benefits of letter spacing and coloured overlays（颜色标注）among children.Two groups of children, dyslexic（有阅读障碍的）and non-dyslexic children, were asked to read four texts with either standard or extra-large letter spacing, both with and without a coloured overlay. They were instructed to read the text out loud while being recorded. The recording was used to measure the number of errors they made — specifically missed words, added words, wrong words, and pronunciation —as well as the participants’ reading time. The study discovered that text with increased space between letters provided a benefit for both groups. On average, the dyslexia group showed a 13% increase in reading speed, while the group of non-dyslexic children showed a 5% increase. In addition to improving reading speed, it also resulted in a significant decline in the number of words missed by the children with dyslexia. However, the study found that coloured overlays had no significant impact on reading speed or the reduction of errors for either group.Dr Stagg, Senior Lecturer in Psychology at ARU, said:“We believe extra-large letter spacing works by reducing what is known as the ‘crowding effect’, which can hamper the recognition of letters and reduce reading speed.”“Our findings strongly suggest that teachers can be confident that all children would be helped by increased letter spacing in reading materials.”Dr Stagg added. “While we found that coloured overlays provided little benefit, we suggest children should be encouraged to use overlays if they find these help their reading. Coloured overlays may not increase reading speeds，but they may extend reading stamina(持久力). Previous research suggests their benefits may not become apparent if reading time is less than 10 minutes and the short reading duration of the tests in our study may have put coloured overlays at a disadvantage.28. Which of the following best describes the research process in Paragraph 2?A. Illustrate, group and report.B. Read, instruct and measure.C. Question, assume and prove.D. Categorize, record and count.29. What can we learn about increased letter spacing from Paragraph 3?A. It reduces reading errors.B. It sharpens reading skills.C. It arouses reading interest.D. It enhances reading comprehension.30. What does the underlined word “hamper”in Paragraph 4 probably mean?A. Obtain.B. Prevent.C. Change.D. Involve.31. What can be inferred about coloured overlays from Paragraph 5?A. They might not have been evaluated fairly.B. They might not be recommended by experts.C. They might work better in short reading.D. They might have some disadvantages.DLike any doctor, Jacques Fellay wants to give his patients the best care. Hidden inside our bodies are genetic(基因的) markers that can tell him which patients could probably have diseases such as AIDS and provide early treatments. However, there are worries for Fellay: they contain sensitive details that could lead to embarrassment, discrimination or even worse.A new kind of encryption, fully homomorphic encryption（FHE）(全同态加密), is making it possible for data users to run multiple operations on genetic data without seeing the contents. This can help end big data’s privacy problem, and Fellay's patients can be some of the first to benefit.In 1978, the concept of homomorphic encryption （HE）was firstly proposed. According to it, one could encrypt and share data with others, who could analyze and perform calculations on the data with no idea what it means. After getting the data back, the data user could simply arrive at the result using the secret key and it will make total sense.In 2009, Craig Gentry firstly provided a workable FHE program. Like HE, it rests on a mathematical idea called a homomorphism，which mostly relies on using algebra(代数) to map data from one form to another without changing its underlying structure. However, it supports multiple operations on encrypted data, rather than only one calculation in HE.Later, Gentry went on to work at IBM, which now has complicated FHE tools to run encrypted data. In the medical field, for example, it detected signs of COVID-19 infection by applying FHE to analyze the encrypted CT scans of more than 1, 500 people’s lungs. In a paper published in October 2021, the team used FHE to collect data from multiple sources and predicted the effect of cancer treatments or the process of HIV infection, as in Fellay’s case.假定你是李华，校英语学习俱乐部负责人。
2024届广东省大亚湾区普通高中毕业高三第一次联合模拟考试(一)英语试卷
2024届广东省大亚湾区普通高中毕业高三第一次联合模拟考试（一）英语试卷一、听力选择题1. How much does Mike need for the computer?A．1,000 dollars.B．800 dollars.C．200 dollars.2. What is Dr.Freeth going to do?A．Make an appointment.B．Go travelling.C．Attend a wedding.3.A．Go out to look for her dog.B．Read a book about gardening.C．Find some work to do in the garden.D．Stop taking her mind off the dog.4.A．Plan his budget carefully.B．Buy a gift for his mother.C．Ask someone else for advice.D．Give her more information.5. What will the speakers do next?A．Check the map.B．Leave the restaurant.C．Park the car.二、听力选择题6. 听下面一段较长对话，回答以下小题。
1. What did the woman watch?A．A soap opera.B．A documentary.C．A news program.2. What was the news mainly about?A．President Barack Obama’s schedule.B．The release of CO₂ by power plants.C．A plan to fight against global warming.3. What is the speakers’ attitude toward global warming?A．Indifferent.B．Positive.C．Negative.4. What will the speakers do next?A．Go to have supper.B．Go to have a drink.C．Keep talking at home.7. 听下面一段较长对话，回答以下小题。
广东省韶关市2024届高三上学期第一次模拟考试 数学试题(含解析)
A ．π3
2f ⎛⎫
=- ⎪⎝⎭
B ．将()y f x =的图象向右平移
C ．12,R x x ∀∈，都有(f ()2f x m =⎡
⊥；
(1)证明：A B B C
【详解】
P 在第一象限，由题意将x c =（其中222a b c =+）代入椭圆方程
12．AB
【分析】利用周期函数的定义判断A ；求出导数，利用轴对称的意义判断点判断C ；利用导数探讨单调性并确定极值点判断【详解】依题意，cos(2π)sin((2π)e e x f x ++=-确；
π
由M 在以12F F 为直径的圆上可得：故2
2
2
1212MF MF F F +=，且四边形由双曲线2
2:13
x C y -=可知：2a 即2216m n +=，
则()()()(110,0,3,0,3,0,0,3,3,3,0,0A B B C 所以()()110,3,3,3,3,3A B B C =-=-- ，
所以110A B B C ⋅=
，
所以11A B B C ⊥.
（2）因为点P 在棱1CC 上，1:C P PC 又()()()(110,0,0,0,0,3,0,3,0,3,0,3A A B C
2
⎝
【点睛】方法点睛：求最值或范围问题的基本解法
（1）几何法:根据已知的几何量之间的相互关系、平面几何和解析几何知识加以解决物线上的点到某个定点和焦点的距离之和、光线反射问题等
（2）代数法:建立求解目标关于某个
基本不等式方法、导数方法等。
广东省肇庆市2025届高中毕业班第一次模拟考试一模物理试卷(含答案)
肇庆市2025届高中毕业班第一次模拟考试物理本试题共6页，考试时间75分钟，满分100分2024.11注意事项：1. 答题前，考生先将自己的信息填写清楚、准确，将条形码准确粘贴在条形码粘贴处。
2. 请按照题号顺序在答题卡各题目的答题区域内作答，超出答题区域书写的答案无效。
3. 答题时请按要求用笔，保持卡面清洁，不要折叠，不要弄破、弄皱，不得使用涂改液、修正带、刮纸刀。
考试结束后，请将本试题及答题卡一并交回。
一、单项选择题：本题共7小题，每小题4分，共28分。
在每小题给出的四个选项中只有一项是符合题目要求的。
1. 地铁是一种“绿色”的公共交通工具. 如图，某次地铁上连接左侧圆环的拉绳呈竖直状态，人拉着的右侧圆环的拉绳与竖直方向成一定角度，且处于绷紧状态，人与地铁车厢保持相对静止.下列说法正确的是A. 由左侧拉绳的状态可知地铁车厢处于静止状态B. 由右侧拉绳的状态可知地铁车厢可能正在向右加速C. 人对圆环的拉力小于圆环对人的拉力D. 人受到车厢地面水平向左的摩擦力作用2. 2024年巴黎奥运会上，中国选手获得了自由式小轮车女子公园赛冠军. 如图，比赛中运动员骑行小轮车交替通过水平路面和圆弧坡面，并在空中完成各种高难度动作. 下列说法正确的是A. 运动员在圆弧面上的运动是匀变速运动B. 当运动员骑行速度变大时，其惯性也增大C. 在圆弧面骑行时运动员和车所受支持力可能大于其总重力D. 研究运动员在空中的动作时，可将运动员看作质点3. 港珠澳大桥海底隧道是世界上最长的公路沉管隧道，也是我国第一条外海沉管隧道，全长6.7km，隧道限速90km/h，将沉管隧道简化为直线，t=0时刻，甲、乙两车同时由同一t=0入口进入隧道，之后在隧道中运动的速度随时间变化的关系如图所示. 则经过多长时间乙车在遂道中追上甲车A. 5sB. 10sC. 12sD. 20s高三·物理第1页 (共6页)高考直通车4. 如图甲为一种名为“喊泉”的游乐设施，游客对着池边的话筒大声呼喊时，水池中的喷口就会有水倾斜喷出 (可认为水恰好从水面喷出).如图乙，某次呼喊时，水柱在空中的射高为11.25 m，已知喷口与水面的夹角为45°，重力加速度取10m/s²,不计空气阻力.下列说法正确的是A. 在最高点的水的速率不为零B. 在最高点的水的重力的瞬时功率不为零C. 水在空中运动时先超重后失重D. 水柱的水平射程为22.5m5. 如图甲为小青设计的加速度计，一个质量为0.1kg的滑块穿在一个光滑水平直滑杆上，两根完全相同的弹簧两端分别与挡板和滑块侧面栓接. 初始时，两根弹簧均处于原长，然后将这个装置沿运动方向水平固定在一辆汽车上，当汽车以5m/s²的加速度做匀变速直线运动时，两弹簧的形变量均为1cm，如图乙，弹簧始终在弹性限度内. 下列说法正确的是A. 此时单根弹簧的弹力大小为0.5NB. 弹簧的劲度系数为25N/mC. 此时汽车一定向右做匀加速运动D. 汽车在运动时，两侧弹簧的弹性势能可能不相等6. 物体在引力场中具有的势能称为引力势能，在无其他星体影响的情况下，若取距离地球无穷远,其中G为引力常量，M为地处的引力势能为零，则质量为m的物体的引力势能为E p=−GMmr球的质量，r为物体到地心的距离(r大于等于地球半径R). 如图，虚线圆Ⅰ和Ⅱ为人造地球卫星的两个圆轨道，已知轨道Ⅰ是近地轨道，轨道Ⅱ的半径为轨道Ⅰ半径的2倍. 关于质量为m ₀的人造地球卫星，下列说法正确的是A. 未发射的人造卫星位于赤道上时，其引力势能为零B. 在轨道Ⅰ上的动能是在轨道Ⅱ上的动能的2C. 在轨道Ⅰ上的周期是在轨道Ⅱ上的周期的14D. 从轨道Ⅰ到轨道Ⅱ需要的能量至少为GMm04R高三·物理第2页 (共6页)7. 新能源汽车以其环保、智能等优势受到广大消费者的青睐. 某款新能源汽车在t=0时刻沿平直公路由静止开始以恒定加速度启动，发动机在t₁时刻达到额定功率，然后保持功率不变继续加速，t₂时刻达到最大速度后匀速行驶. 假设汽车所受的阻力大小恒定，则此过程中汽车的加速度a、动量p、牵引力F、功率P随时间t的变化规律正确的是二、多项选择题：本题共3小题，每小题6分，共18分。
2024届广东省大亚湾区普通高中毕业高三第一次联合模拟考试(一)英语模拟试题(含答案)
2024届广东省大亚湾区普通高中毕业高三第一次联合模拟考试英语模拟试题注意事项：1. 答卷前, 考生务必将自己的学校、班级、姓名、考场号、座位号和准考证号填写在答题卡上, 将条形码横贴在答题卡“条形码粘贴处”。
2. 作答选择题时, 选出每小题答案后, 用2B铅笔在答题卡上将对应题目选项的答案信息点涂黑；如需改动, 用橡皮擦干净后, 再选涂其他答案。
答案不能答在试卷上。
3. 非选择题必须用黑色字迹的钢笔或签字笔作答, 答案必须写在答题卡各题目指定区域内相应位置上；如需改动, 先划掉原来的答案, 然后再写上新答案；不准使用铅笔和涂改液。
不按以上要求作答无效。
4. 考生必须保证答题卡的整洁。
考试结束后, 将试卷和答题卡一并交回。
第二部分阅读（共两节, 满分50分）第一节（共15小题；每小题2. 5分, 满分37. 5分）阅读下列短文, 从每题所给的A、B、C、D四个选项中选出最佳选项。
ACritics’ choice: The year’s best novels. . .1 The Love Songs of W. E. B. Du Bois by Honorée Fanonne Jeffers （Harper, $29）The novel from a celebrated poet, this 800-page work is many things at once: a moving coming- of-age story, an examination of race, and an exploration of American history. Out of curiosity about the secrets in her family, Ailey uncovers stories that trace back to her Native American, Scottish, and enslaved African-American ancestors.2 Matrix by Lauren Groff （Knopf, $28）If Matrix were written by anyone else, it would be a hard sell. But Lauren Groff has reimagined the life of 12th-century poet Marie de France so that we can’t help but stay with this royal castoff as she’s sent away, at 17, to run a poor church, which she eventually changes into a shelter for women and a challenge to the male-dominated world.3 Klara and the Sun by Kazuo Ishiguro （Knopf, $28）A robot that has been programmed to be a sickly teenager’s ＂Artificial Friend, ＂Klara serves as our eyes on a troubling near future in which technological progress is pushing moral boundaries. But her outsider view allows the author of Remains of the Day, now a Nobel Prize winner, to explore what makes people irreplaceable.4 Cloud Cuckoo Land by Anthony Doerr （Scribner, $30）Doerr's first novel since 2014 covers several hundred years-from 1453 Constantinople to a 22nd-century ceship escaping from Earth. The book runs 640 pages, but without a wasted word. As engaging as a great bedtime story and as readable as Harry Potter, it's above all a celebration of the human spirit and its power to dream impossible dreams.21. Which book is written by a poet?A. The Love Songs of W. E.B. Du Bois. B. Matrix.C. Klara and the Sun.D. Cloud Cuckoo Land.22. What is the central theme explored in Klara and the Sun?A. Historical events.B. Pursuit of dreams.C. Human irreplaceability.D. Teenagers' friendship.23. What do the four novels have in common?A. They all cover hundreds of years.B. They are all chosen by critics.C. They are all the author's first novel.D. The main characters are all humans.BThe world of underwater slumber is unlike what we see on the surface. For mammals （哺乳动物）, a good night's sleep sends an animal into a state of unconsciousness （无知觉） during hours of restorative sleeping that helps heal the body and solidify memories firmly in the mind. For fish, however, sleep isn't such a well-defined process. Instead, many fish species have taken to spending short periods in a state of reduced activity or rest while keeping their eyes open to scan the water for potential threats. During these periods of reduced activity, a fish's bodily functions slow down and its watchfulness drops.The breadth of animals in the ocean that exhibit this type of reduced activity is widereaching. Among the hundreds of thousands of creatures in the sea, some merely ride along with the current,such as jellyfish. Others find rocky bedrooms for the night, while those such as stingrays bury themselves in the sand to rest. However, it must be really hard to get a good night's sleep when you breathe air but live in water. This is something that the world's underwater mammals have to deal with.To prevent themselves from drowning （溺死） in their sleep, underwater mammals such as dolphins and manatees have developed the ability to "turn off' one side of their brain to rest, known as unihemispheric sleep. While one side is fast asleep, the other keeps all the animal's vital functions running, such as breathing. Manatees, for example, love to sleep, spending up to 12 hours a day in a deep slumber. However, as air-breathing animals, manatees need to surface every 20 minutes or so for oxygen. To achieve this, the half of the brain that's still awake sends the sleeping manatee to the surface to get some air.24. What does the underlined word "slumber" in paragraph 1 mean?A. Rest.B. Sleep.C. Danger.D. Activity.25. Why is it hard to define sleep for fish?A. Because fish remain partially awake.B. Because fish need short resting times.C. Because fish possess no consciousness.D. Because fish pretend not to be sleeping.26. What is a challenge faced by underwater mammals?A. Hiding in sand.B. Riding the current.C. Finding spots to rest.D. Breathing while sleeping.27. How do manatees ensure their oxygen supply during sleep?A. By sleeping less.B. By adapting to the sea.C. By surfacing periodically.D. By controlling their breath.CProfessor Lum seems to be under the spell of ChatGPT, but let's not get carried away with the AI chatbot craze. While it does have some benefits, his arguments are like a leaky boat in a stormy sea- full of holes.According to Professor Lum, ChatGPT offers students a wealth of information. Sure, it's like having a walking encyclopedia （百科全书）, but learning is not about copying an answer. It's aboutthe journey of discovery. ChatGPT, armed with its vast database, denies students the opportunity to explore and learn through their own efforts. It's like awarding a participation medal for a race that hasn't even begun!He also claims that ChatGPT improves communication skills. Seriously? Chatting with an AI is not the same as having a real conversation. How can we expect students to handle the complexities of the real world if all they know is how to talk to a chatbot? Let's not turn them into socially awkward bots themselves.Oh, and let's not forget about critical thinking. Professor Lum thinks ChatGPT challenges students to think critically. But come on, folks! It's merely a machine! It's not going to teach them how to question assumptions, analyze evidence, or come up with innovative solutions. We need human interaction and lively debates for that.Lastly, Professor Lum believes ChatGPT can provide support and guidance. Again, seriously? Are we going to replace human emotions with lines of code （代码）？ ChatGPT might offer some comforting words, but it can't replace the warmth and understanding that can only be found in human interaction. Let us not abandon our students, leaving them to feel more alone and disconnected than ever before.Professor Lum's opinion regarding ChatGPT warns us against too much reliance on technology. Rather than prioritizing ChatGPT, the focus should be on developing critical thinking, promoting genuine human interaction, and emphasizing the core aspects of education.28. What is compared to a participation medal in paragraph 2?A. A walking encyclopedia.B. A journey to explore.C. A vast database.D. An answer.29. What can be inferred about ChatGPT from paragraphs 3～5?A. It is not good at socializing.B. It is a skilled critical thinker.C. It offers nothing helpful to humans.D. It forms a warm bond with humans.30. What is the author's attitude towards ChatGPT?A. Totally negative.B. Fully supportive.C. Highly skeptical.D. Quite indifferent.31. What is the main purpose of the passage?A. To criticize a viewpoint.B. To question an authority.C. To clarify two arguments.D. To compare two chatbots.DIf there was an app on your phone that claimed to improve your memory, would you be willing to try it? It's hard to resist the idea of having a better memory. After all, our memories are delicate and can be affected by various factors such as diseases, injuries, mental health conditions, and aging.In response to this perceived need, the brain training industry has grown rapidly, offering a wide range of accessible and affordable mental exercises through smartphone and tablet apps. Many researchers have found evidence suggesting that well-designed exercises can improve mental abilities and even lead to better scores on IQ tests. They compare the brain to muscles, suggesting that targeted repetition of memory exercises can strengthen and condition our memory processes, similar to how athletes engage in strength and conditioning by repeatedly exercising specific muscle groups.In contrast to everyday activities like reading a detailed article or solving a complex puzzle, memory training apps are specifically designed to provide a more challenging workout for the brain. These apps typically involve tasks that require. tracking a large number of objects while being distracted by a secondary task, such as mental calculations or exploring a game's landscape. If this brain training proves effective, it could be beneficial for individuals with brain-related disabilities or those recovering from conditions like cancer.Critics, though, argue that while the concept is appealing, the overall evidence fails to demonstrate significant improvement in core brain processes. Additionally, despite the claims made by many apps and brain training companies, scientists have yet to identify the key factors that make an intervention truly effective or determine the best approaches to address the diverse needs of individuals seeking help.While the question is still open to debate, there is evidence suggesting that short-term working memory training can benefit high-functioning individuals, such as college students. However, it's important to note that brain training, like diet and exercise, is unlikely to have identical effects on every individual.32. In what way is the brain similar to muscles according to many researchers?A. In testing results.B. In work processes.C. In bodily functions.D. In training methods.33. What is the main feature of the typical tasks in memory training apps?A. Detailed.B. Effective.C. Demanding.D. Complicated.34. What do critics think memory training apps lack?A. Legal claims.B. Proven effects.C. Tailored services.D. Appealing concepts.35. What can be a suitable title for the text?A. Can our memory be improved?B. Does brain training actually work?C. What have scientists found about memory?D. What does the brain training industry bring?第二节（共5小题；每小题2. 5分, 满分12. 5分）阅读下面短文, 从短文后的选项中选出可以填入空白处的最佳选项。
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高中阶段学校招生模拟考试
数学试卷（一）
一、选择题（本大题共5小题，每小题3分，共15分）
在每小题给出的四个选项中，只有一个是正确的，请将所选选项的字母写在题目后面的括号内．
1．“一方有难，八方支援”，这是中华民族的传统美德．在“5·12”汶川大地震中，全国共接收 国内外社会各界捐赠款物（截至2008年8月25日12时）总计592．98亿元．这个数字用科学记数法表示并保留三个有效数字，正确的是（ ）
A ．5.93×103亿元
B ．5.93×102亿元
C ．5.9298×102亿元
D ．523×102亿元 2．下列等式成立的是 A ．33=-- B ．551
-=- C ．434322+=+ D ．6)6(2=-
3．若
2=+a
b
b a ，则a 与b 的关系是（ ） A ．相等
B ．互为相反数
C ．互为倒数
D ．不能确定
4．下列图案中，既是轴对称图形，又是中心对称图形的是（ ）
A
B
C
D
5．我市某一周的最高气温统计如下表：
则这组数据的中位数是（ ）
A ．27
B ．28
C ．29
D ．30
二、填空题（本大题共5小题，每小题4分，共20分） 请把下列各题的正确答案填写在横线上． 6．因式分解：=-1823
a _________
7．如图，P 是一个反比例函数的图象在第一象限内的一点，这个反比例函数的解析式是：_______________.
8．若等腰三角形的一个外角是800，则它的顶角是________．
9．如图，PA 切⊙O 于A ，PO 交⊙O 于B ，若∠PAB=350，则∠P___________．
10．如图是一个正方体，它的棱长是1，一只蚂蚁从点A 处出发，沿正方体的表面到点B 处觅食，它要走的最短路程是___________．
三、解答题（本大题共5小题，每小题6分，共30分）
11．计算：0
1060sin 12)3(2)14.3(--÷+--π
12．先将代数式化简，再选择一个你喜欢的实数代人求值：
)1
1
1(12-+-a a a 13．把二次函数762
+-=x x y 化成k h x a y +-=2
)(的形式，并指出它的图象的开口方
向和顶点坐标．
14．有A 、B 、C 三个乡镇，它们的位置如图所示．已测得AB=AC ，BC=24km ，∠BAC=1200．电信局要建一个电视发射塔，使电视信号传送到三个乡镇的距离相等．
（1）请用尺规作图确定电视塔的位置．
（2）计算电视发射塔到各个乡镇的距离．（精确到0.1km ，参考数据：3≈1.73） 15．小华和小娟两位小朋友玩扑克牌游戏，在一副扑克牌中取出如图所示的四张牌，然后将四张牌背朝上，各人随机抽出一张（先抽出的牌不放回）．抽出的两张牌的数字之和若大于7，则小华赢；若小于7，则小娟赢；若等于7，则两人打和．请用列表法或画树状图法求出
各人赢的概率，再说明游戏是否公平．
四、解答题（本大题共4小题，每小题7分，共28分）
16．某校为了进一步了解九年级250名男生的身体素质状况，在九年级随机抽取50名男生送行100米跑测试，以测试数据为样本，绘制出如下的频数分布表和频数分布直方图（均未完成）：
请根据图表数据解答下列问题：
（1）表中数据a=____________．
（2）请把频数分布直方图补充完整．
（3）若九年级男生l00米跑的要求是：时间小于16.3秒为合格；小于14.3秒为优秀．根据以上信息，可估计该校九年级男生100米跑的合格率为________％，全年级大约有______名男生达到优秀．
17．学校为了解决学生上学、放学时出现的交通问题，决定改造学校门前道路，计划用若干天完成．如果单独由甲工程队做，要延期2天完成；如果单独由乙工程队做，要延期6天完
成．现先由甲工程队做3天后，乙工程队再加入做，刚好如期完成．此道路改造工程计划用几天完成?
18．某公司为了激励员工多销售产品，对员工的月分配采取如下方案：每个员工一个月内要完成一定量的销售任务，在完成任务的基础上，每多销售一件产品就给予适当的奖励．已知员工的月工资y（元）与销售产品量x（44）之间的函数关系如图所示．
（1）求出当x≥100时，y与x之间的函数关系式（要注明x的取值范围）．
（2）某员工某月的薪金是2500元，该员工当月的销售量是多少?
19．在一座高为l2米的建筑物CD的顶部C测得对旗杆顶部A的仰角口为300，对旗杆底
（精确到0．1米，供选用数据：2≈1．414，3≈1．732）部B的俯角p为450，求旗杆的高AB．
五、解答题（本大题共3小题，每小题9分，共27分）
20．如图是钢材公司用来放置圆形钢管的支架的横截面．解答下列问题：
（1）找出层数（从最底层数起）与该层的钢管数之间的关系，填表：
（2）把表中的钢管数首尾相加[即第1层与第n层相加，第2层与第（n-l）层相加，第3层与第（n-2）层相加，……]，想想这样的和是多少，共有多少个．若放满第竹层时，则一共可放置钢管________条．
（3）若要放置60条钢管，则要放到第几层?
21．如图，P是⊙O的直径AB延长线上一点，C是⊙O上一点，过点A的直线AD上PC 于D，AC平分∠BAD．
（1）求证：PC是⊙O的切线．
（2）若PB=6，AD=9，求⊙O的半径长及∠P的度数．
22．如图所示，在平面直角坐标系xoy中，四边形OABC是等腰梯形，BC∥OA，OA=7，AB=4，∠COA=600，且点P为x轴上的一个动点，点P不与点O、点A重合．连结CP，过点P作PD交AB于点D．
（1）求点B的坐标．
（2）当点P运动到什么位置时，△OCP为等腰三角形，求这时点P的坐标．
（3）若AD=2.5，当点P运动到什么位置时，使得△OCP∽△APD，求这时点P的坐标．。