2019年北京市高考数学试卷（理科）
适用年级：高三试卷类型：高考真题试题总数：20浏览次数：7860上传日期：2019/6/10
1 . 已知复数
z=2+i，则
A． | B． | C．3 | D．5 |
难度系数：0.85使用：2072次题型：单选题更新：2019/6/10
2 . 执行如图所示的程序框图，输出的
s值为
难度系数：0.85使用：1785次题型：单选题更新：2019/11/19
3 . 已知直线
l的参数方程为
（
t为参数），则点（1,0）到直线
l的距离是
难度系数：0.85使用：1387次题型：单选题更新：2019/6/10
4 . 已知椭圆
（
a＞
b＞0）的离心率为
，则
| A．a2=2b2 | B．3a2=4b2 | C．a=2b | D．3a=4b |
难度系数：0.85使用：2113次题型：单选题更新：2019/6/10
5 . 若
x，
y满足
，且
y≥−1，则3
x+y的最大值为
难度系数：0.85使用：1157次题型：单选题更新：2019/6/10
6 . 在天文学中，天体的明暗程度可以用星等或亮度来描述.两颗星的星等与亮度满足
，其中星等为
mk的星的亮度为
Ek（
k=1,2）.已知太阳的星等是–26.7，天狼星的星等是–1.45，则太阳与天狼星的亮度的比值为
| A．1010.1 | B．10.1 | C．lg10.1 | D．10–10.1 |
难度系数：0.65使用：2085次题型：单选题更新：2019/6/10
7 . 设点
A，
B，
C不共线，则“
与
的夹角为锐角”是“
”的
| A．充分而不必要条件 | B．必要而不充分条件 |
| C．充分必要条件 | D．既不充分也不必要条件 |
难度系数：0.85使用：1577次题型：单选题更新：2019/6/10
8 . 数学中有许多形状优美、寓意美好的曲线，曲线
C：
就是其中之一（如图）.给出下列三个结论：
①曲线
C恰好经过6个整点（即横、纵坐标均为整数的点）；
②曲线
C上任意一点到原点的距离都不超过
；
③曲线
C所围成的“心形”区域的面积小于3.
其中，所有正确结论的序号是
难度系数：0.4使用：1437次题型：单选题更新：2019/6/10
9 . 函数f(x)=sin22x的最小正周期是__________．
难度系数：0.85使用：1950次题型：填空题更新：2019/6/10
10 . 设等差数列{an}的前n项和为Sn，若a2=−3，S5=−10，则a5=__________，Sn的最小值为__________．
难度系数：0.85使用：2426次题型：填空题更新：2019/6/10
11 . 某几何体是由一个正方体去掉一个四棱柱所得，其三视图如图所示．如果网格纸上小正方形的边长为1，那么该几何体的体积为__________．
难度系数：0.65使用：1834次题型：填空题更新：2019/6/10
12 . 已知
l，
m是平面
外的两条不同直线．给出下列三个论断：
①
l⊥
m；②
m∥
；③
l⊥
．
以其中的两个论断作为条件，余下的一个论断作为结论，写出一个正确的命题：__________．
难度系数：0.65使用：2120次题型：填空题更新：2019/11/4
13 . 设函数f（x）=ex+ae−x（a为常数）．若f（x）为奇函数，则a=________；若f（x）是R上的增函数，则a的取值范围是___________．
难度系数：0.65使用：1999次题型：填空题更新：2019/6/10
14 . 李明自主创业，在网上经营一家水果店，销售的水果中有草莓、京白梨、西瓜、桃，价格依次为60元/盒、65元/盒、80元/盒、90元/盒．为增加销量，李明对这四种水果进行促销：一次购买水果的总价达到120元，顾客就少付x元．每笔订单顾客网上支付成功后，李明会得到支付款的80%．
①当x=10时，顾客一次购买草莓和西瓜各1盒，需要支付__________元；
②在促销活动中，为保证李明每笔订单得到的金额均不低于促销前总价的七折，则x的最大值为__________．
难度系数：0.65使用：1759次题型：填空题更新：2019/11/26
15 . 在△
ABC中，
a=3，
b−
c=2，cos
B=
．
（Ⅰ）求
b，
c的值；
（Ⅱ）求sin（
B–
C）的值．
难度系数：0.65使用：2529次题型：解答题更新：2019/6/10
16 . 如图，在四棱锥
P–
ABCD中，
PA⊥平面
ABCD，
AD⊥
CD，
AD∥
BC，
PA=
AD=
CD=2，
BC=3．
E为
PD的中点，点
F在
PC上，且
．
（Ⅰ）求证：
CD⊥平面
PAD；
（Ⅱ）求二面角
F–AE–P的余弦值；
（Ⅲ）设点
G在
PB上，且
．判断直线
AG是否在平面
AEF内，说明理由．
难度系数：0.65使用：2868次题型：解答题更新：2019/6/10
17 . 改革开放以来，人们的支付方式发生了巨大转变．近年来，移动支付已成为主要支付方式之一．为了解某校学生上个月A，B两种移动支付方式的使用情况，从全校学生中随机抽取了100人，发现样本中A，B两种支付方式都不使用的有5人，样本中仅使用A和仅使用B的学生的支付金额分布情况如下：
 交付金额（元）
支付方式
| （0,1000]
| （1000,2000]
| 大于2000
|
仅使用A
| 18人
| 9人
| 3人
|
仅使用B
| 10人
| 14人
| 1人
|
（Ⅰ）从全校学生中随机抽取1人，估计该学生上个月A，B两种支付方式都使用的概率；
（Ⅱ）从样本仅使用A和仅使用B的学生中各随机抽取1人，以
X表示这2人中上个月支付金额大于1000元的人数，求
X的分布列和数学期望；
（Ⅲ）已知上个月样本学生的支付方式在本月没有变化．现从样本仅使用A的学生中，随机抽查3人，发现他们本月的支付金额都大于2000元．根据抽查结果，能否认为样本仅使用A的学生中本月支付金额大于2000元的人数有变化？说明理由．
难度系数：0.65使用：1247次题型：解答题更新：2019/6/10
18 . 已知抛物线C：x2=−2py经过点（2，−1）．
（Ⅰ）求抛物线C的方程及其准线方程；
（Ⅱ）设O为原点，过抛物线C的焦点作斜率不为0的直线l交抛物线C于两点M，N，直线y=−1分别交直线OM，ON于点A和点B.求证：以AB为直径的圆经过y轴上的两个定点．
难度系数：0.65使用：1792次题型：解答题更新：2019/6/10
19 . 已知函数
.
（Ⅰ）求曲线
的斜率为1的切线方程；
（Ⅱ）当
时，求证：
；
（Ⅲ）设
，记
在区间
上的最大值为
M（
a），当
M（
a）最小时，求
a的值．
难度系数：0.65使用：2405次题型：解答题更新：2019/11/4
20 . 已知数列
，从中选取第
项、第
项、…、第
项
，若
，则称新数列
为
的长度为
的递增子列．规定：数列
的任意一项都是
的长度为1的递增子列．
（Ⅰ）写出数列1，8，3，7，5，6，9的一个长度为4的递增子列；
（Ⅱ）已知数列
的长度为
的递增子列的末项的最小值为
，长度为
的递增子列的末项的最小值为
.若
，求证：
；
（Ⅲ）设无穷数列
的各项均为正整数，且任意两项均不相等.若
的长度为
的递增子列末项的最小值为
，且长度为
末项为
的递增子列恰有
个
，求数列
的通项公式．
难度系数：0.4使用：882次题型：解答题更新：2019/6/10
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id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27933\" type=\"1\">平面解析几何</category><category id=\"29043\" type=\"1\">直线与方程</category><category id=\"29048\" type=\"1\">直线的交点坐标与距离公式</category><category id=\"29087\" type=\"1\">点到直线的距离公式</category><category id=\"29102\" type=\"1\">求点到直线的距离</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27939\" type=\"1\">坐标系与参数方程</category><category id=\"29835\" type=\"1\">直线的参数方程</category><category id=\"29871\" type=\"1\">直线的参数方程</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79714\" type=\"0\">选修4-4</category><category id=\"79716\" type=\"0\">第二章 参数方程</category><category id=\"79723\" type=\"0\">§2 直线和圆锥曲线的参数方程</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80452\" type=\"0\">选修4-4</category><category id=\"80454\" type=\"0\">第二讲 参数方程</category><category id=\"80471\" type=\"0\">三 直线的参数方程</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80935\" type=\"0\">选修4-4</category><category id=\"80937\" type=\"0\">第二章 参数方程</category><category id=\"80954\" type=\"0\">2.2 直线和圆的参数方程</category><category id=\"80960\" type=\"0\">2.2.1 直线的参数方程</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81445\" type=\"0\">选修4-4</category><category id=\"81449\" type=\"0\">4.4 参数方程</category><category id=\"81459\" type=\"0\">4.4.2 参数方程与普通方程的互化</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"82093\" type=\"0\">选修4-4</category><category id=\"82095\" type=\"0\">第2章 参数方程</category><category id=\"82105\" type=\"0\">2.2 直线的参数方程</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:35","QuesBody":"<div>【题文】已知直线<i>l</i>的参数方程为<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/b056316b-2fa3-4d34-b323-c3a46a68169d/paper.files/image001.png\" alt=\"eqId26c073a3703e44fc8a209e91db8b81d0\">（<i>t</i>为参数），则点（1,0）到直线<i>l</i>的距离是<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tbody><tr><td width=\"25%\">A．<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/b056316b-2fa3-4d34-b323-c3a46a68169d/paper.files/image002.png\" alt=\"eqIde1490df2d2c84688942d45fd01c90a85\"></td><td width=\"25%\">B．<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/b056316b-2fa3-4d34-b323-c3a46a68169d/paper.files/image003.png\" alt=\"eqIdaf22e0e393474044907f7074dad72e76\"></td><td width=\"25%\">C．<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/b056316b-2fa3-4d34-b323-c3a46a68169d/paper.files/image004.png\" alt=\"eqId170361f2c13a48679439924a405d8282\"></td><td width=\"25%\">D．<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/b056316b-2fa3-4d34-b323-c3a46a68169d/paper.files/image005.png\" alt=\"eqIdaad9151f16a6437bad2f932b5230dc49\"></td></tr></tbody></table></div>","QuesAnswer":"","QuesParse":"","UseSum":1387,"AvgScore":5.0,"ParseAnswerKey":"8452fa62b285e0e20f733f25eff5c441","QuesAttribute":3,"QuesAttributes":{"ID":3,"Name":"跨章节综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196633,"Guid":"ed38b1b6-0e33-44eb-8fca-ae855ca6d2fe","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2701,"Name":"单选题","IsSelectType":true,"OrderIndex":2,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":2,"DifValue":0.85,"Name":"较易"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27933\" type=\"1\">平面解析几何</category><category id=\"29045\" type=\"1\">圆锥曲线</category><category id=\"29171\" type=\"1\">椭圆</category><category id=\"29194\" type=\"1\">椭圆的离心率</category><category id=\"29217\" type=\"1\">求椭圆的离心率或离心率的取值范围</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79458\" type=\"0\">选修1-1</category><category id=\"79460\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"79479\" type=\"0\">§1 椭圆</category><category id=\"79484\" type=\"0\">1.2 椭圆的简单性质</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79540\" type=\"0\">选修2-1</category><category id=\"79543\" type=\"0\">第三章 圆锥曲线与方程</category><category id=\"79573\" type=\"0\">§1 椭圆</category><category id=\"79579\" type=\"0\">1.2 椭圆的简单性质</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80101\" type=\"0\">选修1-1</category><category id=\"80103\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80122\" type=\"0\">2.1 椭圆</category><category id=\"80127\" type=\"0\">2.1.2 椭圆的简单几何性质</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80171\" type=\"0\">选修2-1</category><category id=\"80173\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80193\" type=\"0\">2.2 椭圆</category><category id=\"80200\" type=\"0\">2.2.2 椭圆的简单几何性质</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80699\" type=\"0\">选修1-1</category><category id=\"80701\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80714\" type=\"0\">2.1 椭圆</category><category id=\"80719\" type=\"0\">2.1.2 椭圆的几何性质</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80763\" type=\"0\">选修2-1</category><category id=\"80765\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80779\" type=\"0\">2.2 椭圆</category><category id=\"80787\" type=\"0\">2.2.2 椭圆的几何性质</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81167\" type=\"0\">选修1-1</category><category id=\"81169\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81181\" type=\"0\">2.2 椭圆</category><category id=\"81187\" type=\"0\">2.2.2 椭圆的几何性质</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81228\" type=\"0\">选修2-1</category><category id=\"81230\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81242\" type=\"0\">2.2 椭圆</category><category id=\"81249\" type=\"0\">2.2.2 椭圆的几何性质</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81878\" type=\"0\">选修1-1</category><category id=\"81880\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81891\" type=\"0\">2.1 椭圆</category><category id=\"81897\" type=\"0\">2.1.2 椭圆的简单几何性质</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81945\" type=\"0\">选修2-1</category><category id=\"81947\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81958\" type=\"0\">2.1 椭圆</category><category id=\"81965\" type=\"0\">2.1.2 椭圆的简单几何性质</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:37","QuesBody":"<div>【题文】已知椭圆<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/79531fc3-28ae-42cd-b09a-a2513c8dcda3/paper.files/image001.png\" alt=\"eqIdc9a60e7d8af84065a800dc67cb2e2e1b\">（<i>a</i>＞<i>b</i>＞0）的离心率为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/79531fc3-28ae-42cd-b09a-a2513c8dcda3/paper.files/image002.png\" alt=\"eqIddb6b28cbd8a14caa9cf1146f6f7916c0\">，则<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tbody><tr><td width=\"25%\">A．<i>a</i><sup>2</sup>=2<i>b</i><sup>2</sup></td><td width=\"25%\">B．3<i>a</i><sup>2</sup>=4<i>b</i><sup>2</sup></td><td width=\"25%\">C．<i>a</i>=2<i>b</i></td><td width=\"25%\">D．3<i>a</i>=4<i>b</i></td></tr></tbody></table></div>","QuesAnswer":"","QuesParse":"","UseSum":2113,"AvgScore":5.0,"ParseAnswerKey":"c4a5a0c94be3cbefeddf66755d9dacaf","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·吉林长春市实验中学高二月考（文）","PaperSource":[{"Id":580554,"PaperTitle":"吉林省长春市实验中学2019-2020学年高二上学期10月月考数学（文）试题","LearnGrade":11,"PaperType":3,"Province":220000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-10-22T11:30:53","SchoolId":14187,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196634,"Guid":"564c1bea-8489-4281-b7fe-7a05caf18da2","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2701,"Name":"单选题","IsSelectType":true,"OrderIndex":2,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":2,"DifValue":0.85,"Name":"较易"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27931\" type=\"1\">不等式</category><category id=\"28743\" type=\"1\">线性规划</category><category id=\"28771\" type=\"1\">简单的线性规划问题</category><category id=\"28784\" type=\"1\">根据线性规划求最值或范围</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79422\" type=\"0\">必修5</category><category id=\"79425\" type=\"0\">第三章 不等式</category><category id=\"79448\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80669\" type=\"0\">必修5</category><category id=\"80672\" type=\"0\">第三章 不等式</category><category id=\"80694\" type=\"0\">本章复习与测试</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81818\" type=\"0\">必修第四册</category><category id=\"81821\" type=\"0\">第10章 不等式</category><category id=\"81835\" type=\"0\">10.4 简单线性规划</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:38","QuesBody":"<div>【题文】若<i>x</i>，<i>y</i>满足<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/7fc603c7-eb82-43bc-be4a-8d99e7203beb/paper.files/image001.png\" alt=\"eqIdd9971073b9064844835f12741c8d2cf6\">，且<i>y</i>≥−1，则3<i>x+y</i>的最大值为<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tbody><tr><td width=\"25%\">A．−7</td><td width=\"25%\">B．1</td><td width=\"25%\">C．5</td><td width=\"25%\">D．7</td></tr></tbody></table></div>","QuesAnswer":"","QuesParse":"","UseSum":1157,"AvgScore":5.0,"ParseAnswerKey":"3e93b88594ee0659a3935dedaa787e33","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196618,"Guid":"9ffd8c30-f6e7-47ee-b3a2-0e088550a96c","Title":"2019年北京市高考数学试卷（文科）","From":"","Knowledge":"","QuesType":{"ID":2701,"Name":"单选题","IsSelectType":true,"OrderIndex":2,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27927\" type=\"1\">函数与导数</category><category id=\"28042\" type=\"1\">指对幂函数</category><category id=\"28124\" type=\"1\">对数函数</category><category id=\"28169\" type=\"1\">对数函数的应用</category><category id=\"28198\" type=\"1\">利用对数函数的性质综合解题</category></item><item><category id=\"79775\" type=\"0\">沪教版（上海）</category><category id=\"79804\" type=\"0\">高一第二学期</category><category id=\"79805\" type=\"0\">第四章 幂函数、指函数和对数函数（下）</category><category id=\"79811\" type=\"0\">4.6 对数函数的图像与性质</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"79904\" type=\"0\">必修1</category><category id=\"79906\" type=\"0\">第二章 基本初等函数（1）</category><category id=\"79921\" type=\"0\">2.2 对数函数</category><category id=\"79927\" type=\"0\">2.2.2 对数函数及其性质</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80513\" type=\"0\">必修1</category><category id=\"80516\" type=\"0\">第三章 基本初等函数(Ⅰ)</category><category id=\"80541\" type=\"0\">3.2 对数与对数函数</category><category id=\"80549\" type=\"0\">3.2.3 指数函数与对数函数的关系</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81005\" type=\"0\">必修1</category><category id=\"81008\" type=\"0\">第3章 指数函数、对数函数和幂函数</category><category id=\"81023\" type=\"0\">3.2 对数函数</category><category id=\"81030\" type=\"0\">3.2.2 对数函数</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81721\" type=\"0\">必修第一册</category><category id=\"81723\" type=\"0\">第2章 指数函数、对数函数和幂函数</category><category id=\"81740\" type=\"0\">2.2 对数函数</category><category id=\"81749\" type=\"0\">2.2.3 对数函数的图象和性质</category></item>","ChildNum":1,"Time":"2019-06-10T14:24:42","QuesBody":"<div>【题文】在天文学中，天体的明暗程度可以用星等或亮度来描述.两颗星的星等与亮度满足<img style=\"vertical-align:middle;\" src=\"Upload/2019-09/21/411df7f9-15f3-4405-bbf0-3db542ffe655/paper.files/image001.png\">，其中星等为<i>m<sub>k</sub></i>的星的亮度为<i>E<sub>k</sub></i>（<i>k</i>=1,2）.已知太阳的星等是–26.7，天狼星的星等是–1.45，则太阳与天狼星的亮度的比值为<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tbody><tr><td width=\"25%\">A．10<sup>10.1</sup></td><td width=\"25%\">B．10.1</td><td width=\"25%\">C．lg10.1</td><td width=\"25%\">D．10–<sup>10.1</sup></td></tr></tbody></table></div>","QuesAnswer":"","QuesParse":"","UseSum":2085,"AvgScore":5.0,"ParseAnswerKey":"495aab8e82f62d045022c2826ba4c9ea","QuesAttribute":3,"QuesAttributes":{"ID":3,"Name":"跨章节综合题"},"TitleAbbreviation":"2019·甘肃高三月考（理）","PaperSource":[{"Id":569310,"PaperTitle":"2020年甘肃省会宁县第一中学高三上学期第一次月考数学（理）试题","LearnGrade":12,"PaperType":3,"Province":620000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-09-28T16:15:35","SchoolId":0,"MatchingDegree":0},{"Id":550490,"PaperTitle":"甘肃省兰州市第一中学2018-2019学年高二下学期期末数学（文）试题","LearnGrade":11,"PaperType":5,"Province":620000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-08-13T15:34:18","SchoolId":0,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0},{"Id":525063,"PaperTitle":"2019年北京市高考数学试卷（文科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:24:08","SchoolId":0,"MatchingDegree":0}]},{"ID":8196635,"Guid":"36767df9-1b59-4cc9-bdd7-ad3843cdb6da","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2701,"Name":"单选题","IsSelectType":true,"OrderIndex":2,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":2,"DifValue":0.85,"Name":"较易"},"Categories":"<item><category 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type=\"0\">人教B版</category><category id=\"80763\" type=\"0\">选修2-1</category><category id=\"80764\" type=\"0\">第一章 常用逻辑用语</category><category id=\"80770\" type=\"0\">1.3 充分条件、必要条件与命题的四种形式</category><category id=\"80776\" type=\"0\">1.3.1 推出与充分条件、必要条件</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81167\" type=\"0\">选修1-1</category><category id=\"81168\" type=\"0\">第1章 常用逻辑用语</category><category id=\"81172\" type=\"0\">1.1 命题及其关系</category><category id=\"81177\" type=\"0\">1.1.2 充分条件和必要条件</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81228\" type=\"0\">选修2-1</category><category id=\"81229\" type=\"0\">第1章 常用逻辑用语</category><category id=\"81233\" type=\"0\">1.1 命题及其关系</category><category id=\"81238\" type=\"0\">1.1.2 充分条件与必要条件</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81486\" type=\"0\">高中一年级第一学期</category><category id=\"81487\" type=\"0\">第1章 集合和命题</category><category 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充要条件</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:39","QuesBody":"<div>【题文】设点<i>A</i>，<i>B</i>，<i>C</i>不共线，则“<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/cad34659-b231-4f75-9819-a3221899ff4c/paper.files/image001.png\" alt=\"eqId7bc6fc8794944e57a77be639d9d491b5\">与<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/cad34659-b231-4f75-9819-a3221899ff4c/paper.files/image002.png\" alt=\"eqId4ddba6db3da948fbad12a7b08aa05161\">的夹角为锐角”是“<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/cad34659-b231-4f75-9819-a3221899ff4c/paper.files/image003.png\" alt=\"eqIde113bc4898d546d283e5757a50f9cec3\">”的<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tbody><tr><td width=\"50%\">A．充分而不必要条件</td><td width=\"50%\">B．必要而不充分条件</td></tr><tr><td width=\"50%\">C．充分必要条件</td><td width=\"50%\">D．既不充分也不必要条件</td></tr></tbody></table></div>","QuesAnswer":"","QuesParse":"","UseSum":1577,"AvgScore":5.0,"ParseAnswerKey":"cd3acb7fabd43c5f08c778971017fe4d","QuesAttribute":3,"QuesAttributes":{"ID":3,"Name":"跨章节综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196636,"Guid":"ab6e21c6-d8c0-4937-aac5-8737017c265f","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2701,"Name":"单选题","IsSelectType":true,"OrderIndex":2,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":4,"DifValue":0.4,"Name":"较难"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27933\" type=\"1\">平面解析几何</category><category id=\"29045\" type=\"1\">圆锥曲线</category><category id=\"29176\" type=\"1\">圆锥曲线综合</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79458\" type=\"0\">选修1-1</category><category id=\"79460\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"79482\" type=\"0\">本章复习与测试</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79540\" type=\"0\">选修2-1</category><category id=\"79543\" type=\"0\">第三章 圆锥曲线与方程</category><category id=\"79577\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80101\" type=\"0\">选修1-1</category><category id=\"80103\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80125\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80171\" type=\"0\">选修2-1</category><category id=\"80173\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80196\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81167\" type=\"0\">选修1-1</category><category id=\"81169\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81185\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81228\" type=\"0\">选修2-1</category><category id=\"81230\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81247\" type=\"0\">本章复习与测试</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81878\" type=\"0\">选修1-1</category><category id=\"81880\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81895\" type=\"0\">本章复习与测试</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:41","QuesBody":"<div>【题文】数学中有许多形状优美、寓意美好的曲线，曲线<i>C</i>：<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/20/af414bf9-893a-48b1-9030-f51bc20233eb/paper.files/image001.png\" alt=\"eqId95d5090cc542408bb07d83bbdfcf681a\">就是其中之一（如图）.给出下列三个结论：<br><img style=\"vertical-align:middle;\" src=\"Upload/2019-11/20/af414bf9-893a-48b1-9030-f51bc20233eb/paper.files/image002.png\" alt=\"figure\"><br>①曲线<i>C</i>恰好经过6个整点（即横、纵坐标均为整数的点）；<br>②曲线<i>C</i>上任意一点到原点的距离都不超过<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/20/af414bf9-893a-48b1-9030-f51bc20233eb/paper.files/image003.png\" alt=\"eqId65021072e3ee4edc934739d219c8c5e2\">；<br>③曲线<i>C</i>所围成的“心形”区域的面积小于3.<br>其中，所有正确结论的序号是<table name=\"optionsTable\" cellpadding=\"0\" cellspacing=\"0\" width=\"100%\"><tbody><tr><td width=\"25%\">A．①</td><td width=\"25%\">B．②</td><td width=\"25%\">C．①②</td><td width=\"25%\">D．①②③</td></tr></tbody></table></div>","QuesAnswer":"","QuesParse":"","UseSum":1437,"AvgScore":5.0,"ParseAnswerKey":"5de38cb72eca932024ee615c8e18bae8","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·甘肃高二期末（文）","PaperSource":[{"Id":550490,"PaperTitle":"甘肃省兰州市第一中学2018-2019学年高二下学期期末数学（文）试题","LearnGrade":11,"PaperType":5,"Province":620000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-08-13T15:34:18","SchoolId":0,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196637,"Guid":"f22c97aa-babf-4f38-ae03-51ee4a4a43e1","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":2,"DifValue":0.85,"Name":"较易"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27928\" type=\"1\">三角函数与解三角形</category><category id=\"28321\" type=\"1\">三角函数</category><category id=\"28328\" type=\"1\">三角函数的图象与性质</category><category id=\"28400\" type=\"1\">余弦函数的周期性</category><category id=\"28453\" type=\"1\">求余弦（型）函数的最小正周期</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27928\" type=\"1\">三角函数与解三角形</category><category id=\"28322\" type=\"1\">三角恒等变换</category><category id=\"28506\" type=\"1\">二倍角公式</category><category id=\"28527\" type=\"1\">二倍角的余弦公式</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79370\" type=\"0\">必修4</category><category id=\"79373\" type=\"0\">第三章 三角恒等变形</category><category id=\"79417\" type=\"0\">§3 二倍角的三角函数</category></item><item><category id=\"79775\" type=\"0\">沪教版（上海）</category><category id=\"79804\" type=\"0\">高一第二学期</category><category id=\"79807\" type=\"0\">第六章 三角函数</category><category id=\"79818\" type=\"0\">6.1 正弦函数和余弦函数的图像与性质</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80028\" type=\"0\">必修4</category><category id=\"80031\" type=\"0\">第三章 三角恒等变换</category><category id=\"80067\" type=\"0\">3.1 两角和与差的正弦、余弦和正切公式</category><category id=\"80072\" type=\"0\">3.1.3 二倍角的正弦、余弦、正切公式</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80624\" type=\"0\">必修4</category><category id=\"80627\" type=\"0\">第三章 三角恒等变换</category><category id=\"80661\" type=\"0\">3.2 倍角公式和半角公式</category><category id=\"80667\" type=\"0\">3.2.1 倍角公式</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81103\" type=\"0\">必修4</category><category id=\"81106\" type=\"0\">第3章 三角恒等变换</category><category id=\"81133\" type=\"0\">3.2 二倍角的三角函数</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81524\" type=\"0\">高中一年级第二学期</category><category id=\"81527\" type=\"0\">第6章 三角函数</category><category id=\"81553\" type=\"0\">一 三角函数的图像与性质</category><category id=\"81556\" type=\"0\">6.1 正弦函数和余弦函数的图像与性质</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81756\" type=\"0\">必修第二册</category><category id=\"81759\" type=\"0\">第5章 三角恒等变换</category><category id=\"81787\" type=\"0\">5.2 二倍角的三角函数</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:42","QuesBody":"<div>【题文】函数<i>f</i>(<i>x</i>)=sin<sup>2</sup>2<i>x</i>的最小正周期是__________．</div>","QuesAnswer":"","QuesParse":"","UseSum":1950,"AvgScore":5.0,"ParseAnswerKey":"c9e82abeccf9b44719bde74ea7d9a317","QuesAttribute":0,"QuesAttributes":{"ID":0,"Name":""},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0},{"Id":515359,"PaperTitle":"江西省上饶市玉山县第一中学2018-2019高一下学期期中考试数学（理）试题","LearnGrade":10,"PaperType":4,"Province":360000,"PaperYear":2019,"PaperLevel":51,"AddTimeOn":"2019-05-21T19:37:17","SchoolId":35318,"MatchingDegree":0}]},{"ID":8196638,"Guid":"4aac7207-4c63-4299-80a9-f602a886b829","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":2,"DifValue":0.85,"Name":"较易"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27930\" type=\"1\">数列</category><category id=\"28643\" type=\"1\">等差数列</category><category id=\"28666\" type=\"1\">等差数列及其通项公式</category><category id=\"28678\" type=\"1\">等差数列通项公式的基本量计算</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27930\" type=\"1\">数列</category><category id=\"28643\" type=\"1\">等差数列</category><category id=\"28673\" type=\"1\">等差数列前n项和的函数特性</category><category id=\"28697\" type=\"1\">求等差数列前n项和的最值</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79422\" type=\"0\">必修5</category><category id=\"79423\" type=\"0\">第一章 数列</category><category id=\"79428\" type=\"0\">§2 等差数列</category><category id=\"79435\" type=\"0\">2.2 等差数列的前n项和</category></item><item><category id=\"79775\" type=\"0\">沪教版（上海）</category><category id=\"79821\" type=\"0\">高二第一学期</category><category id=\"79822\" type=\"0\">第七章 数列与数学归纳法</category><category id=\"79827\" type=\"0\">7.1 数列</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80073\" type=\"0\">必修5</category><category id=\"80075\" type=\"0\">第二章 数列</category><category id=\"80085\" type=\"0\">2.3 等差数列的前n项和</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80669\" type=\"0\">必修5</category><category id=\"80671\" type=\"0\">第二章 数列</category><category id=\"80680\" type=\"0\">2.2 等差数列</category><category id=\"80686\" type=\"0\">2.2.2 等差数列的前n项和</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81139\" type=\"0\">必修5</category><category id=\"81141\" type=\"0\">第2章 数列</category><category id=\"81149\" type=\"0\">2.2 等差数列</category><category id=\"81154\" type=\"0\">2.2.3 等差数列的前n项和</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81565\" type=\"0\">高中二年级第一学期</category><category id=\"81566\" type=\"0\">第7章 数列与数学归纳法</category><category id=\"81571\" type=\"0\">一 数列</category><category id=\"81576\" type=\"0\">7.2 等差数列</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81818\" type=\"0\">必修第四册</category><category id=\"81820\" type=\"0\">第9章 数列</category><category id=\"81828\" type=\"0\">9.2 等差数列</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:43","QuesBody":"<div>【题文】设等差数列{<i>a<sub>n</sub></i>}的前<i>n</i>项和为<i>S<sub>n</sub></i>，若<i>a</i><sub>2</sub>=−3，<i>S</i><sub>5</sub>=−10，则<i>a</i><sub>5</sub>=__________，<i>S<sub>n</sub></i>的最小值为__________．</div>","QuesAnswer":"","QuesParse":"","UseSum":2426,"AvgScore":5.0,"ParseAnswerKey":"af8105b53c8ebbbf561ad52b5e4e2b99","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·全国高三专题练习","PaperSource":[{"Id":578588,"PaperTitle":"专题6.2 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type=\"0\">人教B版</category><category id=\"80550\" type=\"0\">必修2</category><category id=\"80551\" type=\"0\">第一章 立体几何初步</category><category id=\"80556\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81033\" type=\"0\">必修2</category><category id=\"81034\" type=\"0\">第1章 立体几何初步</category><category id=\"81039\" type=\"0\">1.3 空间几何体的表面积和体积</category><category id=\"81050\" type=\"0\">1.3.2 空间几何体的体积</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81792\" type=\"0\">必修第三册</category><category id=\"81793\" type=\"0\">第6章 立体几何初步</category><category id=\"81796\" type=\"0\">6.1 空间的几何体</category><category id=\"81801\" type=\"0\">6.1.3 面积和体积公式</category></item>","ChildNum":1,"Time":"2019-06-10T14:24:47","QuesBody":"<div>【题文】某几何体是由一个正方体去掉一个四棱柱所得，其三视图如图所示．如果网格纸上小正方形的边长为1，那么该几何体的体积为__________．<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/0fbef6da-2d22-4132-b00c-fb3e1364d3f9/paper.files/image001.png\" 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type=\"1\">高中数学综合库</category><category id=\"27932\" type=\"1\">空间向量与立体几何</category><category id=\"28805\" type=\"1\">点、直线、平面之间的位置关系</category><category id=\"28898\" type=\"1\">空间点、直线、平面之间的位置关系</category><category id=\"28907\" type=\"1\">线面关系</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27932\" type=\"1\">空间向量与立体几何</category><category id=\"28805\" type=\"1\">点、直线、平面之间的位置关系</category><category id=\"28898\" type=\"1\">空间点、直线、平面之间的位置关系</category><category id=\"28908\" type=\"1\">面面关系</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79292\" type=\"0\">必修2</category><category id=\"79294\" type=\"0\">第二章 解析几何初步</category><category id=\"79320\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"79935\" type=\"0\">必修2</category><category id=\"79937\" type=\"0\">第二章 点、直线、平面之间的位置关系</category><category id=\"79955\" 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alt=\"eqIdc13953f2514e4c8f9c8aaaf5241c33ac\">；③<i>l</i>⊥<img style=\"vertical-align:middle;\" src=\"Upload/2019-10/30/0fdb2587-8341-4764-83f2-47961302f081/paper.files/image001.png\" alt=\"eqIdc13953f2514e4c8f9c8aaaf5241c33ac\">．<br>以其中的两个论断作为条件，余下的一个论断作为结论，写出一个正确的命题：__________．</div>","QuesAnswer":"","QuesParse":"","UseSum":2120,"AvgScore":5.0,"ParseAnswerKey":"e38465e5b29af5442763cf73afceaadf","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·遵义航天高级中学高二期中","PaperSource":[{"Id":587716,"PaperTitle":"贵州省遵义市汇川区航天高级中学2019-2020学年高二上学期期中数学试题","LearnGrade":11,"PaperType":4,"Province":520000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-11-04T15:39:04","SchoolId":65915,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0},{"Id":525063,"PaperTitle":"2019年北京市高考数学试卷（文科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:24:08","SchoolId":0,"MatchingDegree":0}]},{"ID":8196639,"Guid":"95d6e0fb-ab20-45ef-aad5-c44a4dcdaac8","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2702,"Name":"填空题","IsSelectType":false,"OrderIndex":3,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category 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type=\"0\">必修第一册</category><category id=\"81722\" type=\"0\">第1章 集合与函数</category><category id=\"81726\" type=\"0\">1.2 函数的概念和性质</category><category id=\"81734\" type=\"0\">1.2.4 从解析式看函数的性质</category></item><item><category id=\"134949\" type=\"0\">人教A版（2019）</category><category id=\"135303\" type=\"0\">必修（第一册）</category><category id=\"135309\" type=\"0\">第三章 函数概念与性质</category><category id=\"135338\" type=\"0\">3.2 函数的基本性质</category><category id=\"145208\" type=\"0\">3.2.2 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type=\"0\">第3章 指数函数、对数函数和幂函数</category><category id=\"81025\" type=\"0\">3.4 函数的应用</category><category id=\"81032\" type=\"0\">3.4.2 函数模型及其应用</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81721\" type=\"0\">必修第一册</category><category id=\"81723\" type=\"0\">第2章 指数函数、对数函数和幂函数</category><category id=\"81743\" type=\"0\">2.5 函数模型及其应用</category><category id=\"81755\" type=\"0\">2.5.2 形形色色的函数模型</category></item><item><category id=\"134949\" type=\"0\">人教A版（2019）</category><category id=\"135303\" type=\"0\">必修（第一册）</category><category id=\"135309\" type=\"0\">第三章 函数概念与性质</category></item><item><category id=\"136156\" type=\"0\">人教B版（2019）</category><category id=\"136247\" type=\"0\">必修（第一册）</category><category id=\"150397\" type=\"0\">第三章 函数</category><category id=\"150403\" type=\"0\">3.3 函数的应用（一）</category></item>","ChildNum":1,"Time":"2019-11-26T17:23:27","QuesBody":"<div>【题文】李明自主创业，在网上经营一家水果店，销售的水果中有草莓、京白梨、西瓜、桃，价格依次为60元/盒、65元/盒、80元/盒、90元/盒．为增加销量，李明对这四种水果进行促销：一次购买水果的总价达到120元，顾客就少付<i>x</i>元．每笔订单顾客网上支付成功后，李明会得到支付款的80%．<br>①当<i>x</i>=10时，顾客一次购买草莓和西瓜各1盒，需要支付__________元；<br>②在促销活动中，为保证李明每笔订单得到的金额均不低于促销前总价的七折，则<i>x</i>的最大值为__________．</div>","QuesAnswer":"","QuesParse":"","UseSum":1759,"AvgScore":5.0,"ParseAnswerKey":"5c4e6c9a7e07825f9a604afd1b5a6ec2","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·北京师大附中高一期中","PaperSource":[{"Id":605243,"PaperTitle":"北京市西城区北京师范大学附中2019-2020学年高一上学期期中数学试题","LearnGrade":10,"PaperType":4,"Province":110000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-11-26T17:22:14","SchoolId":198,"MatchingDegree":0},{"Id":590255,"PaperTitle":"2019年山东省新高考备考监测高三上学期10月联考数学试题","LearnGrade":12,"PaperType":3,"Province":370000,"PaperYear":2019,"PaperLevel":61,"AddTimeOn":"2019-11-06T23:15:45","SchoolId":0,"MatchingDegree":0},{"Id":590253,"PaperTitle":"湖北省百校大联盟高三上学期10月数学（理）试题","LearnGrade":12,"PaperType":3,"Province":420000,"PaperYear":2019,"PaperLevel":42,"AddTimeOn":"2019-11-06T23:14:39","SchoolId":0,"MatchingDegree":0},{"Id":583294,"PaperTitle":"山东省临沂市第一中学2019-2020学年高三上学期10月月考数学试题","LearnGrade":12,"PaperType":3,"Province":370000,"PaperYear":2019,"PaperLevel":42,"AddTimeOn":"2019-10-26T20:14:44","SchoolId":0,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0},{"Id":525063,"PaperTitle":"2019年北京市高考数学试卷（文科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:24:08","SchoolId":0,"MatchingDegree":0}]},{"ID":8196640,"Guid":"5800d109-d8d2-4d74-82bb-3fcbade25646","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27928\" type=\"1\">三角函数与解三角形</category><category id=\"28323\" type=\"1\">解三角形</category><category id=\"28546\" type=\"1\">正弦定理和余弦定理</category><category id=\"28548\" type=\"1\">正弦定理</category><category id=\"28552\" type=\"1\">正弦定理解三角形</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27928\" type=\"1\">三角函数与解三角形</category><category id=\"28323\" type=\"1\">解三角形</category><category id=\"28546\" type=\"1\">正弦定理和余弦定理</category><category id=\"28550\" type=\"1\">余弦定理</category><category id=\"28559\" type=\"1\">余弦定理解三角形</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79422\" type=\"0\">必修5</category><category id=\"79424\" type=\"0\">第二章 解三角形</category><category id=\"79438\" type=\"0\">§1 正弦定理与余弦定理</category><category id=\"79443\" type=\"0\">1.2 余弦定理</category></item><item><category id=\"79775\" type=\"0\">沪教版（上海）</category><category id=\"79804\" type=\"0\">高一第二学期</category><category id=\"79806\" type=\"0\">第五章 三角比</category><category id=\"79816\" type=\"0\">5.3 同角三角比的关系和诱导公式</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80073\" type=\"0\">必修5</category><category id=\"80074\" type=\"0\">第一章 解三角形</category><category id=\"80078\" type=\"0\">1.1 正弦定理和余弦定理</category><category id=\"80082\" type=\"0\">1.1.2 余弦定理</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80669\" type=\"0\">必修5</category><category id=\"80670\" type=\"0\">第一章 解三角形</category><category id=\"80674\" type=\"0\">1.1 正弦定理和余弦定理</category><category id=\"80678\" type=\"0\">1.1.2 余弦定理</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81139\" type=\"0\">必修5</category><category id=\"81140\" type=\"0\">第1章 解三角形</category><category id=\"81145\" type=\"0\">1.2 余弦定理</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81524\" type=\"0\">高中一年级第二学期</category><category id=\"81526\" type=\"0\">第5章 三角比</category><category id=\"81543\" type=\"0\">三 解斜三角形</category><category id=\"81550\" type=\"0\">5.6 正弦定理、余弦定理和解斜三角形</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81818\" type=\"0\">必修第四册</category><category id=\"81819\" type=\"0\">第8章 解三角形</category><category id=\"81824\" type=\"0\">8.2 余弦定理</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:46","QuesBody":"<div>【题文】在△<i>ABC</i>中，<i>a</i>=3，<i>b</i>−<i>c</i>=2，cos<i>B</i>=<img style=\"vertical-align:middle;\" src=\"Upload/2019-08/26/29d46c2a-d80f-4e7b-a4ab-d14b274a61ca/paper.files/image001.png\" alt=\"eqIdfcc58431683b452ca65d1d9977a280bd\">．<br>（Ⅰ）求<i>b</i>，<i>c</i>的值；<br>（Ⅱ）求sin（<i>B</i>–<i>C</i>）的值．</div>","QuesAnswer":"","QuesParse":"","UseSum":2529,"AvgScore":5.0,"ParseAnswerKey":"0e63a496ffac3e8cfbbb469a9c76b8b0","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·武威市第六中学高一月考","PaperSource":[{"Id":569856,"PaperTitle":"甘肃省武威市第六中学2018-2019学年高一下学期第三次学段考试数学试题","LearnGrade":10,"PaperType":3,"Province":620000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-09-30T08:38:49","SchoolId":79414,"MatchingDegree":0},{"Id":568454,"PaperTitle":"上海市格致中学2019-2020高三9月开学考数学","LearnGrade":12,"PaperType":8,"Province":310000,"PaperYear":2019,"PaperLevel":62,"AddTimeOn":"2019-09-26T08:14:00","SchoolId":18778,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196641,"Guid":"27f2a70e-bc07-49b5-abd3-aa89c7ee6136","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27932\" type=\"1\">空间向量与立体几何</category><category id=\"28807\" type=\"1\">立体几何综合</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79540\" type=\"0\">选修2-1</category><category id=\"79542\" type=\"0\">第二章 空间向量与立体几何</category><category id=\"79566\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80171\" type=\"0\">选修2-1</category><category id=\"80174\" type=\"0\">第三章 空间向量与立体几何</category><category id=\"80207\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80763\" type=\"0\">选修2-1</category><category id=\"80766\" type=\"0\">第三章 空间向量与立体几何</category><category id=\"80794\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81228\" type=\"0\">选修2-1</category><category id=\"81231\" type=\"0\">第3章 空间向量与立体几何</category><category id=\"81259\" type=\"0\">本章复习与测试</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:48","QuesBody":"<div>【题文】如图，在四棱锥<i>P</i>–<i>ABCD</i>中，<i>PA</i>⊥平面<i>ABCD</i>，<i>AD</i>⊥<i>CD</i>，<i>AD</i>∥<i>BC</i>，<i>PA</i>=<i>AD</i>=<i>CD</i>=2，<i>BC</i>=3．<i>E</i>为<i>PD</i>的中点，点<i>F</i>在<i>PC</i>上，且<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/7d7daba4-06e9-4803-b239-28766bff0d9f/paper.files/image001.png\" alt=\"eqIde286632b7e414694ae1f0e04fbfdd12a\">．<br>（Ⅰ）求证：<i>CD</i>⊥平面<i>PAD</i>；<br>（Ⅱ）求二面角<i>F–AE–P</i>的余弦值；<br>（Ⅲ）设点<i>G</i>在<i>PB</i>上，且<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/7d7daba4-06e9-4803-b239-28766bff0d9f/paper.files/image002.png\" alt=\"eqIdcbf1a61a4243427cb78adf2132330806\">．判断直线<i>AG</i>是否在平面<i>AEF</i>内，说明理由．<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/7d7daba4-06e9-4803-b239-28766bff0d9f/paper.files/image003.jpg\" alt=\"figure\"></div>","QuesAnswer":"","QuesParse":"","UseSum":2868,"AvgScore":5.0,"ParseAnswerKey":"118ced287506ac6187dcea3f605afbba","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196642,"Guid":"9a80c09c-f1cc-4162-839e-9a52f7f09624","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27934\" type=\"1\">计数原理与概率统计</category><category id=\"29369\" type=\"1\">概率</category><category id=\"29555\" type=\"1\">概率综合</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79332\" type=\"0\">必修3</category><category id=\"79335\" type=\"0\">第三章 概率</category><category id=\"79364\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"79992\" type=\"0\">必修3</category><category id=\"79995\" type=\"0\">第三章 概率</category><category id=\"80020\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80584\" type=\"0\">必修3</category><category id=\"80587\" type=\"0\">第三章 概率</category><category id=\"80615\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81066\" type=\"0\">必修3</category><category id=\"81069\" type=\"0\">第3章 概率</category><category id=\"81100\" type=\"0\">本章复习与测试</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81837\" type=\"0\">必修第五册</category><category id=\"81840\" type=\"0\">第13章 概率</category><category id=\"81873\" type=\"0\">本章复习与测试</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:48","QuesBody":"<div>【题文】改革开放以来，人们的支付方式发生了巨大转变．近年来，移动支付已成为主要支付方式之一．为了解某校学生上个月A，B两种移动支付方式的使用情况，从全校学生中随机抽取了100人，发现样本中A，B两种支付方式都不使用的有5人，样本中仅使用A和仅使用B的学生的支付金额分布情况如下：<br><table border=\"1\" cellpadding=\"0\" cellspacing=\"0\"><tbody><tr style=\"height:16.5pt\"><td style=\"width:107.25pt;border:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"143\"><img style=\"vertical-align:middle;\" src=\"Upload/2019-06/18/c80a4f2f-caf4-438b-9ef2-aeee1c34e79f/paper.files/image001.png\" alt=\"figure\">交付金额（元）<br>支付方式<br></td><td style=\"width:111.0pt;border:solid black 1.0pt;border-left:none; padding:3.75pt 5.4pt 3.75pt 5.4pt;height:16.5pt\" width=\"148\">（0,1000]<br></td><td style=\"width:111.0pt;border:solid black 1.0pt;border-left:none; padding:3.75pt 5.4pt 3.75pt 5.4pt;height:16.5pt\" width=\"148\">（1000,2000]<br></td><td style=\"width:96.75pt;border:solid black 1.0pt;border-left:none; padding:3.75pt 5.4pt 3.75pt 5.4pt;height:16.5pt\" width=\"129\">大于2000<br></td></tr><tr style=\"height:16.5pt\"><td style=\"width:107.25pt;border:solid black 1.0pt;border-top:none; padding:3.75pt 5.4pt 3.75pt 5.4pt;height:16.5pt\" width=\"143\">仅使用A<br></td><td style=\"width:111.0pt;border-top:none;border-left:none; border-bottom:solid black 1.0pt;border-right:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"148\">18人<br></td><td style=\"width:111.0pt;border-top:none;border-left:none; border-bottom:solid black 1.0pt;border-right:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"148\">9人<br></td><td style=\"width:96.75pt;border-top:none;border-left:none; border-bottom:solid black 1.0pt;border-right:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"129\">3人<br></td></tr><tr style=\"height:16.5pt\"><td style=\"width:107.25pt;border:solid black 1.0pt;border-top:none; padding:3.75pt 5.4pt 3.75pt 5.4pt;height:16.5pt\" width=\"143\">仅使用B<br></td><td style=\"width:111.0pt;border-top:none;border-left:none; border-bottom:solid black 1.0pt;border-right:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"148\">10人<br></td><td style=\"width:111.0pt;border-top:none;border-left:none; border-bottom:solid black 1.0pt;border-right:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"148\">14人<br></td><td style=\"width:96.75pt;border-top:none;border-left:none; border-bottom:solid black 1.0pt;border-right:solid black 1.0pt;padding:3.75pt 5.4pt 3.75pt 5.4pt; height:16.5pt\" width=\"129\">1人<br></td></tr></tbody></table> <br>（Ⅰ）从全校学生中随机抽取1人，估计该学生上个月A，B两种支付方式都使用的概率；<br>（Ⅱ）从样本仅使用A和仅使用B的学生中各随机抽取1人，以<i>X</i>表示这2人中上个月支付金额大于1000元的人数，求<i>X</i>的分布列和数学期望；<br>（Ⅲ）已知上个月样本学生的支付方式在本月没有变化．现从样本仅使用A的学生中，随机抽查3人，发现他们本月的支付金额都大于2000元．根据抽查结果，能否认为样本仅使用A的学生中本月支付金额大于2000元的人数有变化？说明理由．</div>","QuesAnswer":"","QuesParse":"","UseSum":1247,"AvgScore":5.0,"ParseAnswerKey":"0ffd19e7d3bc85b434330fb3067df4cd","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196643,"Guid":"a2a5c79e-2879-4817-ad04-35e3d9561080","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27933\" type=\"1\">平面解析几何</category><category id=\"29044\" type=\"1\">圆与方程</category><category id=\"29118\" type=\"1\">圆的方程</category><category id=\"29122\" type=\"1\">圆的一般方程</category><category id=\"29132\" type=\"1\">圆过定点问题</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27933\" type=\"1\">平面解析几何</category><category id=\"29045\" type=\"1\">圆锥曲线</category><category id=\"29173\" type=\"1\">抛物线</category><category id=\"29274\" type=\"1\">抛物线标准方程的求法</category><category id=\"29287\" type=\"1\">根据抛物线上的点求标准方程</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79458\" type=\"0\">选修1-1</category><category id=\"79460\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"79482\" type=\"0\">本章复习与测试</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79540\" type=\"0\">选修2-1</category><category id=\"79543\" type=\"0\">第三章 圆锥曲线与方程</category><category id=\"79576\" type=\"0\">§4 曲线与方程</category><category id=\"79586\" type=\"0\">4.3 直线与圆锥曲线的交点</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80101\" type=\"0\">选修1-1</category><category id=\"80103\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80125\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80171\" type=\"0\">选修2-1</category><category id=\"80173\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80196\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80699\" type=\"0\">选修1-1</category><category id=\"80701\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80717\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80763\" type=\"0\">选修2-1</category><category id=\"80765\" type=\"0\">第二章 圆锥曲线与方程</category><category id=\"80782\" type=\"0\">2.5 直线与圆锥曲线</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81228\" type=\"0\">选修2-1</category><category id=\"81230\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81247\" type=\"0\">本章复习与测试</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81878\" type=\"0\">选修1-1</category><category id=\"81880\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81894\" type=\"0\">2.4 圆锥曲线的应用</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81945\" type=\"0\">选修2-1</category><category id=\"81947\" type=\"0\">第2章 圆锥曲线与方程</category><category id=\"81961\" type=\"0\">2.4 圆锥曲线的应用</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:49","QuesBody":"<div>【题文】已知抛物线<i>C</i>：<i>x</i><sup>2</sup>=−2<i>py</i>经过点（2，−1）．<br>（Ⅰ）求抛物线<i>C</i>的方程及其准线方程；<br>（Ⅱ）设<i>O</i>为原点，过抛物线<i>C</i>的焦点作斜率不为0的直线<i>l</i>交抛物线<i>C</i>于两点<i>M</i>，<i>N</i>，直线<i>y</i>=−1分别交直线<i>OM</i>，<i>ON</i>于点<i>A</i>和点<i>B</i>.求证：以<i>AB</i>为直径的圆经过<i>y</i>轴上的两个定点．</div>","QuesAnswer":"","QuesParse":"","UseSum":1792,"AvgScore":5.0,"ParseAnswerKey":"bc2ac254fb3936a74cbc06190b1975cb","QuesAttribute":2,"QuesAttributes":{"ID":2,"Name":"章内综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]},{"ID":8196631,"Guid":"940da83c-e435-44c5-83a0-5318f9c852c2","Title":"2019年北京市高考数学试卷（文科）","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":3,"DifValue":0.65,"Name":"一般"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27927\" type=\"1\">函数与导数</category><category id=\"28044\" type=\"1\">导数及其应用</category><category id=\"28257\" type=\"1\">导数的概念和几何意义</category><category id=\"28263\" type=\"1\">导数的几何意义</category><category id=\"28271\" type=\"1\">已知切线（斜率）求参数</category></item><item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27927\" type=\"1\">函数与导数</category><category id=\"28044\" type=\"1\">导数及其应用</category><category id=\"28260\" type=\"1\">导数的综合应用</category><category id=\"28294\" type=\"1\">导数在函数中的其他应用</category><category id=\"28296\" type=\"1\">利用导数证明不等式</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79458\" type=\"0\">选修1-1</category><category id=\"79461\" type=\"0\">第三章 变化率与导数</category><category id=\"79490\" type=\"0\">§2 导数的概念及其几何意义</category><category id=\"79495\" type=\"0\">2.2 导数的几何意义</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79458\" type=\"0\">选修1-1</category><category id=\"79462\" type=\"0\">第四章 导数应用</category><category id=\"79500\" type=\"0\">本章复习与测试</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79587\" type=\"0\">选修2-2</category><category id=\"79589\" type=\"0\">第二章 变化率与导数</category><category id=\"79604\" type=\"0\">§2 导数的概念及其几何意义</category><category id=\"79610\" type=\"0\">2.2 导数的几何意义</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79587\" type=\"0\">选修2-2</category><category id=\"79590\" type=\"0\">第三章 导数应用</category><category id=\"79615\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80101\" type=\"0\">选修1-1</category><category id=\"80104\" type=\"0\">第三章 导数及其应用</category><category id=\"80136\" type=\"0\">本章复习与测试</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80101\" type=\"0\">选修1-1</category><category id=\"80104\" type=\"0\">第三章 导数及其应用</category><category id=\"80132\" type=\"0\">3.1 变化率与导数</category><category id=\"80139\" type=\"0\">3.1.3 导数的几何意义</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80213\" type=\"0\">选修2-2</category><category id=\"80214\" type=\"0\">第一章 导数及其应用</category><category id=\"80218\" type=\"0\">1.1 变化率与导数</category><category id=\"80228\" type=\"0\">1.1.3 导数的几何意义</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80699\" type=\"0\">选修1-1</category><category id=\"80702\" type=\"0\">第三章 导数及其应用</category><category id=\"80727\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80699\" type=\"0\">选修1-1</category><category id=\"80702\" type=\"0\">第三章 导数及其应用</category><category id=\"80724\" type=\"0\">3.1 导数</category><category id=\"80730\" type=\"0\">3.1.3 导数的几何意义</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80804\" type=\"0\">选修2-2</category><category id=\"80805\" type=\"0\">第一章 导数及其应用</category><category id=\"80813\" type=\"0\">本章复习与测试</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80804\" type=\"0\">选修2-2</category><category id=\"80805\" type=\"0\">第一章 导数及其应用</category><category id=\"80809\" type=\"0\">1.1 导数</category><category id=\"80816\" type=\"0\">1.1.3 导数的几何意义</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81167\" type=\"0\">选修1-1</category><category id=\"81170\" type=\"0\">第3章 导数及其应用</category><category id=\"81196\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81167\" type=\"0\">选修1-1</category><category id=\"81170\" type=\"0\">第3章 导数及其应用</category><category id=\"81192\" type=\"0\">3.1 导数的概念</category><category id=\"81198\" type=\"0\">3.1.2 瞬时变化率——导数</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81268\" type=\"0\">选修2-2</category><category id=\"81269\" type=\"0\">第1章 导数及其应用</category><category id=\"81278\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81268\" type=\"0\">选修2-2</category><category id=\"81269\" type=\"0\">第1章 导数及其应用</category><category id=\"81273\" type=\"0\">1.1 导数的概念</category><category id=\"81280\" type=\"0\">1.1.2 瞬时变化率——导数</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81878\" type=\"0\">选修1-1</category><category id=\"81881\" type=\"0\">第3章 导数及其应用</category><category id=\"81906\" type=\"0\">本章复习与测试</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81878\" type=\"0\">选修1-1</category><category id=\"81881\" type=\"0\">第3章 导数及其应用</category><category id=\"81902\" type=\"0\">3.1 导数概念</category><category id=\"81909\" type=\"0\">3.1.3 导数的概念和几何意义</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81980\" type=\"0\">选修2-2</category><category id=\"81981\" type=\"0\">第4章 导数及其应用</category><category id=\"81990\" type=\"0\">本章复习与测试</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81980\" type=\"0\">选修2-2</category><category id=\"81981\" type=\"0\">第4章 导数及其应用</category><category id=\"81985\" type=\"0\">4.1 导数概念</category><category id=\"81993\" type=\"0\">4.1.3 导数的概念和几何意义</category></item>","ChildNum":1,"Time":"2019-11-04T14:49:08","QuesBody":"<div>【题文】已知函数<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image001.png\" alt=\"eqId51cdf4868027499b9a4b78f7966b85d5\">.<br>（Ⅰ）求曲线<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image002.png\" alt=\"eqId144eff5c0e4649138a21f2b1b6a06907\">的斜率为1的切线方程；<br>（Ⅱ）当<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image003.png\" alt=\"eqId649d99a7111a46318498e8b9d8ef5feb\">时，求证：<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image004.png\" alt=\"eqId11def370b16c409db756234905bbe69a\">；<br>（Ⅲ）设<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image005.png\" alt=\"eqIde0e337a0b6124699a799a7b5055d8cbd\">，记<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image006.png\" alt=\"eqId9c705d01261145f1a3b3d15d276106ff\">在区间<img style=\"vertical-align:middle;\" src=\"Upload/2019-11/05/3813ab8f-ed3d-4d09-bcd6-c74b96d1a3e0/paper.files/image007.png\" alt=\"eqId2725c0a898494669a165a207f7a0605a\">上的最大值为<i>M</i>（<i>a</i>），当<i>M</i>（<i>a</i>）最小时，求<i>a</i>的值．</div>","QuesAnswer":"","QuesParse":"","UseSum":2405,"AvgScore":5.0,"ParseAnswerKey":"a1d6762b5cdc7127d9a2fc6506193641","QuesAttribute":3,"QuesAttributes":{"ID":3,"Name":"跨章节综合题"},"TitleAbbreviation":"2019·甘肃省武威第一中学高三月考（理）","PaperSource":[{"Id":587678,"PaperTitle":"甘肃省武威市第一中学2019年高三上学期10月月考数学（理）试题","LearnGrade":12,"PaperType":3,"Province":620000,"PaperYear":2019,"PaperLevel":42,"AddTimeOn":"2019-11-04T14:47:54","SchoolId":71957,"MatchingDegree":0},{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0},{"Id":525063,"PaperTitle":"2019年北京市高考数学试卷（文科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:24:08","SchoolId":0,"MatchingDegree":0}]},{"ID":8196644,"Guid":"99a7483b-12ee-4ef0-a409-5504593c1716","Title":"2019年北京市高考数学试卷（理科）","From":"","Knowledge":"","QuesType":{"ID":2703,"Name":"解答题","IsSelectType":false,"OrderIndex":4,"ParentId":0},"QuesAbility":{"ID":1,"Name":"了解和识记"},"QuesDiff":{"ID":4,"DifValue":0.4,"Name":"较难"},"Categories":"<item><category id=\"27925\" type=\"1\">高中数学综合库</category><category id=\"27935\" type=\"1\">推理与证明</category><category id=\"29672\" type=\"1\">数学归纳法</category><category id=\"29702\" type=\"1\">数学归纳法的应用</category><category id=\"29706\" type=\"1\">数学归纳法证明数列问题</category></item><item><category id=\"79243\" type=\"0\">北师大版</category><category id=\"79587\" type=\"0\">选修2-2</category><category id=\"79588\" type=\"0\">第一章 推理与证明</category><category id=\"79597\" type=\"0\">§4 数学归纳法</category></item><item><category id=\"79775\" type=\"0\">沪教版（上海）</category><category id=\"79821\" type=\"0\">高二第一学期</category><category id=\"79822\" type=\"0\">第七章 数列与数学归纳法</category><category id=\"79828\" type=\"0\">7.2 等差数列</category></item><item><category id=\"79903\" type=\"0\">人教A版</category><category id=\"80213\" type=\"0\">选修2-2</category><category id=\"80215\" type=\"0\">第二章 推理与证明</category><category id=\"80241\" type=\"0\">2.3 数学归纳法</category></item><item><category id=\"80512\" type=\"0\">人教B版</category><category id=\"80804\" type=\"0\">选修2-2</category><category id=\"80806\" type=\"0\">第二章 推理与证明</category><category id=\"80827\" type=\"0\">2.3 数学归纳法</category><category id=\"80834\" type=\"0\">2.3.2 数学归纳法应用举例</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81204\" type=\"0\">选修1-2</category><category id=\"81206\" type=\"0\">第2章 推理与证明</category><category id=\"81215\" type=\"0\">本章复习与测试</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81268\" type=\"0\">选修2-2</category><category id=\"81270\" type=\"0\">第2章 推理与证明</category><category id=\"81292\" type=\"0\">2.3 数学归纳法</category></item><item><category id=\"81004\" type=\"0\">苏教版</category><category id=\"81565\" type=\"0\">高中二年级第一学期</category><category id=\"81566\" type=\"0\">第7章 数列与数学归纳法</category><category id=\"81572\" type=\"0\">二 数学归纳法 </category><category id=\"81579\" type=\"0\">7.5 数学归纳法的应用</category></item><item><category id=\"81720\" type=\"0\">湘教版</category><category id=\"81980\" type=\"0\">选修2-2</category><category id=\"81983\" type=\"0\">第6章 推理与证明</category><category id=\"82011\" type=\"0\">6.3 数学归纳法</category></item>","ChildNum":1,"Time":"2019-06-10T14:25:51","QuesBody":"<div>【题文】已知数列<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">，从中选取第<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image002.png\" alt=\"eqId1ec38518c5ac4ee3a5275ce334d33b12\">项、第<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image003.png\" alt=\"eqId85175c55152f4b94ab0066c7bd2aa31d\">项、…、第<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image004.png\" alt=\"eqIda76d8d5f13ca41c7b7cc132ec44267f9\">项<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image005.png\" alt=\"eqIdfe9ca748a12e4091ab1c2eb9d97ac483\">，若<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image006.png\" alt=\"eqIdf143a15634e546c7a388e94eb85086a7\">，则称新数列<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image007.png\" alt=\"eqId92da9ff5dd2a4a8cbda55e5d6ee58a0a\">为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的长度为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image008.png\" alt=\"eqId3a4208e0040c42ed8b09de2cae9d562e\">的递增子列．规定：数列<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的任意一项都是<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的长度为1的递增子列．<br>（Ⅰ）写出数列1，8，3，7，5，6，9的一个长度为4的递增子列；<br>（Ⅱ）已知数列<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的长度为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image009.png\" alt=\"eqId0c243e5dded24989821daf9bef2676a6\">的递增子列的末项的最小值为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image010.png\" alt=\"eqId508fe952d9b64b1b9ff4dc8dbc95bec2\">，长度为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image011.png\" alt=\"eqIda5d0fc594d4048549d5bbda484fbecc8\">的递增子列的末项的最小值为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image012.png\" alt=\"eqId310318f5e7f34c8d89beec7f96e88bb2\">.若<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image013.png\" alt=\"eqId237d0298be9a494abc2bd4a6824195a1\">，求证：<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image014.png\" alt=\"eqId4ec648b777d443409c859ab0bbd5caae\">；<br>（Ⅲ）设无穷数列<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的各项均为正整数，且任意两项均不相等.若<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的长度为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image015.png\" alt=\"eqId08191f2c12324f94b4ddfe2f324a71cf\">的递增子列末项的最小值为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image016.png\" alt=\"eqId3e11ecfdb53b4b6e8d71b315fbed857c\">，且长度为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image015.png\" alt=\"eqId08191f2c12324f94b4ddfe2f324a71cf\">末项为<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image016.png\" alt=\"eqId3e11ecfdb53b4b6e8d71b315fbed857c\">的递增子列恰有<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image017.png\" alt=\"eqIdd8cbe3aed9684422bd4dd29dac96530a\">个<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image018.png\" alt=\"eqId77eeffcf800d4e1db83d163c5c1d7804\">，求数列<img style=\"vertical-align:middle;\" src=\"Upload/2019-06/29/37ca7b2e-d8ac-48b1-b55c-c4408bb060d6/paper.files/image001.png\" alt=\"eqId93e38ecd74a24cb59da79181b95bfd3a\">的通项公式．</div>","QuesAnswer":"","QuesParse":"","UseSum":882,"AvgScore":5.0,"ParseAnswerKey":"24370630bcb32cc9bdad78f032aa9423","QuesAttribute":3,"QuesAttributes":{"ID":3,"Name":"跨章节综合题"},"TitleAbbreviation":"2019·北京高考真题（理）","PaperSource":[{"Id":525064,"PaperTitle":"2019年北京市高考数学试卷（理科）","LearnGrade":12,"PaperType":12,"Province":110000,"PaperYear":2019,"PaperLevel":41,"AddTimeOn":"2019-06-10T14:25:06","SchoolId":0,"MatchingDegree":0}]}],"bankId":0,"PaperQuesCount":20,"ReadSum":7860,"schoolId":0}
