1 . 已知数列
有递推关系![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c6217221298b1bddf76f34b3c21286d.png)
(1)记
若数列
的递推式形如
且
,也即分子中不再含有常数项,求实数
的值;
(2)求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c6217221298b1bddf76f34b3c21286d.png)
(1)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b95f505b82ec97db6f5c170d251df963.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05cb52f95f72f1fc1c0d7e60f542c6ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57672a57201fe561d9fcf971f9cd73a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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2023-02-21更新
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2卷引用:上海市实验学校2021-2022学年高二上学期期中数学试题
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2 . 空间内水平放置的两个封闭图形分别为(i)长为
、宽为
的矩形;(ii)边长为
的正三角形,记(i)中原图形面积为
,斜二测画法得到的直观图面积为
,(ii)中原图形面积为
,斜二测画法得到的直观图面积为
,对以下两个命题:①
;②
,以下判断正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5484ad8796435dbd5d069a5f89bc5c69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4abcedf1a2a3022f01029dccad6cb961.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc3db1bc329a2ab147c9d08f930d0d36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a9ee2eb542fcfea9c78da919b33ee8b.png)
A.①为真命题,②为假命题 | B.①为假命题,②为真命题 |
C.①为真命题,②为真命题 | D.①为假命题,②为假命题 |
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解题方法
3 . 如图,曲线是一个圆心位于
,半径为
得四分之一圆弧,
是直线
上的线段,两者交于
,
,
与
轴共同构造一个封闭区域
,将
绕
轴旋转一周得到几何体
,现已知:过点
作
的水平截面,所得的截面积
与
之间的函数关系式为
,利用
的表达式与祖暅原理,考虑一个长方体,一个四棱锥和一个平放的半圆柱,计算几何体
的体积为
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解题方法
4 . 已知等差数列
满足,
,等比数列
的公比
,令![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b57b19c8d7ce597a64b147e06ef12c4.png)
的前
项和为
,若“
”是“
”的充分条件,则正整数
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1579e5e7d44cb8215a6d87280371be7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b1a1b910ff5ef0d244d659ccd6bc07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b57b19c8d7ce597a64b147e06ef12c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c29bfcb2e31e3c21967ede660eaa0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dd643a1cdf1bc2d2740c393746d75aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
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5 . 在两平面平行的判定定理中,假设
为两不同平面,
为两不同直线,若要得到
,则需要在条件“
”之外补充条件______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48ccc19a183b9ce7f82d2609a14b9a43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35f747152f006301e03b643afb80195c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f7ccda1d5e4f06b50deed70891012a9.png)
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5卷引用:上海市实验学校2021-2022学年高二上学期期中数学试题
上海市实验学校2021-2022学年高二上学期期中数学试题(已下线)第一章 点线面位置关系 专题一 空间平行关系的判定与证明 微点1 空间平行关系的判定与证明【培优版】(已下线)专题04平面与平面的位置关系(2个知识点8种题型)-【倍速学习法】2023-2024学年高二数学核心知识点与常见题型通关讲解练(沪教版2020必修第三册)(已下线)专题19 平面与平面平行-《重难点题型·高分突破》(人教A版2019必修第二册)(已下线)8.5.3 平面与平面平行【第一练】“上好三节课,做好三套题“高中数学素养晋级之路
名校
6 . 在平面直角坐标系中,二元方程
的曲线为
,若存在一个定点
和一个定角
,使得曲线
上的所有点以
为中心顺时针(或逆时针)旋转角
,所得到的图形与原曲线重合,则称曲线
为旋转对称曲线.给出以下方程及其对应的曲线:
; ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/199da209c1d449712901f9d8fa751d10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d69accbf3a6fb381616ae0a7c43b160.png)
其中是旋转对称曲线的是__ (填上所有符合题意的曲线).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a6af0c85c1a59c2b77b6e550e1ef54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de72548ccc62e6e15d1be4afcdf09ad5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f036026cd92e9ad059c3f22a7658638.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/199da209c1d449712901f9d8fa751d10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55411ad607653d2077c4e6ca583f985.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d69accbf3a6fb381616ae0a7c43b160.png)
其中是旋转对称曲线的是
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解题方法
7 . 在
中,
,
,点
在
所在平面外,
平面
,且
,设
分别是线段
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/17/5c4393de-3ca6-4b65-95ad-2780d61ceb6c.png?resizew=184)
(1)求证:
是异面直线
与
的公垂线段.
(2)若过点
分别作
的垂线
,其中
分别是垂足,求四面体
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/829018a6ca0aff95d89e3f7cd943274e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/829f9180ddd9aa1a0ee0dc520f4e0b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3a51949f48ee8cf746851ba779b078e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cca777c664ecc22e40dff4ccae6b248.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/17/5c4393de-3ca6-4b65-95ad-2780d61ceb6c.png?resizew=184)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
(2)若过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ace5e8c3769ad8f370c86f879246c174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b14fa212bbddd28310d463fcdef7e62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6670479a0083dd2dfd5ad55b47b1ab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a406f24b5131eb7da9127750319e52.png)
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8 . 本次数学期末考试共三种题型:填空题、选择题、解答题,其中填空题满分54分,共有12道小题,前6题每小题4分,后6题每小题5分,每小题答对得满分,答错得零分,则学生解答填空题共有______ 种不同的可能分值.
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2023-02-16更新
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816次组卷
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6卷引用:上海市进才中学2020-2021学年高二下学期期末数学试题
上海市进才中学2020-2021学年高二下学期期末数学试题(已下线)第六章 计数原理 讲核心 01(已下线)第6章 计数原理(A卷·知识通关练)(1)(已下线)第6章 计数原理(基础、常考、易错、压轴)分类专项训练-【满分全攻略】2022-2023学年高二数学下学期核心考点+重难点讲练与测试(沪教版2020选修一+选修二)新疆阿克苏市新疆生产建设兵团第一师高级中学2022-2023学年高二下学期3月月考数学试题(已下线)第六章 计数原理(单元重点综合测试)-2023-2024学年高二数学单元速记·巧练(沪教版2020选择性必修第二册)
20-21高三下·上海浦东新·开学考试
名校
9 . 设等差数列
的公差为
,且
,若设
是从
开始的前
项数列的和,即
(
,
),
(
),如此下去,其中数列
是从第
(
)开始到第
(
)项为止的数列的和,即
(
,
).
(1)若数列
(
,
),试找出一组满足条件的
、
、
,使得:
;
(2)试证明对于数列
(
),一定可通过适当的划分,使所得的数列
中的各数都为平方数;
(3)若等差数列
中
,
,试探索该数列中是否存在无穷整数数列
(
),
,使得
为等比数列,如存在,就求出数列
;如不存在,则说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed47d7adb987d1b8a09eb7916b5df736.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c7eb49a823f757461cd5260757b088.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aedf14667dee4e314edafb7b9ee1c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c712eb92d59690142074ad3a5506260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3c450c252716acec1614ed7fdc88527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83fd8c2034d8006c3b3189d88a3b9bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/691a9cca1f5615b264e5753184d46530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28fc739eafa04616239443b3756c0cdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b590fa4f9abc64845e88da1dac2474f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa79b3013eea4df7c2db80b3a936902e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e90c998886b1483221a5b4941f6e874c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/777ba68a8bcd9801bac28930645512de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b320fc7cf9e9027b6435a34c1bf31564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f2b1e6d79cfa497903d69c43acf8a88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a56cc4f5be6b50145fc30e15abb3e4a.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0e4495f687884e309e0ce8a1b9c39be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df1d9b712b639c8b6809c9f3ae03706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32efe4eff75508cb93e828c735dcb695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7737fba2ccf2eaaffcf10a01f6ba9ef1.png)
(2)试证明对于数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80c1ed7b10ac7ca1cd81cdd39a8fcc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/444b973a18cd8f31d62e2c1868bb6a08.png)
(3)若等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c1344592c925b273f2cb9b9e47ebbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c7375f2b2acc8ffddece91deb6e68a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4870990e9d5da4a12a64b02015745522.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/444b973a18cd8f31d62e2c1868bb6a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/444b973a18cd8f31d62e2c1868bb6a08.png)
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21-22高三上·上海浦东新·期中
名校
解题方法
10 . 贾先生买了一套总价为
万元的商品房,首付
万元,其余
万元(本金)向银行申请贷款,贷款月利率
.从贷款后的第一个月后开始还款(即第一次还款日距贷款发放日正好一个月),
年还清.(精确到
元)
(1)若每月等额偿还本金(
万元),则贷款利息随本金逐月递减,还款额也逐月递减,其计算方法是:每月还款金额
(贷款本金/还款月数)
(本金
已归还本金累计额)
每月利率,请计算第
个月还款金额是多少元?
(2)为图方便,若每月还款金额相等,问每月应还款多少元?(注:如果上个月欠银行贷款
元,则一个月后,应还给银行固定数额
元,此时贷款余额为
元)
(3)请问
年后还清贷款时,用这两种不同还款方式归还贷款,实际还款总额分别是多少元?(不考虑时间价值等因素).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47543f9c51eb71b208b483444a4cff58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b685c556cc423e4833c1dc671a134cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c6bf2782515ceceb1503a845446ad7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f58b9b382aa6c0bb3d4967f13ca9b56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53c7539ed297ea63b9ace6f5cc58ca8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
(1)若每月等额偿还本金(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c6bf2782515ceceb1503a845446ad7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d4cd9a7068de096606d1ab991f5e6da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bfc339cf6dd66599db64fa3fa44e608.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2468403b3eba9e40bfa36f464e927738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
(2)为图方便,若每月还款金额相等,问每月应还款多少元?(注:如果上个月欠银行贷款
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1e165d6b0dfbe3fbe66486ee4e152d0.png)
(3)请问
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53c7539ed297ea63b9ace6f5cc58ca8.png)
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