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2016年普通高等学校招生全国统一考试英语科听力部分

2016年普通高等学校招生全国统一考试英语科听力部分
2016年普通高等学校招生全国统一考试英语科听力部分

江西省南昌市2015-2016学年度第一学期期末试卷

(江西师大附中使用)高三理科数学分析

一、整体解读

试卷紧扣教材和考试说明,从考生熟悉的基础知识入手,多角度、多层次地考查了学生的数学理性思维能力及对数学本质的理解能力,立足基础,先易后难,难易适中,强调应用,不偏不怪,达到了“考基础、考能力、考素质”的目标。试卷所涉及的知识内容都在考试大纲的范围内,几乎覆盖了高中所学知识的全部重要内容,体现了“重点知识重点考查”的原则。 1.回归教材,注重基础

试卷遵循了考查基础知识为主体的原则,尤其是考试说明中的大部分知识点均有涉及,其中应用题与抗战胜利70周年为背景,把爱国主义教育渗透到试题当中,使学生感受到了数学的育才价值,所有这些题目的设计都回归教材和中学教学实际,操作性强。 2.适当设置题目难度与区分度

选择题第12题和填空题第16题以及解答题的第21题,都是综合性问题,难度较大,学生不仅要有较强的分析问题和解决问题的能力,以及扎实深厚的数学基本功,而且还要掌握必须的数学思想与方法,否则在有限的时间内,很难完成。 3.布局合理,考查全面,着重数学方法和数学思想的考察

在选择题,填空题,解答题和三选一问题中,试卷均对高中数学中的重点内容进行了反复考查。包括函数,三角函数,数列、立体几何、概率统计、解析几何、导数等几大版块问题。这些问题都是以知识为载体,立意于能力,让数学思想方法和数学思维方式贯穿于整个试题的解答过程之中。

二、亮点试题分析

1.【试卷原题】11.已知,,A B C 是单位圆上互不相同的三点,且满足AB AC →

=,则A BA C →→

?的最小值为( )

A .1

4- B .12-

C .34-

D .1-

【考查方向】本题主要考查了平面向量的线性运算及向量的数量积等知识,是向量与三角的典型综合题。解法较多,属于较难题,得分率较低。

【易错点】1.不能正确用OA ,OB

,OC 表示其它向量。

2.找不出OB 与OA 的夹角和OB

与OC 的夹角的倍数关系。

【解题思路】1.把向量用OA ,OB

,OC 表示出来。

2.把求最值问题转化为三角函数的最值求解。

【解析】设单位圆的圆心为O ,由AB AC →

=得,22

()()OB OA OC OA -=- ,因为

1OA OB OC ===

,所以有,OB OA OC OA ?=? 则()()AB AC OB OA OC OA ?=-?-

2OB OC OB OA OA OC OA =?-?-?+

21OB OC OB OA =?-?+

设OB 与OA 的夹角为α,则OB

与OC 的夹角为2α

所以,cos22cos 1AB AC αα?=-+ 211

2(cos )22

α=--

即,AB AC ? 的最小值为1

2

-,故选B 。

【举一反三】

【相似较难试题】【2015高考天津,理14】在等腰梯形ABCD 中,已知

//,2,1,60AB DC AB BC ABC ==∠= ,动点E 和F 分别在线段BC 和DC 上,且,1,,9BE BC DF DC λλ

== 则AE AF ? 的最小值为.

【试题分析】本题主要考查向量的几何运算、向量的数量积与基本不等式.运用向量的几何

运算求,AE AF ,体现了数形结合的基本思想,再运用向量数量积的定义计算AE AF ? ,体

现了数学定义的运用,再利用基本不等式求最小值,体现了数学知识的综合应用能力.是思维能力与计算能力的综合体现. 【答案】

2918

【解析】因为1,9DF DC λ= 12

DC AB =

119199918CF DF DC DC DC DC AB λλλλλ

--=-=-== ,

AE AB BE AB BC λ=+=+ ,19191818AF AB BC CF AB BC AB AB BC λλλλ

-+=++=++=+ ,

()

221919191181818AE AF AB BC AB BC AB BC AB BC

λλλλλλλλλ+++?????=+?+=+++?? ? ?????

19199421cos1201818

λλ

λλ++=

?++???

?2117172992181818λλ=

++≥+= 当且仅当2192λλ=即23λ=时AE AF ? 的最小值为

29

18

. 2.【试卷原题】20. (本小题满分12分)已知抛物线C 的焦点()1,0F ,其准线与x 轴的

交点为K ,过点K 的直线l 与C 交于,A B 两点,点A 关于x 轴的对称点为D . (Ⅰ)证明:点F 在直线BD 上; (Ⅱ)设8

9

FA FB →

?=

,求BDK ?内切圆M 的方程. 【考查方向】本题主要考查抛物线的标准方程和性质,直线与抛物线的位置关系,圆的标准方程,韦达定理,点到直线距离公式等知识,考查了解析几何设而不求和化归与转化的数学思想方法,是直线与圆锥曲线的综合问题,属于较难题。

【易错点】1.设直线l 的方程为(1)y m x =+,致使解法不严密。

2.不能正确运用韦达定理,设而不求,使得运算繁琐,最后得不到正确答案。 【解题思路】1.设出点的坐标,列出方程。 2.利用韦达定理,设而不求,简化运算过程。 3.根据圆的性质,巧用点到直线的距离公式求解。

【解析】(Ⅰ)由题可知()1,0K -,抛物线的方程为24y x =

则可设直线l 的方程为1x my =-,()()()112211,,,,,A x y B x y D x y -, 故2

14x my y x =-??

=?整理得2

440y my -+=,故121244

y y m y y +=??=? 则直线BD 的方程为()212221y y y y x x x x +-=--即2

222144y y y x y y ?

?-=- ?-??

令0y =,得1214

y y

x ==,所以()1,0F 在直线BD 上.

(Ⅱ)由(Ⅰ)可知121244

y y m y y +=??=?,所以()()2

12121142x x my my m +=-+-=-,

()()1211111x x my my =--= 又()111,FA x y →=-,()221,FB x y →

=-

故()()()2

1212121211584FA FB x x y y x x x x m →→

?=--+=-++=-,

则2

84

84,93

m m -=

∴=±,故直线l 的方程为3430x y ++=或3430x y -+=

21y y -==

故直线BD 的方程330x -=或330x -=,又KF 为BKD ∠的平分线,

故可设圆心()(),011M t t -<<,(),0M t 到直线l 及BD 的距离分别为3131

,54t t +--------------10分 由

31315

4t t +-=

得1

9t =或9t =(舍去).故圆M 的半径为31253

t r +=

= 所以圆M 的方程为2

21499x y ?

?-+= ??

?

【举一反三】

【相似较难试题】【2014高考全国,22】 已知抛物线C :y 2=2px(p>0)的焦点为F ,直线y =4与y 轴的交点为P ,与C 的交点为Q ,且|QF|=5

4|PQ|.

(1)求C 的方程;

(2)过F 的直线l 与C 相交于A ,B 两点,若AB 的垂直平分线l′与C 相交于M ,N 两点,且A ,M ,B ,N 四点在同一圆上,求l 的方程.

【试题分析】本题主要考查求抛物线的标准方程,直线和圆锥曲线的位置关系的应用,韦达定理,弦长公式的应用,解法及所涉及的知识和上题基本相同. 【答案】(1)y 2=4x.

(2)x -y -1=0或x +y -1=0. 【解析】(1)设Q(x 0,4),代入

y 2=2px ,得

x 0=8

p

所以|PQ|=8p ,|QF|=p 2+x 0=p 2+8

p

.

由题设得p 2+8p =54×8

p ,解得p =-2(舍去)或p =2,

所以C 的方程为y 2=4x.

(2)依题意知l 与坐标轴不垂直,故可设l 的方程为x =my +1(m≠0). 代入y 2=4x ,得y 2-4my -4=0. 设A(x 1,y 1),B(x 2,y 2), 则y 1+y 2=4m ,y 1y 2=-4.

故线段的AB 的中点为D(2m 2+1,2m), |AB|=

m 2+1|y 1-y 2|=4(m 2+1).

又直线l ′的斜率为-m ,

所以l ′的方程为x =-1

m y +2m 2+3.

将上式代入y 2=4x ,

并整理得y 2+4

m y -4(2m 2+3)=0.

设M(x 3,y 3),N(x 4,y 4),

则y 3+y 4=-4

m

,y 3y 4=-4(2m 2+3).

故线段MN 的中点为E ? ????

2m

2+2m 2+3,-2m ,

|MN|=

1+1

m 2|y 3-y 4|=4(m 2+1)2m 2+1

m 2

.

由于线段MN 垂直平分线段AB ,

故A ,M ,B ,N 四点在同一圆上等价于|AE|=|BE|=1

2|MN|,

从而14|AB|2+|DE|2=1

4|MN|2,即 4(m 2+1)2+

? ????2m +2m 2+? ??

??2

m 2+22=

4(m 2+1)2(2m 2+1)

m 4

化简得m 2-1=0,解得m =1或m =-1, 故所求直线l 的方程为x -y -1=0或x +y -1=0.

三、考卷比较

本试卷新课标全国卷Ⅰ相比较,基本相似,具体表现在以下方面: 1. 对学生的考查要求上完全一致。

即在考查基础知识的同时,注重考查能力的原则,确立以能力立意命题的指导思想,将知识、能力和素质融为一体,全面检测考生的数学素养,既考查了考生对中学数学的基础知识、基本技能的掌握程度,又考查了对数学思想方法和数学本质的理解水平,符合考试大纲所提倡的“高考应有较高的信度、效度、必要的区分度和适当的难度”的原则. 2. 试题结构形式大体相同,即选择题12个,每题5分,填空题4 个,每题5分,解答题8个(必做题5个),其中第22,23,24题是三选一题。题型分值完全一样。选择题、填空题考查了复数、三角函数、简易逻辑、概率、解析几何、向量、框图、二项式定理、线性规划等知识点,大部分属于常规题型,是学生在平时训练中常见的类型.解答题中仍涵盖了数列,三角函数,立体何,解析几何,导数等重点内容。

3. 在考查范围上略有不同,如本试卷第3题,是一个积分题,尽管简单,但全国卷已经不考查了。

四、本考试卷考点分析表(考点/知识点,难易程度、分值、解题方式、易错点、是否区分度题)

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