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2016年普通高等学校招生全国统一考试(全国新课标Ⅲ卷)数学试题 (理科)解析版

2016年普通高等学校招生全国统一考试(全国新课标Ⅲ卷)数学试题 (理科)解析版
2016年普通高等学校招生全国统一考试(全国新课标Ⅲ卷)数学试题 (理科)解析版

绝密★启封并使用完毕前

注意事项:

1.本试卷分第Ⅰ卷(选择题)和第Ⅱ卷(非选择题)两部分。第Ⅰ卷1至3页,第Ⅱ卷3至5页。

2.答题前,考生务必将自己的姓名、准考证号填写在本试题相应的位置。

3.全部答案在答题卡上完成,答在本试题上无效。

4. 考试结束后,将本试题和答题卡一并交回。

第Ⅰ卷

选择题:本大题共12小题,每小题5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)设集合

{}{}

|(2)(3)0,|0S x x x T x x =--≥=> ,则S T =( )

(A) [2,3] (B)(-∞ ,2]U [3,+∞) (C) [3,+∞) (D)(0, 2]U [3,+∞) 【答案】

D

考点:1、不等式的解法;2、集合的交集运算.

【技巧点拨】研究集合的关系,处理集合的交、并、补的运算问题,常用韦恩图、数轴等几何工具辅助解题.一般地,对离散的数集、抽象的集合间的关系及运算,可借助韦恩图,而对连续的集合间的运算及关系,可借助数轴的直观性,进行合理转化.

(2)若i 12z =+,则4i

1zz =

-( )

(A)1 (B) -1 (C)i (D) i -

【答案】C 【解析】

试题分析:4i 4i

i

(12i)(12i)11zz ==+---,故选C .

考点:1、复数的运算;2、共轭复数.

【举一反三】复数的加、减法运算中,可以从形式上理解为关于虚数单位

“i ”的多项式合并同类项,复数的乘法与多项式的乘法相类似,只是在结果中把2

i 换成-1.复数除法可类比实数运算的分母有理化.复数加、减法的几何意义可依平面向量的加、减法的几何意义进行理解.

(3)已知向量

13(,)2BA =uu v ,31(,)

2BC =uu u v ,则ABC ∠=( ) (A)30? (B)45? (C)60? (D)120? 【答案】A

考点:向量夹角公式.

【思维拓展】(1)平面向量a

与b 的数量积为

·cos a b a b θ

=,其中θ是a 与b 的夹角,要

注意夹角的定义和它的取值范围:0180θ?≤≤?;(2)由向量的数量积的性质有||=a a a ·

·cos a b a b

θ=

,·

0a b a b ?⊥=,因此,利用平面向量的数量积可以解决与长度、角度、垂直等有关的问题.

(4)某旅游城市为向游客介绍本地的气温情况,绘制了一年中月平均最高气温和平均最低气温的雷达图.图中A 点表示十月的平均最高气温约为15C ?,B 点表示四月的平均最低气温约为5C ?.下面叙述不正确的是( )

(A)各月的平均最低气温都在0C ?以上 (B)七月的平均温差比一月的平均温差大 (C)三月和十一月的平均最高气温基本相同 (D)平均气温高于20C ?的月份有5个

【答案】

D

考点:1、平均数;2、统计图.

【易错警示】解答本题时易错可能有两种:(1)对图形中的线条认识不明确,不知所措,只觉得是两把雨伞重叠在一起,找不到解决问题的方法;(2)估计平均温差时易出现错误,错选B .

(5)若

3

tan 4α=

,则2cos 2sin 2αα+=( )

(A)6425 (B) 4825 (C) 1 (D)1625

【答案】A 【解析】

试题分析:由

3tan 4α=

,得34sin ,cos 55αα==或34

sin ,cos 55αα=-=-,所以2161264

cos 2sin 24252525αα+=+?=

,故选A .

考点:1、同角三角函数间的基本关系;2、倍角公式.

【方法点拨】三角函数求值:①“给角求值”将非特殊角向特殊角转化,通过相消或相约消去

非特殊角,进而求出三角函数值;②“给值求值”关键是目标明确,建立已知和所求之间的联系.

(6)已知43

2a =,25

4b =,13

25c =,则( )

(A )b a c << (B )a b c << (C )b c a << (D )c a b << 【答案】A 【解析】

试题分析:因为4

223

3

5

244a b ==>=,1223

3

3

2554c a ==>=,所以b a c <<,故选A . 考点:幂函数的图象与性质.

【技巧点拨】比较指数的大小常常根据三个数的结构联系相关的指数函数与对数函数、幂函

数的单调性来判断,如果两个数指数相同,底数不同,则考虑幂函数的单调性;如果指数不同,底数相同,则考虑指数函数的单调性;如果涉及到对数,则联系对数的单调性来解决.

(7)执行下图的程序框图,如果输入的46a b ==,,那么输出的n =( )

(A )3 (B )4 (C )5 (D )6 【答案】B

考点:程序框图.

【注意提示】解决此类型时要注意:第一,要明确是当型循环结构,还是直到型循环结构.根据各自的特点执行循环体;第二,要明确图中的累计变量,明确每一次执行循环体前和执行循环体后,变量的值发生的变化;第三,要明确循环体终止的条件是什么,会判断什么时候终止循环体.

(8)在ABC △中,

π4B =

,BC 边上的高等于1

3BC

,则cos A =( )

(A ) (B ) (C )- (D )

-

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