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湖南省衡阳县第四中学2018_2019学年高一数学12月月考试题扫描版无答案_图文

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16.解 利用数轴,分别表示出全集 U 及集合 A,B,先求出?UA 及?UB,再求解.

则?UA={x|x≤-2,或 3≤x≤4}, ?UB={x|x<-3,或 2<x≤4}. 所以 A∩B={x|-2<x≤2}; (?UA)∪B={x|x≤2,或 3≤x≤4}; A∩(?UB)={x|2<x<3}.

17.解:(1)

1 原式=2×32

×???32???13

1 ×126

?1 =21+??-3

1 ??+3
?

1 ×32

1 +3

1 +6

=2×3=6.

??

(2) 原式=(lg 5)2+lg 2(2-lg 2)

=(lg 5)2+(1+lg 5)lg 2

=(lg 5)2+lg 2·lg 5+lg 2

=(lg 5+lg 2)·lg 5+lg 2

=lg 5+lg 2=1.

18.[解析] ∵E、F 分别是 AA1 和 BB1 的中点,∴EF∥AB. 又 AB?平面 EFGH,EF? 平面 EFGH, ∴AB∥平面 EFGH. 又 AB? 平面 ABCD, 平面 ABCD∩平面 EFGH=GH,∴AB∥GH.

19.[解析] 正方体的表面积为 4×4×6=96(cm2), 圆柱的侧面积为 2π ×1×1≈6.28(cm2), 则挖洞后几何体的表面积约为 96+6.28=102.28(cm2).

20.解:(1)因为 f(x)是 R 上的奇函数, 所以 f(0)=0. 当 x<0 时,-x>0,
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f(x)=-f(-x)=-???12???-x=-2x, -2x,x<0,
??0,x=0, ??? 所以 f(x)= ???12???x,x>0.
(2)函数图象如图所示,
通过函数的图象可以知道,f(x)的单调递减区间是(-∞,0),(0,+∞).
四.附加题(共 20 分) 1.[证明] 取 AD 的中点 G,连接 PG,BG.因为 PA=PD,
所以 AD⊥PG. 设菱形 ABCD 边长为 1.
1 在△ABG 中,因为∠GAB=60°,AG=2,AB=1, 所以∠AGB=90°,即 AD⊥GB. 又 PG∩GB=G,所以 AD⊥平面 PGB, 从而 AD⊥PB. 因为 E,F 分别是 BC,PC 的中点,所以 EF∥PB,从而 AD⊥EF. 易证 DE∥GB,且 AD⊥GB, 所以 AD⊥DE,因为 DE∩EF=E, 所以 AD⊥平面 DEF.
2.解:(1)当 x<0 时,f(x)=0; 1
当 x≥0 时,f(x)=2x-2x. 1
由条件可知 2x-2x=2,即 22x-2·2x-1=0,
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解得 2x=1± 2. 因为 2x>0,所以 x=log2(1+ 2).
(2)当 t∈[1,2]时,2t???22t-212t???+m???2t-21t???≥0, 即 m(22t-1)≥-(24t-1). 因为 22t-1>0,所以 m≥-(22t+1). 因为 t∈[1,2],所以-(1+22t)∈[-17,-5], 故 m 的取值范围是[-5,+∞).
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