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xpj1251.com:2017届高三数学理一轮总复习课时跟踪检测:36不等关系与不等式(江苏专用).doc

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课时跟踪检测(三十六) 不等关系与不等式 ?一抓基础,多练小题做到眼疾手快 1.设 a,b∈[0,+∞),A= a+ b,B= a+b,则 A,B 的大小关系是________. 解析:由题意得,B2-A2=-2 ab≤0, 且 A≥0,B≥0,可得 A≥B. 答案:A≥B 2.设 a=2- 5,b= 5-2,c=5-2 5,则 a,b,c 之间的大小关系为________. 解析: a=2- 5= 4- 5<0, 所以 b>0.c=5-2 5= 25- 20>0.b-c=3 5-7= 45 - 49<0.所以 c>b>a. 答案:c>b>a 3.(2016· 西安八校联考 )“x1>3 且 x2>3”是“x1+x2>6 且 x1x2>9”的________条件(填“充 要”“充分不必要”“必要不充分”“既不充分又不必要”). 1 解析:x1>3,x2>3? x1+x2>6,x1x2>9;反之不成立,例如 x1= ,x2=20. 2 答案:充分不必要 3? 4.现给出三个不等式:①a2+1>2a;②a2+b2>2? ?a-b-2?; ③ 7+ 10> 3+ 14.其中恒成立的不等式共有________个.

解析:因为 a2-2a+1=(a-1)2≥0,所以①不恒成立;对于②,a2+b2-2a+2b+3=(a - 1)2 + (b + 1)2 + 1>0 ,所以②恒成立;对于③,因为 ( 7 + 10)2 - ( 3 + 14)2 = 2 70 - 2 42>0,且 7+ 10>0, 3+ 14>0,所以 7+ 10> 3+ 14,即③恒成立. 答案:2 π? β ? π? 5.设 α∈? ?0,2?,β∈?0,2?,那么 2α-3的取值范围是________. β π π β π β 解析:由题设得 0<2α<π,0≤ ≤ ,所以- ≤- ≤0,所以- <2α- <π. 3 6 6 3 6 3 π ? 答案:? ?-6,π?

?二保高考,全练题型做到高考达标 1.若 a1<a2,b1<b2,则 a1b1+a2b2 与 a1b2+a2b1 的大小关系是________. 解析:(a1b1+a2b2)-(a1b2+a2b1)=(a1-a2)(b1-b2)>0. 答案:a1b1+a2b2>a1b2+a2b1 2.设 a=lg e,b=(lg e)2,c=lg e,则三者大小关系是________. 1 1 e = lg e = a<a ,又 b= (lg 2 2

解析: 0<lg e<1 ,即 0<a<1 ; b = (lg e)2 = a2<a ; c= lg

1 e)2<lg 10· lg e= lg e=c,因此 a>c>b. 2 答案:a>c>b π 3.若角 α,β 满足- <α<β<π,则 α-β 的取值范围是________. 2 π π π 解析:∵- <α<π,- <β<π,∴-π<-β< , 2 2 2 3π 3π ∴- <α-β< . 2 2 3π 又∵α<β,∴α-β<0,从而- <α-β<0. 2 3π ? 答案:? ?- 2 ,0? 1 1 4. (2016· 南京名校联考)设 a, b 是实数, 则“a>b>1”是“a+ >b+ ”的________条件(填“充 a b 要”“充分不必要”“必要不充分”“既不充分又不必要”). 1 ?a-b? ? ab-1? 1 1 1 b+ ? = b+ ? = 解析:因为 a + - ? ,若 a>b>1 ,显然 a + - ? a ? b? ab a ? b? ?a-b? ? ab-1? 1 2 1 1 >0,则充分性成立,当 a= ,b= 时,显然不等式 a+ >b+ 成立,但 ab 2 3 a b a>b>1 不成立,所以必要性不成立. 答案:充分不必要 5.某高速公路对行驶的各种车辆的速度 v 的最大值限速为 120 km/h,行驶过程中,同 一车道上的车间距 d 不得小于 10 m,用不等式表示为________. 解析:最大值即为小于或等于, 不小于即为大于或等于.
?v≤120, ? 故用不等式表示为? ?d≥10. ? ?v≤120, ? 答案:? ?d≥10 ?

6.用一段长为 30 m 的篱笆围成一个一边靠墙的矩形菜园,墙长 18 m,要求菜园的面 积不小于 216 m2,靠墙的一边长为 x m,其中的不等关系可用不等式(组)表示为________. 30-x 解析:矩形靠墙的一边长为 x m,则另一边长为 m, 2 0<x≤18, ? ? x 15- ?m,根据题意知? ? 即? x? 2? ? ? ?x?15-2?≥216.

0<x≤18, ? ? 答案:? ? x? ? ?x?15-2?≥216 7.已知存在实数 a 满足 ab2>a>ab,则实数 b 的取值范围是__________. 解析:∵ab2>a>ab,∴a≠0, 当 a>0 时,b2>1>b,
2 ? ?b >1, 即? 解得 b<-1; ?b<1, ?

当 a<0 时,b2<1<b,
?b2<1, ? 即? 此式无解. ?b>1, ?

综上可得实数 b 的取值范围为(-∞,-1). 答案:(-∞,-1) a b 1 1 8.已知 a+b>0,则 2+ 2与 + 的大小关系是________. b a a b 1 1? a-b b-a a-b?2 a b ? 12- 12?=?a+b?2 ? + = 2 + 2 =(a-b)· 解析: 2+ 2-? . 2 ?b a ? b a ?a b? b a ab ∵a+b>0,(a-b)2≥0, ∴ ?a+b? ? a-b?2 ≥0. 2 2 ab

a b 1 1 ∴ 2+ 2≥ + . b a a b a b 1 1 答案: 2+ 2≥ + b a a b e e 9.若 a>b>0,c<d<0,e<0.求证: > . ?a-c?2 ?b-d?2 证明:∵c<d<0,∴-c>-d>0. 又∵a>b>0,∴a-c>b-d>0.∴(a-c)2>(b-d)2>0. ∴0< 1 1 e e < .又∵e<0,∴ > . ?a-c?2 ?b-d?2 ?a-c?2 ?b-d?2

1 1 1 10.(2016· 南京学情调研)(1)设 x≥1,y≥1,证明:x+y+ ≤ + +xy; xy x y (2)设 1<a≤b≤c,证明:logab+logbc+logca≤logba+logcb+logac. 证明:(1)由于 x≥1,y≥1, 1 1 1 所以 x+y+ ≤ + +xy?xy(x+y)+1≤y+x+(xy)2. xy x y 将上式中的右式减左式, 得[y+x+(xy)2]-[xy(x+y)+1]

=[(xy)2-1]-[xy(x+y)-(x+y)] =(xy+1)(xy-1)-(x+y)(xy-1) =(xy-1)(xy-x-y+1) =(xy-1)(x-1)(y-1). 因为 x≥1,y≥1,所以(xy-1)(x-1)(y-1)≥0, 从而所要证明的不等式成立. 1 1 (2)设 logab=x,logbc=y,x≥1,y≥1,由对数的换底公式得 logca= ,logba= ,logcb xy x 1 = ,logac=xy. y 1 1 1 于是,所要证明的不等式即为 x+y+ ≤ + +xy. xy x y 由(1)知所要证明的不等式成立.

?三上台阶,自主选做志在冲刺名校 1.若 1<α<3,-4<β<2,则 α-|β|的取值范围是________. 解析:∵-4<β<2,∴0≤|β|<4.∴-4<-|β|≤0. ∴-3<α-|β|<3. 答案:(-3,3) c 2.(2016· 合肥质检)已知△ABC 的三边长分别为 a,b,c,且满足 b+c≤3a,则 的取值 a 范围为________. a<b+c≤3a, ? ? 解析:由已知及三角形三边关系得?a+b>c, ? ?a+c>b,

? ? bc ∴?1+a>a, c b ? > , ?1+a a
答案:(0,2)

b c 1< + ≤3, a a

?1<a+a≤3, ∴? c b ?-1<a-a<1,

b

c

c c 两式相加得,0<2× <4,∴ 的取值范围为(0,2). a a

3.某单位组织职工去某地参观学习需包车前往.甲车队说:“如果领队买一张全票,其 余人可享受 7.5 折优惠.”乙车队说:“你们属团体票,按原价的 8 折优惠.”这两个车队的 原价、车型都是一样的,试根据单位去的人数比较两车队的收费哪家更优惠.

解:设该单位职工有 n 人(n∈N*),全票价为 x 元,坐甲车需花 y1 元,坐乙车需花 y2 元, 3 1 3 4 则 y1=x+ x· (n-1)= x+ xn,y2= nx. 4 4 4 5 1 3 4 所以 y1-y2= x+ xn- nx 4 4 5 1 1 = x- nx 4 20 n 1 1- ? . = x? 4 ? 5? 当 n=5 时,y1=y2; 当 n>5 时,y1<y2; 当 n<5 时,y1>y2. 因此当单位去的人数为 5 人时,两车队收费相同; 多于 5 人时,甲车队更优惠; 少于 5 人时,乙车队更优惠.


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