人教版选修【1-2】2.2.2《反证法》习题及答案

数学·选修1－2(人教A版)

2．1合情推理与演绎推理

2．2.2 反证法

?达标训练

1．“实数a，b，c不全为0”的意思为( )

A．a，b，c均不为0

B．a，b，c中至多有一个为0

C．a，b，c至少有一个为0

D．a，b，c至少有一个不为0

答案：D

2．下列命题中错误的是( )

A．三角形中至少有一个内角不小于60°

B．四面体的三组对棱都是异面直线

C．区间(a，b)上单调函数f(x)至多有一个零点

D．设a，b∈Z，若a＋b是奇数，则a，b中为奇数的一个也没有

答案：D

3．用反证法证明命题“如果a＞b，则3

a＞

b”时，假设内容

应是( )

A.3

a＝

b B.

a＜

C.3

a＝

b且

a＜

b D.

a＝

b或

a＜

解析：容易知道，“3

a＞

b”的否定是“

a＜

b或

a＝

b”，所以选D.

答案：D

4．如果两个实数之和为正数，则这两个数( )

A．至少有一个是正数

B．两个都是正数

C．一个是正数，一个是负数

D．两个都是负数

解析：假设两个都是负数，其和必为负数，矛盾，所以选A.

答案：A

5．用反证法证明命题“一个三角形不能有两个直角”的过程归纳为以下三个步骤：

①∠A＋∠B＋∠C＝90°＋90°＋∠C＞180°，这与三角形内角和为180°矛盾，所以∠A＝∠B＝90°不成立②所以一个三角形中不能有两个直角③假设∠A，∠B，∠C中有两个直角，不妨设∠A ＝∠B＝90°

其中顺序正确的是( )

A．①②③ B．①③②

C．③①② D．③②①

解析：根据反证法的步骤，容易知道选C.

答案：C

6．否定结论“至多有两个解”的说法中，正确的是( )

A．有一个解 B．有两个解

C．至少有三个解 D．至少有两个解

答案：C

?素能提高

1．a＞0，b＞0，c＞0，则三个数a＋1

，b＋

，c＋

( )

A．都大于2

B．都小于2

C．至少有一个数不大于2

D ．至少有一个数不小于2

解析：a ＋1b ＋b ＋1c ＋c ＋1a ＝? ????a ＋1a ＋? ????b ＋1b ＋? ????

c ＋1c ≥2＋2＋2＝

6.若三个数均小于2，则a ＋1b ＋b ＋1c ＋c ＋1

a ＜6，矛盾，故选D.

答案：D

2．设椭圆x 2a 2＋y 2

2＝1(a ＞b ＞0)的离心率为，右焦点为F (c,0)，

方程ax 2＋bx －c ＝0的两个实根分别为x 1和x 2，则点P (x 1，x 2)( )

A ．必在圆x 2＋y 2＝2上

B ．必在圆x 2＋y 2＝2外

C ．必在圆x 2＋y 2＝2内

D ．以上三种情形都有可能

解析：∵e ＝c a ＝1

，

∴a ＝2c ，∴b 2＝a 2－c 2＝3c 2.

假设点P (x 1，x 2)不在圆x 2＋y 2＝2内，

则x 21＋x 22≥2，但x 21＋x 22＝(x 1＋x 2)2

－2x 1x 2＝? ??

??－b a 2＋2c a ＝3c 24c 2＋

2c 2c ＝7

＜2，矛盾． ∴假设不成立，∴点P 必在圆x 2＋y 2＝2内．故选C. 答案：C

3．在用反证法证明数学命题时，如果原命题的否定项不止一个时，必须将结论的否定情况逐一驳倒，才能肯定原命题的结论是正确的．例如：在△ABC 中，若AB ＝AC ，P 是△ABC 内一点，∠APB ＞∠APC ，求证：∠BAP ＜∠CAP .用反证法证明时应分：假设________和________两类．

解析：因为小于的否定是不小于，所以应填∠BAP ＝∠CAP 和∠BAP ＞∠CAP .

答案：∠BAP ＝∠CAP ∠BAP ＞∠CAP

4．完成下面的反证法证题的全过程．

已知：设a1，a2，…，a7是1,2，…，7的一个全排列．

求证：乘积p＝(a1－1)(a2－2)…(a7－7)为偶数．

证明：假设p为奇数，则______①______均为奇数，

因为奇数个奇数之和为奇数，故有奇数

＝________②________

＝________③________

＝0.

但奇数≠偶数，这一矛盾说明，p为偶数．

答案：①a1－1，a2－2，…，a7－7

②(a1－1)＋(a2－2)＋…＋(a7－7)

③(a1＋a2＋...＋a7)－(1＋2＋ (7)

5．用反证法证明命题：“a，b∈N，ab可以被5整除，那么a，b中至少有一个能被5整除．”那么假设的内容是______________．解析：“至少有n个”的否定是“最多有n－1个”．

答案：a，b中没有一个能被5整除

6．用反证法证明：如果a＞b＞0，那么a＞b.

证明：假设a不大于b，即a＜b或a＝b.

∵a＞0，b＞0，∴a＜b?a·a＜b·a，a·b＜b·b?a ＜ab，ab＜b?a＜b，a＝b?a＝b.这些都同已知条件a＞b＞0矛盾，即a＞b.

7．已知a，b，c均为实数，且a＝x2－2y＋π

2，b＝y

2－2z＋

3，c

＝z2－2x＋π

6.求证：a，b，c中至少有一个大于0.

证明：假设a，b，c都不大于0，即a≤0，b≤0，c≤0，

得a＋b＋c≤0，

而a＋b＋c＝(x－1)2＋(y－1)2＋(z－1)2＋π－3≥π－3>0，即a＋b＋c>0，与a＋b＋c≤0矛盾，

∴a ，b ，c 中至少有一个大于0.

8．已知f (x )＝a x

＋x －2x ＋1

(a ＞1)，证明方程f (x )＝0没有负数根．

证明：假设x 0是f (x )＝0的负数根，则x 0＜0且x 0≠－1且ax 0＝－x 0－2x 0＋1

，

∴0＜ax 0＜1?0＜－x 0－2x 0＋1＜1，解得1

＜x 0＜2，这与x 0＜0矛盾，

故方程f (x )＝0没有负数根．

点评：(1)凡是“至少”、“唯一”或含有否定词的命题从正面突破往往比较困难，适宜用反证法．即“正难则反”．(2)反证法步骤：假设结论不成立→推出矛盾→假设不成立．

?品味高考 1．(2013·四川卷)设函数f (x )＝e x ＋x －a (a ∈R ，e 为自然对数的底数)．若存在b ∈[0,1]，使f (f (b ))＝b 成立，则a 的取值范围是( )

A ．[1，e]

B ．[1,1＋e]

C ．[e,1＋e]

D ．[0,1]

解析：易知f (x )＝e x ＋x －a 在定义域内是增函数，由f (f (b ))＝b ，猜想f (b )＝b .

反证法：若f (b )＞b ，则f (f (b ))＞f (b )＞b ，与题意不符，若f (b )＜b ，则f (f (b ))＜f (b )＜b ，与题意也不符，故f (b )＝b ，

即f (x )＝x 在[0,1]上有解． ∴e x ＋x －a ＝x ， a ＝e x －x 2＋x ，

令g (x )＝e x －x 2＋x ，g ′(x )＝e x －2x ＋1＝(e x ＋1)－2x ，当x ∈[0,1]时，e x ＋1≥2,2x ≤2，

g ′(x )≥0，∴g (x )在[0,1]上是增函数， ∴g (0)≤g (x )≤g (1)?1≤g (x )≤e，即1≤a ≤e，故选A. 答案：A

2．设直线l 1：y ＝k 1x ＋1，l 2：y ＝k 2x －1，其中实数k 1，k 2满足k 1k 2＋2＝0.

(1)证明l 1与l 2相交；

(2)证明l 1与l 2的交点在椭圆2x 2＋y 2＝1上．

分析：本题考查直线与直线的位置关系、线线相交的判断与证明、点在曲线上的判断与证明、椭圆方程等基本知识，考查推理论证能力和运算求解能力．

证明：(1)假设l 1与l 2不相交，则l 1与l 2平行，有k 1＝k 2，代入k 1k 2＋2＝0，得k 21＋2＝0，这与k 1为实数的事实相矛盾，从而k 1≠k 2，即l 1与l 2相交．

(2)证法一由方程组?????

y ＝k 1x ＋1，

y ＝k 2

x －1解得交点P 的坐标(x ，y )为

? ????2k 2

－k 1，k 2＋k 1k 2－k 1. 而2x 2

＋y 2

＝22

-k k ?? ?

2＋2

+-k k k k ?? ???

＝

8＋k 22＋k 21＋2k 1k 2k 22＋k 2

1－2k 1k 2＝k 21＋k 2

2＋4

k 21＋k 22

＋4＝1. 此即表明交点P (x ，y )在椭圆2x 2＋y 2＝1上．

证法二交点P 的坐标(x ，y )满足?????

y －1＝k 1x ，

y ＋1＝k 2x .

故知x ≠0，从而???

k 1＝y －1x ，

k 2

＝y ＋1

x .

代入k 1k 2＋2＝0，得 y －1x ·y ＋1

x ＋2＝0，

整理后，得2x 2＋y 2＝1，

所以交点P在椭圆2x2＋y2＝1上．s

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高中生物选修一生物技术实践知识点总结专题一传统发酵技术的应用课题一果酒和果醋的制作 1、发酵：通过微生物技术的培养来生产大量代谢产物的过程。 2、有氧发酵：醋酸发酵谷氨酸发酵无氧发酵：酒精发酵乳酸发酵 3、酵母菌是兼性厌氧菌型微生物真菌酵母菌的生殖方式：出芽生殖(主要) 分裂生殖孢子生殖 4、在有氧条件下，酵母菌进行有氧呼吸，大量繁殖。 C 6H 12O 6＋6O 2→6CO 2＋6H 2O 5、在无氧条件下，酵母菌能进行酒精发酵。 C 6H 12O 6→2C 2H 5OH ＋2CO 2 6、20℃左右最适宜酵母菌繁殖酒精发酵时一般将温度控制在18℃-25℃ 7、在葡萄酒自然发酵的过程中,起主要作用的是附着在葡萄皮表面的野生型酵母菌。在发酵过程中, 随着酒精浓度的提高,红葡萄皮的色素也进入发酵液,使葡萄酒呈现深红色。在缺氧呈酸性的发酵液中，酵母菌可以生长繁殖，而绝大多数其他微生物都因无法适应这一环境而受到制约。 8、醋酸菌是单细胞细菌(原核生物),代谢类型是异养需氧型,生殖方式为二分裂 9、当氧气、糖源都充足时，醋酸菌将葡萄汁中的糖分解成醋酸；当缺少糖源时，醋酸菌将乙醇变为乙醛，再将乙醛变为醋酸。 2C 2H 5OH ＋4O 2→CH 3COOH ＋6H 2O 10、控制发酵条件的作用①醋酸菌对氧气的含量特别敏感，当进行深层发酵时，即使只是短时间中断通入氧气，也会引起醋酸菌死亡。②醋酸菌最适生长温度为30～35℃，控制好发酵温度，使发酵时间缩短，又减少杂菌污染的机会。③有两条途径生成醋酸：直接氧化和以酒精为底物的氧化。 11、实验流程：挑选葡萄→冲洗→榨汁→酒精发酵→果酒（→醋酸发酵→果醋） 12、酒精检验：果汁发酵后是否有酒精产生，可以用重铬酸钾来检验。在酸性条件下，重铬酸钾与酒精反应呈现灰绿色。先在试管中加入发酵液2ｍL ，再滴入物质的量浓度为3mol/L 的H 2SO 43滴，振荡混匀，最后滴加常温下饱和的重铬酸钾溶液3滴，振荡试管，观察颜色 13、充气口是在醋酸发酵时连接充气泵进行充气用的；排气口是在酒精发酵时用来排出二氧化碳的；出料口是用来取样的。排气口要通过一个长而弯曲的胶管与瓶身相连接，其目的是防止空气中微生物的污染。开口向下的目的是有利于二氧化碳的排出。使用该装置制酒时，应该关闭充气口；制醋时，应该充气口连接气泵，输入氧气。疑难解答（1）你认为应该先冲洗葡萄还是先除去枝梗？为什么？应该先冲洗，然后再除去枝梗，以避免除去枝梗时引起葡萄破损，增加被杂菌污染的机会。（2）你认为应该从哪些方面防止发酵液被污染？如：要先冲洗葡萄，再除去枝梗；榨汁机、发酵装置要清洗干净，并进行酒精消毒；每次排气时只需拧松瓶盖，不要完全揭开瓶盖等。（3）制葡萄酒时，为什么要将温度控制在18～25℃？制葡萄醋时，为什么要将温度控制酶酶酶

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人教版英语选修六课文原文

Unit 1A SHORT HISTORY OF WESTERN PAINTING Art is influenced by the customs and faith of a people. Styles in Western art have changed many times. As there are so many different styles of Western art, it would be impossible to describe all of them in such a short text. Consequently, this text will describe only the most important ones. Starting from the sixth century AD. The Middle Ages(5th to the 15th century AD) During the Middle Ages, the main aim of painters was to represent religious themes. A conventional artistof this period was not interested in showing nature and people as they really were. A typical picture at this time was full of religious symbols, which created a feeling of respect and love for God. But it was evident that ideas were changing in the 13th century when painters like Giotto di Bondone began to paint religious scenes in a more realistic way. The Renaissance(15th to 16th century) During the Renaissance, new ideas and values gradually replaced those held in the Middle Ages. People began to concentrate less on religious themes and adopt a more humanistic attitude to life.At the same time painters returned to classical Roman and Greek ideas about art. They tried to paint people and nature as they really were. Rich people wanted to possess their own paintings, so they coule decorate their superb palaces and great houses. They paid famous artists to paint pictures of themselves, their houses and possessions as well as their activities and achievements. One of the most important discoveries during this period was how to draw things in perspective. This technique was first used by Masaccio in 1428. When people first saw his paintings, they were convinced that they were looking through a hole in a wall at a real scene. If the rules of perspective had not been discovered, no one would have been able to paint such realistic pictures. By coincidence,oil paints were also developed at this time, which made the colours used in paintings look richer and deeper. Without the new paints and the new technique, we would not be able to see the many great masterpieces for which this period is famous. Impressionism(late 19h to early 20th century) In the late 19th century, Europe changed a great deal,from a mostly agricultural society to a mostly industrial one. Many people moved from the countryside to the new cities. There were many new inventions and social changes also led to new painting styles. Among the painters who broke away from the traditional style of painting were the Impressionists, who lived and worke in Paris. The Impressionists were the first painters to work outdoors. They were eager to show how light and shadow fell on objects at different times of day. However, because natural light changes so quickly, the Impressionists had to paint quickly. Their paintings were not as detailed as those of earlier painters. At first, many people disliked this style of painting and became very angry about it. They said that the painters were careless and their paintings were ridiculous. Modern Art(29th century to today) At the time they were created, the Impressionist paintings were controversial, but today they are accepeted as the beginning of what we call "modern art". This is

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选修6 Unit 1 Art-Reading A SHORT HISTORY OF WESTERN PAINTING Art is influenced by the customs and faith of a people. Styles in Western art have changed many times. As there are so many different styles of Western art, it would be impossible to describe all of them in such a short text. Consequently, this text will describe only the most important ones, starting from the sixth century AD. The Middle Ages (5th to the 15th century AD) During the Middle Ages, the main aim of painters was to represent religious themes. A conventional artist of this period was not interested in showing nature and people as they really were. A typical picture at this time was full of religious symbols, which created feeling of respect and love for God. But it was evident that ideas were changing in the 13th century when painters like Giotto di Bondone began to paint religious scenes in a more realistic way. The Renaissance (15th to 16th century) During the Renaissance, new ideas and values graduallv replaced those held in the Middle Ages.People began to concentrate less on religious themes and adopt a more humanistic attitude to life. At the same time painters returned to classical Roman and Greek ideas about art. They tried to paint people and nature as they really were. Rich people wanted to possess t heir own paintings, so they could decorate their superb palaces and great houses. They paid famous artists to paint pictures of themselves, their houses and possessions as well as their activities and achievements. One of the most important discoveries during this period was how to draw things in perspective. This technique was first used by Masaccio in 1428. When people first saw his paintings, they were convinced that they were looking through a hole in a wall at a real scene. If the roles of perspective had not been discovered, no one would have been able to paint such realistic pictures. By coincidence, oil paints were also developed at this time, which made the colours used in paintings look richer and deeper. Without the new paints and the new technique, we would not be able to see the many great masterpieces for which this period is famous. Impressionism (late 19th to early 20th century) In the late 19th century, Europe changed a great deal. from a mostly agricultural society to a mostly industrial one. Many people moved from the countryside to the new cities. There were many new inventions and social changes. Naturally, these changes also led to new painting styles. Among the painters who broke away from the traditional style of painting were the Impressionists, who lived and worked in Paris.

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2.酵母菌繁殖的最适温度20℃左右，酒精发酵一般控制在18～25℃。 3.自然发酵过程中，起作用的主要是附着于葡萄皮上的野生型酵母菌。也可以在果汁中加入人工培养的酵母菌。(二)果醋制作的原理 1.菌种是醋酸菌，属于原核生物，新陈代谢类型为异养需氧型。只有在氧气充足时，才能进行旺盛的生命活动。变酸的酒表面观察到的菌膜就是醋酸菌在液面大量繁殖形成的。 2.当氧气、糖源都充足时，醋酸菌将葡萄汁中的糖分解成醋酸，当缺少糖源时，醋酸菌将乙醇变为乙醛，再将乙醛变为醋酸，反应简式为C 2H 5OH+O 2→CH 3COOH+H 2O 。 3.醋酸菌的最适合生长温度为30～35℃。 4.菌种来源：到生产食醋的工厂或菌种保藏中心购买，或从食醋中分离醋酸菌。二.实验设计 1.流程图挑选葡萄→冲洗→榨汁→酒精发酵→醋酸发酵 ↓↓ 果酒果醋 2.制作实例 (1)实验材料葡萄、榨汁机、纱布、醋酸菌(或醋曲)、发酵瓶(如右图)、气泵、体积分数为70%的酒精等。 (2)实验步骤酶

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1专题一传统发酵技术的应用课题1 果酒和果醋的制作 1、发酵：通过微生物技术的培养来生产大量代谢产物的过程。 2、有氧发酵：醋酸发酵谷氨酸发酵· 无氧发酵：酒精发酵乳酸发酵 3、酵母菌是兼性厌氧菌型微生物真菌·酵母菌的生殖方式：出芽生殖(主要)分裂生殖孢子生殖 4、在有氧条件下，酵母菌进行有氧呼吸，大量繁殖。 C6H12O6+6O2→6CO2+6H2O 5、在无氧条件下，酵母菌能进行酒精发酵。 C6H12O6→2C2H5OH+2CO2 6、20℃左右最适宜酵母菌繁殖酒精发酵时一般将温度控制在18℃-25℃ 7、在葡萄酒自然发酵的过程中,起主要作用的是附着在葡萄皮表面的野生型酵母菌.在发酵过程中,随着酒精浓度的提高,红葡萄皮的色素也进入发酵液,使葡萄酒呈现深红色.在缺氧呈酸性的发酵液中，酵母菌可以生长繁殖，而绝大多数其他微生物都因无法适应这一环境而受到制约。 8、醋酸菌是单细胞细菌(原核生物),代谢类型是异养需氧型,生殖方式为二分裂 9、当氧气、糖源都充足时，醋酸菌将葡萄汁中的糖分解成醋酸;当缺少糖源时，醋酸菌将乙醇变为乙醛，再将乙醛变为醋酸。 1

2C2H5OH+4O2→CH3COOH+6H2O 10、控制发酵条件的作用①醋酸菌对氧气的含量特别敏感，当进行深层发酵时，即使只是短时间中断通入氧气，也会引起醋酸菌死亡。②醋酸菌最适生长温度为30～35℃，控制好发酵温度，使发酵时间缩短，又减少杂菌污染的机会。③有两条途径生成醋酸：直接氧化和以酒精为底物的氧化。 11、实验流程：挑选葡萄→冲洗→榨汁→酒精发酵→果酒(→醋酸发酵→果醋) 12、酒精检验：果汁发酵后是否有酒精产生，可以用重铬酸钾来检验。在酸性条件下，重铬酸钾与酒精反应呈现灰绿色。先在试管中加入发酵液2mL，再滴入物质的量浓度为3mol/L的H2SO43滴，振荡混匀，最后滴加常温下饱和的重铬酸钾溶液3滴，振荡试管，观察颜色. 13、充气口是在醋酸发酵时连接充气泵进行充气用的;排气口是在酒精发酵时用来排出二氧化碳的;出料口是用来取样的。排气口要通过一个长而弯曲的胶管与瓶身相连接，其目的是防止空气中微生物的污染。开口向下的目的是有利于二氧化碳的排出。使用该装置制酒时，应该关闭充气口;制醋时，应该充气口连接气泵，输入氧气。课题2 腐乳的制作 1、多种微生物参与了豆腐的发酵，如青霉、酵母、曲霉、毛霉等，其中起主要作用的是毛霉。毛霉是一种丝状真菌。代谢类型是异养需氧型。生殖方式是孢子生殖。营腐生生活。 2、原理：毛霉等微生物产生的蛋白酶能将豆腐中的蛋白质分解成小分子的肽和氨基酸;脂肪酶可将脂肪水解为甘油和脂肪酸。 3、实验流程：让豆腐上长出毛霉→加盐腌制→加卤汤装瓶→密封腌制 4、酿造腐乳的主要生产工序是将豆腐进行前期发酵和后期发酵。前期发酵的主要作用：（1）创造条件让毛霉生长。 2

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