Question
Fri April 03, 2015 By: Chaitali Arlekar

respected sir,
please give step by step explaination for the following questions:-
1.find the number of positive integral roots of the equation ax=a+10x
2.show that there is no rational number whose square is 2.

Expert Reply
Vimala Ramamurthy
Sat April 04, 2015
 
 
1. C o n s i d e r space t h e space g i v e n space e q u a t i o n colon a x equals a plus 10 x rightwards double arrow a x minus a minus 10 x equals 0 rightwards double arrow a x minus a minus 10 x plus 10 equals 10 rightwards double arrow a open parentheses x minus 1 close parentheses minus 10 open parentheses x minus 1 close parentheses equals 10 rightwards double arrow open parentheses a minus 10 close parentheses open parentheses x minus 1 close parentheses equals 10 W e space n e e d space p o s i t i v e space i n t e g r a l space v a l u e s space o f space x comma space x minus 1 greater or equal than 0 space s u c h space t h a t space x minus 1 i s space a space f a c t o r space o f space 10. T h u s comma space x minus 1 equals 1 comma 2 comma 5 space a n d space 10 rightwards double arrow x equals 2 comma 3 comma 6 comma 11 T h u s space t h e r e space a r e space 4 space p o s i t i v e space i n t e g r a l space v a l u e s space o f space x  2. W e space s h a l l space p r o v e space t h i s space b y space t h e space m e t h o d space o f space c o n t r a d i c t i o n. S u p p o s e space a s s u m e space t h a t space t h e r e space i s space a space r a t i o n a l space n u m b e r space p over q comma space w h e r e space p space a n d space q space d o n t space h a v e space c o m m o n space f a c t o r s comma space s u c h t h a t space open parentheses p over q close parentheses squared equals 2 rightwards double arrow p squared over q squared equals 2 rightwards double arrow p squared equals 2 q squared rightwards double arrow p squared space i s space e v e n comma space a n d space h e n c e comma space space 2 q squared space i s space a l s o space e v e n rightwards double arrow p space i s space e v e n...... left parenthesis 1 right parenthesis  rightwards double arrow t h e r e space e x i s t s space s o m e space apostrophe m apostrophe space s u c h space t h a t comma space p equals 2 m rightwards double arrow p squared equals open parentheses 2 m close parentheses squared rightwards double arrow 2 q squared equals 4 m squared rightwards double arrow q squared equals 2 m squared rightwards double arrow q squared space i s space e v e n rightwards double arrow q space i s space e v e n...... left parenthesis 2 right parenthesis  F r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space g o t space t h a t space 2 space i s space a space c o m m o n space f a c t o r space o f space p space a n d space q. B u t space t h i s space c o n t r a d i c t s space t h e space f a c t space t h a t space p space a n d space q space h a v e space n o space c o m m o n space f a c t o r space o t h e r space t h a n space 1. H e n c e space o u r space a s s u m p t i o n space t h a t space t h e r e space i s space a space r a t i o n a l space n u m b e r space w h o s e space s q u a r e space i s space 2 space i s space w r o n g. T h u s comma space i t space h a s space b e e n space p r o v e d space t h a t space t h e r e space i s space n o space r a t i o n a l space n u m b e r space w h o s e space s q u a r e space i s space 2.
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