For each natural number k, let  denotes the circle with radius k centemeters and center atorigin. On the circle , a particle moves k centimeters in the counter clockwise direction. After completing its motion on the particle moves to +1 in the radial direction. The motionof the particle continues in this manner. The particle starts at (1,0). If the particle crosses the positive direction of the x-axis for the first time on the circle  then n=....

Asked by Sunil Soni | 21st Sep, 2015, 11:57: PM

Expert Answer:

T h e space m o t i o n space o f space t h e space p a r t i c l e space i s space s h o w n space i n space t h e space f i g u r e space a b o v e. space I t space s t a r t s space f r o m space p o i n t space left parenthesis 1 comma space 0 right parenthesis space a n d space m o v e s space i n space a n t i space c l o c k w i s e space d i r e c t i o n space a l o n g space t h e space c i r c l e space o f space r a d i u s space 1 space a space d i s tan c e space o f space 1. H e n c e comma space t h e space a n g l e space s u b t e n d e d space b y space t h e space p a r t i c l e space a t space t h e space c e n t e r equals 1 over 1 equals 1 to the power of c S i m i l a r l y comma space w h e n space t h e space p a r t i c l e space m o v e s space i n space t h e space s e c o n d space c i r c l e space o f space r a d i u s space 2 comma a n g l e space s u b t e n d e d space a t space t h e space c e n t e r space i s space 2 over 2 equals 1 to the power of c. H e n c e comma space w h i l e space m o v i n g space i n space e v e r y space c i r c l e comma space t h e space p a r t i c l e space s u b t e n d s space a n space a n g l e space o f space 1 to the power of c space a t space t h e space c e n t e r. T o space c r o s s space X minus a x i s space f o r space t h e space f i r s t space t i m e comma space t h e space p a r t i c l e space m u s t space c r o s s space 2 pi to the power of c space a n g u l a r space d i s tan c e space a t space c e n t e r. 2 pi almost equal to 6.28. H e n c e comma space t h e space p a r t i c l e space c o m p l e t e s space t r a v e l l i n g space a l o n g space 6 space c i r c l e s space b e f o r e space c r o s sin g space X minus a x i s. H e n c e comma space w h i l e space t h e space p a r t i c l e space i s space t r a v e l l i n g space a l o n g space t h e space c i r c l e space o f space r a d i u s space 7 comma space i t space w i l l space c r o s s space X minus a x i s. H e n c e comma space r a d i u s space o f space t h e space c i r c l e space k equals 7 comma space w h e n space t h e space p a r t i c l e space c r o s s e s space X minus a x i s. left parenthesis n space i s space n o t space s p e c i f i e d space i n space t h e space q u e s t i o n. space A s s u m i n g space t h a t space w e space n e e d space t o space f i n d space t h e space v a l u e space o f space apostrophe k apostrophe space f o r space w h i c h space t h e space p a r t i c l e space c r o s s e s space X minus a x i s. space right parenthesis

Answered by satyajit samal | 22nd Sep, 2015, 01:30: PM