functions

This is one of the best questions i have seen in a long time..

someone pls try it!

i dont think so priyam!

let f(x) ={-1 + sin (k1pie x) } , x is rational
{1 + cos (k2pie x) } , x is irrational
if f(x) is periodic function , then which of the following is correct
(a) either k1 , k2 ε rational or k1 , k2 ε irrational
(b) k1 , k2 ε rational only
(c) k1 , k2 ε irational only
(d) k1 , k2 ε irational such that k1/k2 is irrational

This is a good question... A starting hint is ::

let the period of the function be P
{-1 + sin (k1pie (x+P)) } ={-1 + sin (k1pie x) }
for rational x
and

{1 + cos (k2pie x) } = {1 + cos (k2pie (x+P)) } for irrational

Well sky.. that is not the way.. i cant think of a good way to say what is wrong.. cos this is all too much together.. but i will try to in my next post!

If the period is P and from the above post if u remove the "1+" part, then

sin (k1pie (x+P))=sin (k1pie (x)) for rational x

and

cos (k2pie x) = cos (k2pie (x+P)) for irrational x

so k1pie.P is a multiple of 2pi = 2pi.n

and k2pie.P is a multiple of 2pi = 2pi.m

isnt it?

So

ki/k2 is a rational number!