Question

let ( f(x)={-1+sin (k 1 text { pie } x)}, x ) is rational ( {1+cos (k 2 text { pie } x)}, x ) is irrational
if ( f(x) ) is periodic function, then which of the following is correct
(a) either ( mathrm{k} 1, mathrm{k} 2 varepsilon ) rational or ( mathrm{k} 1, mathrm{k} 2 varepsilon ) irrational
(b) ( mathrm{k} 1, mathrm{k} 2 varepsilon ) rational only
(c) ( mathrm{k} 1, mathrm{k} 2 varepsilon ) irational only
(d) ( mathrm{k} 1, mathrm{k} 2 mathrm{varepsilon} ) irational such that ( mathrm{k} 1 / mathrm{k} 2 ) is irrational
This is a good question... A starting hint is : :
let the period of the function be ( P ). ( {-1+sin (k 1 p i e(x+P))}={-1+sin (k 1 p i e x)} )
for rational ( x )
and
( {1+cos (k 2 p i e x)}={1+cos (k 2 p i e(x+P))} ) for irrational

# I Example 86 Let f(x) - ) -1+sin Ky Ttx, x is rational. DOUBTAS (1+cos K27x, x is irrational. If f(x) is a periodic function, then (a) either ki, K2 E rational or K,K2 e irrational (b) K1, K2 e rational only (C) K1, K2 E irrational only (d) Ky, K, e irrational such that is rational K2

Solution