If `f(x)=sinx+cosa x`is a periodic function, show that `a`is a rational numberA. `a in Z `B. `a in N `C. `a in Q`D. `a in R `

If `f(x)=sinx+cosa x`is a periodic function, show that `a`is a rationa

1.	If `f(x)=sinx+cosa x`is a periodic function, show that `a`is a rational numberA. `a in Z `B. `a in N `C. `a in Q`D. `a in R `
Answer» Correct Answer - C Let T be the period of f(x). Then, `f(x+T)=f(x)` for all x. `implies sin(T+x) + cos a (T+x) = sin x + cos ax ` for all `x in R ` Putting x=0 and x=-T respectively , we get ` sinT+ cos a T =1` and ` -sin T + cos a T=1` Solving these two equations, we get `sin T =0 and cos a T =1` `implies T n pi and a T =2m pi `, where ` m, n in Z`. `implies (aT)/(T)=(2mpi)/(npi)` `implies a=(2m)/(n)`, which is a rational number `=a in Q`

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