Question

Statement-1: The period of the function `f(x)=cos[2pi]^(2)x+cos[-2pi^(2)]x+[x]` is ` pi, [x]` being greatest integer function and [x] is a fractional part of x, is `pi` .
Statement-2: The cosine function is periodic with period ` 2pi`
A. Statement-1 is True, Statement-2 is True, statement-2 is a correct explanation for the statement-1 .
B. Statement-1 is True, Statement-2 is True, statement-2 is not a correct explanation for the statement-1 .
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False , Statement-2 is True.

Answer 1

Correct Answer - D
We have ,
`f(x)=cos[2pi^(2)]x+cos[-2pi^(2)]x+[x]`
`implies f(x)=cos19x+cos20x+x-[x]`
We observe that ` cos19x, cos20x and x-[x]` are periodic function with periods ` (2pi)/(19),(pi)/(10)` and 1 respectively.
But , `(2pi)/(19) and (pi)/(10)` are irrational numbers. So, LCM of `""(1pi)/(19),(pi)/(10)` and 1 does not exist. So, f(x) is not periodic. Hence, statement-1 is not true . However, statement-2 is true.