Question

Which one is not periodic?

(a) |sin3x| + sin²x 

(b) cos√x + cos²x

(c) cos4x + tan²x 

(d) cos2x + sinx.

Answer 1

Correct option (b) : Period of |sin3x| is π/3 and period of sin²x is π

(a) Same as the period of |sinx| or (1 - cos2x)/2 whose period is π
Now period of |sin 3x| + sin²x is the L.C.M of their periods

Therefore, L.C.M of (π/3,π) =(LCMπ,π)/HCF(3,1) = π

(c, d) Similarly we can say that cos 4x + tan²x and cos2x + sinx are periodic function.
(b) Now cos²x is periodic with period π and for period of cos√x let us take.
f (x) = cos√x
Let f (x + T) = f (x)
=> cos√(T + x) = cos√x
=> √(T + x) = 2nπ ±√x
which gives no value of T independent of x

Therefore, f(x) cannot be periodic
now say g(x) = cos²x + cos√x which is sum of a periodic and non periodic function and such function have no period.
So, cos√x + cos²x is non periodic function.​