A) Rational
B) Irrational
C) Positive real number
D) None of these
Correct Answer: A
Solution :
Let \[f(x)\] be periodic with period \[\lambda ,\] then \[\sin \,(x+\lambda )+\cos p\,(x+\lambda )=\sin x+\cos px,\,\,\forall \,\,x\in R\] Putting \[x=0\] and replace \[\lambda \] by \[-\lambda \], we have \[\sin \lambda +\cos p\lambda =1\] and \[-\sin \lambda +\cos p\lambda =1\] Solving these, we get \[\sin \lambda =0\] so \[\lambda =n\pi \] and \[\cos p\lambda =1\] so \[p\lambda =2m\pi .\] As \[\lambda \ne 0,\,\,m\] and \[n\] are non-zero integers. Hence \[p=\frac{2m\pi }{\lambda },\]which is rational.
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