Why cosine is not an eigen signal? According to this website:


  If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system.


but in this question it is saying only option $(c)$ is eigen function.


My question is why can't it  be option $(b)$ and $(d)$ as it will also obey the defintion of eigen function as given in the website?

PS: Please let me if know my question is not clear i will edit it but please don't downvote.
That definition is rubbish.

An eigenfunction of a system is a function that passes that system, only scaled by a scalar value.

Now, suppose your system is a simple complex band pass filter, which has a pass band at $f_0$, but stop bands everywhere else, especially at $-f_0$.

Now your cosine $\cos(2\pi f_0 t)$ has spectral components at both $f_0$ and $-f_0$. The output of that filter contains only a multiple if the positive part. Hence, the output is not a scalar multiple of the input cosine. Done.