Which one of these signals will be least distorted by this communication channel
Normally, a signal is composed of several frequencies, which strengths are given by the Fourier Transform of the signal. 


$x_1(t)=\delta(t) \Leftrightarrow \mathcal{F}(x_1(t))=C$, i.e. the Fourier transform is a constant (value of constant depends on the actual normalization of the Fourier Transform). Hence, it has all frequencies contained, with equal power.
$x_2(t)=5 \Leftrightarrow \mathcal{F}(x_2(t))=\delta(f)$, i.e. the Fourier Transform of a constant is a Dirac impulse at frequency zero. Hence, this signal has only one frequency component, the zero-frequency.
$x_3(t)=10e^{j1000t} \Leftrightarrow \mathcal{F}(x_3(t))=C\delta(f-1000/2\pi)$ i.e. the Fourier Transform of a complex exponential is a Dirac impulse at the position of the frequency of the exponential. Hence, it has only one component, at frequency $1000/2\pi$
$x_4(t)=1/t \Leftrightarrow \mathcal{F}(x_4(t))=C\text{sign}(f)$, i.e. the Fourier transform of the inverse is a sign function. Hence, it contains all frequencies.


Accordingly, the answer to your question would be signal 2, as it only contains the DC frequency ($f=0$).