Let's break the whole thing down :
$X$ is $1 \hspace{0.1cm} km$ northeast of $Y$
$Y$ is $1 \hspace{0.1cm} km$ southeast of $Z$.
$W$ is $1\hspace{0.1cm} km$ west of $Z$
$P$ is $1 \hspace{0.1cm} km$ south of $W$
$Q$ is $1 \hspace{0.1cm} km$ east of $P$
The distance between $X$ & $Z$ will be
Applying pythagoras theorem on $\triangle XYZ$ triangle, we get-
$(XZ)^2 = (XY)^2+(YZ)^2$
Or, $(XZ)^2 = 1+1$ $\qquad \left [ ∵ XY = 1 \hspace{0.1cm} km \hspace{0.1cm} \& \hspace{0.1cm}YZ = 1 \hspace{0.1cm} km \right ]$
Or, $XZ = \sqrt{2} \hspace{0.1cm} km$
Now, we need to find the distance between $X$ & $Q$ in km
By applying Pythagoras Theorem on $\triangle XQZ$ we get -
$(XQ)^2 = (QZ)^2+(ZX)^2$
Or, $(XQ)^2 = 1^2 + (\sqrt{2})^2$ $\qquad \left[ ∵ QZ = 1\hspace{0.1cm} km \hspace{0.1cm} \& \hspace{0.1cm} ZX = \sqrt{2} \hspace{0.1cm} km\right ]$
Or, $(XQ)^2 = 1+2 = 3$
Or, $(XQ) = \sqrt{3} \hspace{0.1cm} km$
Correct Answer: $C$