"In a circle, suppose we draw any central angle at all, then draw a second central angle which is larger than the first. Will the chord of the second central angle be longer than the chord of the first?"
I assume that if both angles are less than or equal to 180 degrees, then the larger the central angle the longer the chord. However, if the larger angle is greater than 180 degrees, say 300 degrees, and the smaller angle is 180 degrees, then the smaller angle has a longer chord.
I'm not so sure about the concept of chords, does it only apply to angles less than or equal to 180 degrees?
In a circle of radius $r$, if a chord intercepts an arc whose central angle is $\theta$, then the chord has length $2r\sin\dfrac{\theta}{2}$. To see why this fact is true, draw the radii to the endpoints of the chord and a segment from the center of the circle to the midpoint of the chord. Then use basic right angle trigonometry to find the length of the chord.
The function $\sin x$ is increasing over $x \in [0^{\circ},90^{\circ}]$ and is decreasing over $x \in [90^{\circ},180^{\circ}]$. Hence, for any fixed $r > 0$, the function $2r\sin\dfrac{\theta}{2}$ is increasing over $\theta \in [0^{\circ},180^{\circ}]$ and is decreasing over $\theta \in [180^{\circ},360^{\circ}]$. So for angles less than $180^{\circ}$, a larger central angle yields a longer chord.