Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
As $|x|$ increases, the distance between the nodes increases This happens because the particle slows down near the edges.
And since it's moving slower, the particle’s de Broglie wavelength increases
$$\lambda=\dfrac{h}{mv}$$
and this means an increase in the space between the nodes.
Thus, Spacing between nodes $\bf increases $ when $|x|$ increases.
$$\color{blue}{\bf [b]}$$
The antinode's height increases as $|x|$ increases.
This is because the particle is spending more time at the edges, and there's a higher chance to find the particle there. So, the taller waves represent where you're more likely to find the particle.
Also because of the slowing down of the particle while $|x|$ increases, it makes it spend more time at higher $|x|$. This also increases the probability of finding the particle at larger $|x|$.
Thus, the antinode's height $\bf increases $ as $|x|$ increases.
$$\color{blue}{\bf [c]}$$
See the graph below.
