How to determine the magnetic character of heteroleptic complexes?
Based on the available information, we would start by figuring out the 3D geometry of the complex. $\ce{Cr^3+}$ should be octahedral, with two possible "combinations", cis or trans. Let's start with the trans version. The symmetry of this system is $D_{4h}$ so we call this "tetragonal".

With a tetragonal geometry, the MO's would be arranged in such a way that there will be two degenerate orbitals at the lowest energy, and three higher-energy orbitals. The MO diagram typically looks something like this:



Chromium(III) has an [Ar] $3d^3$ electronic configuration so we need to fill up three levels. Now the question is whether or not the splitting caused by the ligands are large enough to cause pairing.

We know for a fact that the trans version of the example is a green solution, so it absorbs red light (700 nm or ~14,300 $cm^{-1}$). Pairing energy is typically in the range of 20,000 - 30,000 $cm^{-1}$ so the 'gap' between yz and xy is probably about only ~14,000 $cm^{-1}$. As such, I would say the trans isomer is magnetic and not paired, because it is energetically more favorable to promote the 3rd electron to the higher energy level than to force it to pair.

Similarly, we know for a fact that the cis version is violet. This means it absorbs yellow light, about 570 nm or 17,000 $cm^{-1}$. This is still lower than the pairing energy, so I'd say that The tetraamminedichlorochromium(III) is magnetic.