The Cairo pentagonal lattice, consisting of an irregular pentagonal tiling of magnetic ions on two inequivalent sites (3- and 4-co-ordinated ones), represents a fascinating example for studying geometric frustration effects in two-dimensions. In this
work, we investigate the spin $S$ = $1/2$ Cairo pentagonal lattice with respect to selective exchange coupling (which effectively corresponds to a virtual doping of $x$ = $0, 1/6, 1/3$), in a nearest-neighbour antiferromagnetic Ising model. We also develop a simple method to quantify geometric frustration in terms of a frustration index $phi(beta,T)$, where $beta$ = $J/tilde{J}$, the ratio of the two exchange couplings required by the symmetry of the Cairo lattice. At $T = 0$, the undoped Cairo pentagonal lattice shows antiferromagnetic ordering for $beta le beta_{crit} = 2$, but undergoes a first-order transition to a ferrimagnetic phase for $beta >$ $beta_{crit}$. The results show that $phi(beta,T = 0)$ tracks the transition in the form of a cusp maximum at $beta_{crit}$. While both phases show frustration, the obtained magnetic structures reveal that the frustration originates in different bonds for the two phases. The frustration and ferrimagnetic order get quenched by selective exchange coupling, and lead to robust antiferromagnetic ordering for $x$ = 1/6 and 1/3. From mean-field calculations, we determine the temperature-dependent sub-lattice magnetizations for $x$ = $0, 1/6$ and $1/3$. The calculated results are discussed in relation to known experimental results for trivalent Bi$_2$Fe$_4$O$_9$ and mixed valent BiFe$_2$O$_{4.63}$. The study identifies the role of frustration effects, the ratio $beta$ and selective exchange coupling for stabilizing ferrimagnetic versus anti-ferromagnetic order in the Cairo pentagonal lattice.
