  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.
