  We have carried out detailed $k$-mode analysis of single-inflaton slow-roll
inflationary perturbations including quantum fluctuations by setting up
non-linear coupled differential equations of inflaton field ($\phi$), its
perturbation ($\delta\phi$) and the metric perturbation (the Bardeen potential,
$\Phi_B$), and calculated the number of e-folds ($N$), scalar spectral index
($n_s$), tensor spectral index ($n_h$), scalar power spectrum ($\Delta_s$),
tensor power spectrum ($\Delta_h$), tensor-to-scalar ratio ($r$) and the Hubble
parameter ($H$) for different $k$ values at the horizon crossing. In these
calculations we have employed the $E$-model $\alpha$-attractor potentials which
are found to display slow-roll behaviour. The values of $n_s$ and $r$ obtained
by us are consistent with those given by the well-known universal
$\alpha$-attractor formulae. We got $n_s = 0.956908$ and $r= 0.005571 $ at $k =
10^6$ Planck unit for the value of the potential parameter $n = 1$. These can
be compared with the Planck-2018 data viz., $n_s = 0.9649\pm 0.0042$ at $68\%$
CL, $r<0.064$ at $95\%$ CL and ACT-2020 data viz., $n_s = 0.9691\pm 0.0041$ at
$68\%$ CL.
