  The largest K-band flux-limited sample of luminous quasars to date has been
constructed from the UKIDSS Large Area Survey Early Data Release, covering an
effective area of 12.8 deg^2. Exploiting the K-band excess of all quasars with
respect to foreground stars, including quasars experiencing dust reddening and
objects with non-standard SEDs, a list of targets suitable for spectroscopic
follow-up observations with the AAOmega multi-object spectrograph is
constructed, resulting in more than 200 confirmed AGN. KX-selection
successfully identifies as quasar candidates objects that are excluded from the
SDSS quasar selection algorithm due to their colours being consistent with the
stellar locus in optical colour space (with the space density of the excluded
objects agreeing well with results from existing completeness analyses). Nearly
half of the KX-selected quasars with K<17.0 at z<3 are too faint in the i-band
to have been targeted by the SDSS quasar selection algorithm, revealing a large
population of quasars with red i-K colours. The majority of these objects have
significant amounts of host galaxy light contributing to their K-band
magnitudes, consistent with previous predictions. The remaining objects are
morphologically stellar and have colours consistent with quasars experiencing
SMC-type reddening with 0.10<E(B-V)<0.25. The i-K colour distribution indicates
that <10 per cent of the quasar population is missing from this K-band selected
sample due to dust reddening, and comparisons with simulations strongly favour
an obscured fraction of <20 per cent. (Abridged)
