  We have considered and alternating Heisenberg spin chain with
nearest-neighbor ($J_1$), next-nearest neighbor ($J_2$) antiferromagnetic
couplings along with z-component of the Dzyaloshinskii-Moriya(DM) ($D_z$)
interactions. The Hamiltonian has been studied using (a) Linear Spin-Wave
Theory(LSWT) and (b) Density Matrix Renormalization Group (DMRG). The system
had been reported earlier as a classical ferrimagnet only when nearest neighbor
exchange interactions are present. Both the antiferromagnetic next-nearest
neighbor interactions and DM interactions introduce strong quantum fluctuations
and due to which all the signatures of ferrimagnetism vanishes. We find that
the nonzero $J_2$ introduces strong quantum fluctuations in each of the spin
sites due to which the z-components of both spin-1 and spin-1/2 sites average
out to be zero. The ground state becomes a singlet. The presence of $J_1$ along
with $D_z$ introduces a short range order but develops long range order along
the XY plane. $J_1$ along with $J_2$ induces competing phases with structure
factor showing sharp and wide peaks, at two different angles reflecting the
spin spiral structure locally as well as in the underlying lattice.
Interestingly, we find that the $D_z$ term removes the local spin spiral
structure in z-direction, while developing a spiral order in the XY plane.
