We examine theoretically the time-evolution of NAREX (non-adiabatic rotational excitation) and molecular 2-D alignment (2DA) interacting with a pair of elliptically polarized laser pulses. The pulse shapes taken are half-cycle pulse (HCP) and square pulse (SQP). By choosing the proper value of elliptically polarized field parameters, we demonstrate that efficient field-free 2DA alignment can be achieved. It is also shown that NAREX can be controlled by various laser parameters, out of which pulse shape plays the most significant role. The effect of pulse width along with elliptic parameter on probabilities of rotational states is also under concern. The delay time between the two pulses decides the maximum in 2DAs. (C) 2016 Elsevier B.V. All rights reserved.