For a simple harmonic motion with given angular frequency $\mathrm{\omega }$, two arbitrary initial conditions are necessary and sufficient to determine the motion completely. These initial conditions may be

Description of motion is completely specified if we know the variation of x as a function of time. For simple harmonic motion, the general equation of motion is $\mathrm{x}=\mathrm{A}(\mathrm{\omega t}+\mathrm{\delta })$ As $\mathrm{\omega }$ is given, to describe the motion completely, we need the values of A and $\delta$. From options (2) and (4), we can have the values of A and $\delta$directly.
For option (1), we can find A and $\delta$ if we know initial velocity and initial position. Option (3) cannot give the values of A and $\delta$ so it is not the correct condition.