Determine the molarity containing 10.6g of NaCO3 dissolved in:a)250cm^3 of distilled water.i got 1.06. b)500cm^3 of distilled water.i got 0.53.plz correct where i might be wrong

Neither is correct for two reasons, one of which is not your fault.

mols Na2CO3 = grams/molar mass = 10.6/106 = 0.1 mol. Then
M = mols/L = 0.1/0.250 = ?
b is done the same way.
The other problem is the problem itself. M is defined as mols/L of SOLUTION and when you place 10.6 g (or ANY amount) in 250 cc water or 500 cc water the final volume is NOT 250 cc or 500 cc. In this case it is greater volume but in some cases it is less. This is the fault of the person who made up the problem. SO, if you want to make up a 0.4 M solution of Na2CO3 you weigh out 10.6 g Na2CO3, place the dry solid in a 250 cc volumetric flask, add some water, swirl until all of the solid is dissolved, then add distilled water to the 250 cc level marked on the flask. Mix thoroughly. Label. This way you have 250 cc OF SOLUTION and that's the definition. Students make this mistake all the time; instructors should NEVER make this mistake.