The correct option is D (1) vain, (2) vein, (3) vane
In the first sentence, "I tried in vain to convince her that I was telling the truth", vain is correct because it implies withoutrealsignificance,value, or importance. Vane implies a device, as a rod to which a freelyrotatingpointer is attached,forindicatingthedirection of thewind. Vein implies one of thesystem of branchingvessels or tubesconveyingbloodfromvariousparts of the body to theheart.
In the second sentence, "Suddenly, the child went blue and the vein in her wrist became silent", vein implies one of thesystem of branchingvessels or tubesconveyingbloodfromvariousparts of the body to theheart.
In the third sentence, "The weather vane needs to be replaced", vane implies a device, as a rod to which a freelyrotatingpointer is attached,forindicatingthedirection of thewind.