Statement 1 - Apparent position of the object keeps fluctuating when seen through a hot air
Statement2 - Hotter air and cooler air have same refractive index, so no dispersion happens.
statement 1 is correct, but statement 2 is incorrect
Since hot air is less dense than the cold air. So light rays when entering, from hot air to cold air, undergoes refraction and bends .Thus object when seen through hot air appears to flicker.
so , statement 2 is wrong. But statement 1 is correct.