The above problem is from TIFR-GS examination paper.I solved this as follows:
Soln. The option $(c)$ is the only correct option ,consider a bijection $f:[0,1]\to \mathbb{R-Q}$,then $f:[0,1]\to \mathbb R$ is an injective map with no rational points in range and hence not surjective.But any injective map from $[0,1]$ to $\mathbb R$ must contain an irrational,otherwise the range set would be countable,since $f$ is one-one,so $[0,1]$ is countable,which is absurd.
Is my solution correct?Can someone provide me with some similar problems?