Which of the four resistances P, Q, R and S generate the greatest amount of heat when a current flows from A to B?

Because these branches are in parallel, the branch with lower resistance draws more current. Since the bottom branch ($$3\Omega$$) has half the resistance of the top branch ($$6\Omega$$), it carries double the current. Let the top current be $$I$$ and the bottom current be $$2I$$.

Heat generated:

P ($$2\Omega$$): $$I^2 \times 2 = 2I^2$$

Q ($$4\Omega$$): $$4I^2 \times 4 = 4I^2$$

R ($$1\Omega$$): $$(2I)^2 \times 1 = 4I^2$$

S ($$2\Omega$$): $$(2I)^2 \times 2 = \mathbf{8I^2}$$