Step 1: Calculate velocity of car P and Q
VP=d[xP(t)]dt =ddt(at+bt2) =a+2bt ....(1)
VQ=d[xQ(t)]dt=ddt(ft−t2)=f−2t ....(2)
Step 2: Time at which both have the same velocity
Let t0 be the time when VP and VQ are same
Equate equation (1) and (2)
VP|t=t0=VQ|t=t0
⇒a+2bt0=f−2t0
⇒ t0=f−a2(b+1)
Hence, Option(A) is correct.