Answer: Option B
Let's assume that P can run 1 km in x minutes, which means P's speed is 1/x km per minute. Using the given information, we can deduce that Q can run 1 km in (x + 0.5) minutes (as P runs half a minute faster than Q). Therefore, Q's speed is 1/(x+0.5) km per minute.
Now, we are given that in a 1 km race, Q gets a start of 100 m, which means Q only needs to run 900 m to finish the race. However, despite the head start, Q loses the race by 100 m, which means P finishes the race in 800 m (i.e., P covers 1 km - 100 m).
Let's use the formula distance = speed x time to calculate the time taken by P and Q to run a kilometre:

• P's time to run 1 km = time taken to run 800 m + head start of 200 m = (800/1/x) + (200/1/(x+0.5)) = (800x+400)/(2x+1) minutes
• Q's time to run 1 km = time taken to run 900 m = (900/1/(x+0.5)) = (1800x+900)/2(x+0.5) minutes

We know that Q loses the race by 100 m, so we can equate the time taken by P and Q for running 1 km and add 100 m to Q's time:
(800x+400)/(2x+1) = (1800x+900)/2(x+0.5) + (1/2)Simplifying this equation, we get:4x^2 - 3x - 1 = 0Solving this quadratic equation, we get:x = 1 or x = -1/4
Since x cannot be negative, we choose x = 1, which means P takes 1 minute to run 1 km. Using this value, we can calculate Q's time to run 1 km:
Q's time to run 1 km = (1800 + 900)/2.5 = 2.5 minutes
Therefore, the correct answer is option B, i.e., P takes 1 minute and Q takes 2 minutes 30 seconds to run a kilometre.