Two persons, A and B are running on a circular track. At the start, B is ahead of A and their positions make an angle of 30° at the centre of the circle. When A reaches the point diametrically opposite to his starting point, he meets B. What is the ratio of speeds of A and B, if they are running with uniform speeds?

The correct option is 6 : 5.

Let us analyze the motion of A and B on the circular track step-by-step to find the ratio of their speeds.

Let the starting position of A be at a point which we will refer to as the reference point, representing an angle of 0° at the centre of the circular track.

At the start, B is ahead of A, and their positions make an angle of 30° at the centre. Therefore, the starting position of B corresponds to an angle of 30° at the centre of the track.

We are given that A reaches the point diametrically opposite to his starting point. The angular distance from A's starting point to its diametrically opposite point is half of the full circle, which is 180°.

Therefore, A covers an angular distance of 180° to reach the meeting point.

Since A and B meet at this diametrically opposite point, B must also arrive at this same point in the same duration of time.

Since B started 30° ahead of A, the angular distance B needs to cover to reach the meeting point (which is at 180°) is:
180° - 30° = 150°

Both A and B run with uniform speeds and take the same amount of time, say t, to reach the meeting point.

For uniform motion along a circular track, the speed is directly proportional to the angular distance covered in a given time period. Thus, the ratio of the speed of A to the speed of B is equal to the ratio of the angular distances they cover in the same time interval.

Therefore, we can write the ratio of their speeds as:

$$\frac{{\mathrm{Speed}}_A}{{\mathrm{Speed}}_B}=\frac{\mathrm{Distance\; covered\; by\; A}}{\mathrm{Distance\; covered\; by\; B}}$$

Substituting the angular distances covered by A and B:

$$\frac{{\mathrm{Speed}}_A}{{\mathrm{Speed}}_B}=\frac{180°}{150°}$$

Simplifying the fraction by dividing both the numerator and the denominator by 30°:

$$\frac{{\mathrm{Speed}}_A}{{\mathrm{Speed}}_B}=\frac{6}{5}$$

Hence, the ratio of the speeds of A and B is 6 : 5.