Answer:

Introduction:
The average velocity of a cyclist can be determined by dividing the total displacement by the total time taken. In this case, the cyclist starts from the center O of a circular track, cycles along the circumference, and stops at point Q. We are given that the total time taken is 10 minutes. To find the average velocity, we need to determine the total displacement of the cyclist.

Determining the Total Displacement:
Since the cyclist starts from the center O and stops at point Q, the total displacement can be calculated as the distance between these two points. As the cyclist cycles along the circumference of the circular track, the displacement is equal to the length of the arc between O and Q.

Calculating the Length of the Arc:
To calculate the length of the arc, we need to know the radius of the circular track and the angle between the radii from O to Q. Let's assume the radius of the track is 'r' and the angle is 'θ'.

Calculating the Angle:
To determine the angle θ, we can use the fact that the cyclist takes 10 minutes to complete one full revolution around the circular track. Since there are 360 degrees in a full revolution, the time taken to cover an angle θ is given by the equation:

θ = (10/60) * 360

Calculating the Length of the Arc using the Angle:
The length of the arc can be calculated using the formula:

Arc length = (θ/360) * (2πr)

Substituting the value of θ, we get:

Arc length = ((10/60) * 360/360) * (2πr) = (1/6) * (2πr) = (π/3) * r

Calculating the Average Velocity:
Now that we have determined the total displacement of the cyclist, we can calculate the average velocity. The average velocity is given by the equation:

Average velocity = Total displacement / Total time taken

In this case, the total displacement is equal to the length of the arc (π/3 * r) and the total time taken is 10 minutes.

Average velocity = (π/3 * r) / 10

Conclusion:
The average velocity of the cyclist can be calculated by dividing the length of the arc (π/3 * r) by the total time taken (10 minutes). The exact value of the average velocity depends on the radius of the circular track, which is not provided in the question. Therefore, the average velocity can be expressed as (π/3 * r) / 10.