Given information:
- The train takes 1.75 seconds to cross a telegraph pole.
- The train takes 1.5 seconds to cross a cyclist traveling in the opposite direction at a speed of 10 m/s.

To find:
The length of the train.

Let's break down the problem into steps to find the solution.

Step 1: Determine the relative speed of the train with respect to the cyclist.
- The train is crossing the cyclist, which means the relative speed is the sum of their individual speeds.
- The cyclist's speed is given as 10 m/s.
- Therefore, the relative speed of the train with respect to the cyclist is 10 m/s.

Step 2: Determine the distance covered by the train while crossing the cyclist.
- The time taken by the train to cross the cyclist is 1.5 seconds.
- The relative speed of the train with respect to the cyclist is 10 m/s.
- Therefore, the distance covered by the train while crossing the cyclist is 10 m/s * 1.5 s = 15 meters.

Step 3: Determine the distance covered by the train while crossing the telegraph pole.
- The time taken by the train to cross the telegraph pole is 1.75 seconds.
- The distance covered by the train while crossing the telegraph pole is equal to its length.
- Therefore, the length of the train is 1.75 seconds.

Step 4: Calculate the length of the train.
- The total distance covered by the train while crossing both the cyclist and the telegraph pole is 15 meters + length of the train.
- The total time taken by the train to cross both the cyclist and the telegraph pole is 1.5 seconds + 1.75 seconds = 3.25 seconds.
- Using the formula distance = speed * time, we can write the equation as follows:
15 meters + length of the train = 10 m/s * 3.25 s
- Solving the equation, we get:
length of the train = 32.5 meters - 15 meters
= 17.5 meters

Since the length of the train cannot be negative, the correct answer is option 'D' - 105 meters.