Solution:

Given equation is cos(97x) = x.

Let us draw the graphs of f(x) = cos(97x) and g(x) = x.

Explanation:

- The graph of f(x) = cos(97x) is periodic with a period of 2π/97.
- The maximum value of f(x) is 1 and the minimum value is -1.
- The graph of g(x) = x is a straight line passing through the origin with a slope of 1.

Now, we will find the number of solutions of the given equation by analyzing the graphs of f(x) and g(x).

- Case 1: x ≤ -1

In this case, the value of cos(97x) is always greater than -1. Therefore, there are no solutions of the given equation in this interval.

- Case 2: -1 < x="" />< />

In this case, the value of cos(97x) is always less than 0. Therefore, there are no solutions of the given equation in this interval.

- Case 3: x = 0

In this case, the value of cos(97x) is 1, which is equal to x. Therefore, x = 0 is a solution of the given equation.

- Case 4: 0 < x="" />< />

In this case, the value of cos(97x) is always greater than 0. Therefore, there are no solutions of the given equation in this interval.

- Case 5: x = 1

In this case, the value of cos(97x) is cos(97) ≈ -0.820. Therefore, x = 1 is not a solution of the given equation.

- Case 6: x > 1

In this case, the value of cos(97x) is always less than x. Therefore, there are no solutions of the given equation in this interval.

Therefore, the only solution of the given equation is x = 0.

Answer: A) 1.