To find the maximum height of the projectile, we need to analyze its motion and use the equations of projectile motion. Let's break down the problem into smaller steps:

1. Analyzing the motion:
- The projectile is thrown with an initial velocity u, making an angle θ with the vertical.
- The time taken to cross the top of the first pole is 1 second, and for the second pole is 3 seconds.
- Both poles have the same height h.

2. Finding the time of flight:
- The time of flight, T, is the total time taken by the projectile from launch to landing.
- Since the projectile crosses the top of the first pole at 1 second and the second pole at 3 seconds, the time of flight can be calculated as T = 3 seconds.

3. Finding the vertical component of initial velocity:
- The vertical component of the initial velocity, uy, can be calculated using the equation: uy = u * sin(θ).

4. Finding the maximum height:
- The maximum height, Hmax, can be determined using the equation: Hmax = (uy^2) / (2 * g), where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Now, let's substitute the values and calculate the maximum height:

Given:
- Time of flight, T = 3 seconds
- Initial velocity, u
- Angle with the vertical, θ
- Height of each pole, h

Substituting the values, we have:
uy = u * sin(θ)
Hmax = (uy^2) / (2 * g)

Substituting uy = u * sin(θ) in the second equation, we get:
Hmax = [(u * sin(θ))^2] / (2 * g)

Since we don't have the exact values of u and θ, we cannot determine the exact value of Hmax. However, we can conclude that the maximum height of the projectile is directly proportional to the square of the initial velocity and the square of the sine of the angle with the vertical.

Therefore, with the given options, we can select the one with the highest value of Hmax. In this case, option (d) 49 m would be the correct answer.

Note: The question did not provide the values of u and θ, so we cannot calculate the exact maximum height. However, we can analyze the concept of projectile motion and provide the general method to find the maximum height.