To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of a pendulum is the sum of its potential energy and kinetic energy.

1. Calculate the mechanical energy at the lowest position:
At the lowest position of the pendulum, all of its energy is in the form of kinetic energy. The formula for kinetic energy is given by:
KE = (1/2)mv^2
Where KE is the kinetic energy, m is the mass of the bob, and v is the speed of the bob.

Since the mass of the bob is not given, we can assume it to be constant throughout the motion. Therefore, the mass will cancel out in the subsequent calculations.

Given that the speed of the bob at its lowest position is 3 m/s, we can substitute these values into the formula to find the kinetic energy at its lowest position.

KE = (1/2)(3)^2
KE = (1/2)(9)
KE = 4.5 J

2. Calculate the mechanical energy at the position where the length makes an angle of 60 degrees to the vertical:
At this position, the mechanical energy is the sum of potential energy and kinetic energy. The formula for potential energy is given by:
PE = mgh
Where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the vertical height.

The height can be calculated using trigonometry. Since the length of the pendulum is given as 0.5 m and the angle is 60 degrees, we can use the formula:
h = l(1 - cosθ)
Where h is the height, l is the length, and θ is the angle.

Substituting the given values, we get:
h = 0.5(1 - cos60)
h = 0.5(1 - 0.5)
h = 0.5(0.5)
h = 0.25 m

Therefore, the potential energy at this position can be calculated as:
PE = mgh
PE = (0.25)(9.8)
PE = 2.45 J

Now, we can calculate the total mechanical energy at this position by adding the potential energy and kinetic energy:
Total mechanical energy = PE + KE
Total mechanical energy = 2.45 + 4.5
Total mechanical energy = 6.95 J

3. Calculate the speed of the bob at the position where the length makes an angle of 60 degrees to the vertical:
Since the total mechanical energy is conserved throughout the motion, we can equate the total mechanical energy at the lowest position to the total mechanical energy at the position where the angle is 60 degrees:
Total mechanical energy at lowest position = Total mechanical energy at 60 degrees
4.5 = 6.95 + (1/2)(v^2)

Solving this equation for v, we get:
v^2 = 2(4.5 - 6.95)
v^2 = 2(-2.45)
v^2 = -4.9
v = √(-4.9)

Since the square root of a negative number is not a real number, this result is not physically meaningful. Therefore, we made an error somewhere in our calculations.

Checking the options given, we find that option 'A' (2 m/s)