To determine the discharge across a sluice gate, we can use the equation for flow rate, which is given by:

Q = A * V

Where:
Q = discharge
A = cross-sectional area of flow
V = velocity of flow

In this case, we are given the flow depths across the sluice gate, which can be used to calculate the cross-sectional area of flow. The flow depth is the height of the water above the bottom of the sluice gate. We are also given the width of the sluice gate, which allows us to calculate the cross-sectional area of flow per unit width.

Let's calculate the cross-sectional areas of flow for both flow depths:

Cross-sectional area of flow for flow depth 2.0 m:
A1 = 2.0 m * W
Where W is the width of the sluice gate

Cross-sectional area of flow for flow depth 0.5 m:
A2 = 0.5 m * W

Now, we need to calculate the velocities of flow corresponding to each flow depth. The velocity of flow can be determined using the equation:

V = Q / A

For flow depth 2.0 m:
V1 = Q / A1

For flow depth 0.5 m:
V2 = Q / A2

Since the width of the sluice gate is the same for both flow depths, we can cancel out the width and rewrite the equations as:

V1 = Q / (2.0 m)
V2 = Q / (0.5 m)

Now, let's rearrange the equation for flow rate and substitute the given values:

Q = A * V
Q = (2.0 m * W) * (Q / (2.0 m))
Q = W * Q

Dividing both sides of the equation by Q:

1 = W

Therefore, the discharge per unit width is equal to 1.0 m^2/s. However, this contradicts the given options. Hence, there appears to be an error in the question or the options provided.