Introduction:
In this problem, we have a river with a width of w and a constant flow velocity of v. Swimmer A wants to reach directly from one bank (point A) to the other bank (point B) of the river by swimming with the minimum possible speed with respect to the river. We need to determine the time it takes for the swimmer to cross the river.

Approach:
To calculate the time taken by the swimmer to cross the river, we can consider the relative velocities of the swimmer and the river. The swimmer needs to swim with a velocity that cancels out the effect of the river's flow velocity.

Derivation:
Let's assume the swimmer's speed with respect to the river is u, and the angle between the swimmer's velocity and the direction perpendicular to the river's flow is θ.

Component of the swimmer's velocity perpendicular to the river's flow:
The component of the swimmer's velocity perpendicular to the river's flow is given by:
v_perpendicular = u * sin(θ)

Component of the swimmer's velocity parallel to the river's flow:
The component of the swimmer's velocity parallel to the river's flow is equal to the river's flow velocity:
v_parallel = v

Net velocity of the swimmer:
The net velocity of the swimmer is the vector sum of the perpendicular and parallel components of the swimmer's velocity:
v_net = √(v_perpendicular^2 + v_parallel^2)

Condition for minimum possible speed:
To achieve the minimum possible speed, the net velocity of the swimmer should be zero. Therefore, we can equate the net velocity to zero and solve for u:
v_net = √(v_perpendicular^2 + v_parallel^2) = 0

Squaring both sides, we get:
v_perpendicular^2 + v_parallel^2 = 0

Substituting the values of v_perpendicular and v_parallel, we have:
u^2 * sin^2(θ) + v^2 = 0

Simplifying the equation, we get:
u^2 * sin^2(θ) = -v^2

Since both u and sin^2(θ) are positive, we can remove the negative sign and write:
u^2 * sin^2(θ) = v^2

Taking the square root of both sides, we get:
u * sin(θ) = v

Solving for u, we have:
u = v / sin(θ)

Time taken to cross the river:
The time taken by the swimmer to cross the river can be calculated using the formula:
Time = Distance / Speed

Since the swimmer wants to reach directly at point B on the other bank of the river, the distance to be covered is the width of the river, w.

Substituting the values of u and sin(θ) in the formula, we have:
Time = w / (v / sin(θ))
= w * sin(θ) / v

Final Answer:
The time taken by the swimmer to cross the river is given by:
Time = w * sin(θ) / v

Since sin(θ) can take values between -

I I... think, Question is quite incomplete . But by guess we can choose option 1