A fixed for floating interest rate swap is identical to a series of forward rate agreements coinciding with the settlement periods. For example, a 2 year fixed for floating interest rate swap where the investor pays fixed at 6 monthly rests is equivalent to a long position in 3 FRAs ( a 6 x12, a 12 x 18 and an 18 x 24 FRA). Thus I represents a correct replication of the swap.
An obligation to pay fixed is identical to a short position in a fixed rate bond, and receiving the floating rate is akin to a long position in a floating rate note. Thus II also correctly replicates the swap.
A long position in an FRN correctly replicates the floating cash flow from the swap, but a short position in the FRA obliges the investor to receive fixed and pay floating - which is not equivalent to the flows from the interest rate swap. Thus III is not a correct replication of the interest.
A long position in a cap means the investor will receive the difference between the floating and the cap rate if the floating rate is greater than the cap rate. Effectively, the investor will receive floating. A short postion in a floor at the same strike rate would mean the investor would end up paying the floor rate, effectively equivalent to the obligation to pay a fixed rate. Therefore this replication also correctly replicates the swap, and IV can also be used to replicate the investor's position.
Therefore Choice 'a' is the correct answer and the rest are incorrect.
(For statement IV, let me explain with an example. Recall that the buyer of a cap receives the difference between the current interest rate and the agreed exercise rate if the current interest rate goes above the exercise rate. A cap can be used by a floating rate borrower to limit their exposure to a rise in interest rates. In the same way, the seller of a floor pays the buyer the difference between the current interest rate and the exercise rate if the current interest rate falls below the exercise rate. A floor can be used by a lender to protect against a fall in interest rates.
Now assume the exercise rate is r. Also assume that the current floating interest rate is r'. the holder of the interest rate swap in the example in the question receives r and pays r'. Therefore the net payout is r' - r (pays fixed, ie -r, and receives floating, ie +r).
Now consider the long position in the cap and the short position in the floor.
Case 1: If r > r', the following will happen:
a. The long position in the cap will pay nothing, ie 0.
b. The short position in the floor will require a payment of r - r' to the buyer of the floor, and because it is a payment, the cash flow will be -(r - r') = r' - r
Case 2: If r < r', the following will happen:
a. The long position in the cap will mean the investor receives r' - r.
b. The short position in the floor will pay nothing as the interest rate is higher than the exercise rate.
In both cases, the net cash flow for the investor is r' - r, which is the same as for the swap where the investor pays fixed and receives floating.)
