can someone help me answer the following #Fsharp multiple choice questions along with explanations for each question please.

1. A fraction like 2/3 can be represented in F# as a pair of type int * int. Define infix operators .+ and .* to do addition and multiplication of fractions:

> (1,2) .+ (1,3);;

val it : int * int = (5, 6)

> (1,2) .+ (2,3) .* (3,7);;

val it : int * int = (11, 14)

Note that the F# syntax for defining such an infix operator looks like this:

let (.+) (a,b) (c,d) = ...

Also note that .+ and .* get the same precedences as + and *, respectively, which is why the second example above gives the result it does.

Finally, note that your functions should always return fractions in lowest terms. To implement this, you will need an auxiliary function to calculate the gcd (greatest common divisor) of the numerator and the denominator; this can be done very efficiently using Euclid's algorithm, which can be implemented in F# as follows:

let rec gcd = function

| (a,0) -> a

| (a,b) -> gcd (b, a % b)

2. Write an F# function revlists xs that takes a list of lists xs and reverses all the sub-lists:

> revlists [[0;1;1];[3;2];[];[5]];;

val it : int list list = [[1; 1; 0]; [2; 3]; []; [5]]

Hint: This takes just one line of code, using List.map and List.rev.

3. Write an F# function interleave(xs,ys) that interleaves two lists:

> interleave ([1;2;3],[4;5;6]);;

val it : int list = [1; 4; 2; 5; 3; 6]

Assume that the two lists have the same length.

4. Write an F# function cut xs that cuts a list into two equal parts:

> cut [1;2;3;4;5;6];;

val it : int list * int list = ([1; 2; 3], [4; 5; 6])

Assume that the list has even length.

To implement cut, first define an auxiliary function gencut(n, xs) that cuts xs into two pieces, where n gives the size of the first piece:

> gencut(2, [1;3;4;2;7;0;9]);;

val it : int list * int list = ([1; 3], [4; 2; 7; 0; 9])

Paradoxically, although gencut is more general than cut, it is easier to write! (This is an example of Polya's Inventor's Paradox: "The more ambitious plan may have more chances of success.")

Another Hint: To write gencut efficiently, it is quite convenient to use F#'s local let expression (as in the cos_squared example in the Notes).